Download Physics for non-physicists

Physics for non-physicists:
some basic ideas and some ideas for teaching them
University of Sydney Teachers Workshops, June 15th- 16th, 2006
Dr Kate Wilson
First year coordinator, Department of Physics, Australian National University
Program Directors, Australian Science Olympiads Physics Program
[email protected] or [email protected]
Introduction
People studying physics sometimes try to learn a lot of laws, principles and rules by
memorising a lot of equations. There are actually only a few central ideas in physics which
appear in many different forms. These short workshops will look at a few of these basic ideas,
in particular conservation principles and Newton’s laws.
These principles and laws are often treated as articles of faith – things that are always obeyed
and always true. In fact the laws of physics are reasonable working theories, that have been
shown to work enough times, and never been shown to not-work, that we accept them as true.
In fact it is impossible to prove that they are true. Any law or theory can be disproved, but it
cannot be proved, that is the nature of a scientific theory. This is not to devalue the laws and
theories of physics – to say, “well, it’s just a theory, its no better than any other theory” is
missing the point. Scientific theories are useful – they help us understand our environment and
give us predictive power, and they can be applied to give us useful technology. Many people
find the quantum theory strange and unbelievable – yet all our modern technology, from
transistors to computers and cellular phones is based on quantum theory.
In these workshops we will look at some of these basic laws and discuss ways they can be
taught /learnt in the classroom. The first workshop will deal mainly with conservation
principles, while the second will look at Newton’s laws and forces. These two ideas – forces
and conservation – are related, and this relationship will be stressed throughout.
Suggested Reading
“The Character of Physical Law” by Richard Feynman is an excellent and accessible book for
an introduction to the nature of physics and the basic laws of physics.
Any good text book will have a good explanation of conservation principles and forces,
although some are better than others. I like Tipler’s “Physics” and Kane and Sternheim’s
“Physics”. Kane and Sternheim has lots of good biological examples and Tipler puts quantum
up front, and in context, rather than hiding it in the back as most other text books do. For the
Physics Olympiad Program we use Halliday, Resnick and Krane, which is good but a bit on
the rigorous and succinct side.
Many of the activities and questions presented here are drawn for the “Workshop Tutorials for
Physics” resources published by UniServe. Some of the questions are drawn from Eric
Mazur’s “Peer Instruction”, which has a wealth of excellent conceptual questions.
Session 1: Conservation principles
When I teach an introductory physics course I often start with a concept map such as the one
shown below.
What this course is about…
motion
forces change the state
of motion by causing an
acceleration
every force gives
rise to a potential
energy
there is kinetic
energy associated
with all motion
energy
forces
forces convert one
sort of energy to
another, but the total
is conserved
forces occur
in pairs
conservation
This concept map is from the first lecture of a foundations of physics course I teach at the
ANU. For the first two or three topics I present a more detailed concept map, including any
important equations. The four ovals containing the key ideas; forces, motion, energy and
conservation, appear on each concept map throughout the course – for mechanics, fluids,
thermodynamics, electricity and magnetism, etc. I then refer back to these maps, indicating
every few lectures where we are on the map. Students are encouraged to construct their own
concept maps around these central ideas to help them put together the pieces they learn into a
more coherent picture of what physics is. Our focus in this workshop is the bottom right hand
corner – conservation, and its links to energy and forces.
Conservation principles are a key idea in physics, and much of what you teach and learn in
physics can be reduced to conservation of energy, conservation of mass, conservation of
charge, conservation of linear or angular momentum, and so on. Conservation principles are
the primary theme of most first year physics courses, and for good reason. Almost anything
that happens in an electronic circuit can be described using conservation of energy and
conservation of charge. The way cyclones can pull roofs off houses and why planes can fly
can be explained using conservation of energy and conservation of mass. Conservation of
momentum explains why gases exert pressure and why rockets fly, and conservation of
angular momentum helps explain how our solar system formed. We will start with
conservation of energy.
Conservation of Energy
The principle of Conservation of Energy tells us that the total energy in any isolated system is
constant. So what is an isolated system? An isolated system is on which no external forces are
acting. Its very hard to find a genuine isolated system in real life, unless you take your isolated
system to be the entire universe. But there are systems which are a reasonable approximation
to isolated – ones in which there is very little or no friction. For example, cars skidding on an
icy road or objects falling with little or no air resistance. When an object falls gravitational
energy is converted to kinetic energy, and when it stops due to colliding with the ground that
energy is converted into other forms. In a system like this the total energy remains constant,
although it can change forms.
Energy comes in two different forms – kinetic energy and potential energy. Kinetic energy is
energy associated with movement. A moving car has kinetic energy ! mv2 where m is the
mass of the car and v is the velocity of the car. Thermal energy, often called heat, is also a
form of kinetic energy – it is the average kinetic energy of atoms, which move and vibrate.
The temperature of a material is directly proportional to the average kinetic energy of its
atoms or molecules. The more kinetic energy the atoms of a substance have, the hotter that
material is. There are also other forms of kinetic energy such as energy carried by mechanical
waves such as sound.
Potential energy also comes in different forms, such as gravitational potential energy and
electrical potential energy. For each fundamental force there is a potential energy. For each
type of potential energy there is also a force. This is the link between forces and energy –
forces act to decrease potential energy. We can write this as:
! dU
F=
(1)
dx
In an isolated system the sum of all the different forms of energy remain constant.
In real life we usually deal with systems that are not isolated. In this case we need to take into
account energy coming in and energy going out. The total energy is then the energy coming in,
plus the kinetic energy plus the potential energy minus the work going out. This applies to
mechanical systems including heat engines, and electrical systems.
In thermal physics, we usually call the energy coming in work, and divide the energy going
out into work and heat (thermal energy). It is divided this way because heat is not generally
considered very useful, and a lot of thermal physics is about getting as much useful work out
of a system as possible.
Some statements of conservation of energy:
The first law of thermodynamics states that the change in internal energy of a system is the
heat or energy you put in minus the work done.
"U = Q ! W
(2)
Kirchoff’s loop law for electric circuits tells you that around any loop in a circuit the sum of
changes in electric potential is zero. Electric potential is the potential energy per unit charge,
an energy density.
(3)
" !V = 0
Bernoulli’s equation for ideal fluid flow states that the sum of pressure, kinetic energy density
and gravitational potential energy density is constant. Again, this is a statement of
conservation of energy, written in terms of an energy density, in this case per unit volume.
P + 12 !v 2 + !gh = constant
(4)
The work-energy theorem itself, which states that the work done is equal to the change in
kinetic energy is itself a statement of conservation of energy:
W = " F .dx = !KE
(5)
which we can see from equation (1) is simply saying that:
!U = !KE
(6)
There are many other laws and equations which can also be understood as statements of
conservation of energy for a particular type of system or problem.
It is worth noting that while it is traditional to teach forces and Newton’s laws before the ideas
of energy and conservation of energy, many problems are mathematically easier to solve using
energy than forces. This is because energy is a scalar quantity – is has magnitude but no
direction. Forces are vector quantities, having both magnitude and direction, and hence one
must always be careful to take into account the direction of forces when solving problems
using Newton’s laws. In the initial training prior to team selection for the Physics Olympiads,
we assume a knowledge of kinematics either from school or from our preliminary notes, and
begin our program with energy and energy conservation of energy, using equation (1) as our
definition of force, a discussion of which follows after we have looked at energy and energy
conservation.
Conservation of Momentum
Momentum is a vector quantity and can be defined as:
(7)
p = mv
It depends on both the mass of an object and the velocity at which it moves. The direction of
the momentum is the same as that of the velocity.
The principle of conservation of momentum tells us that the total momentum of all the
particles in an isolated system is constant. If you consider a moving object such as a ball
which has been thrown, it will have a constant momentum unless some force acts on it.
Newton’s first law, which says that an object at rest will remain at rest, and an object in
motion will continue with constant velocity unless acted on by an external force. This is a
statement of conservation of momentum for the simple case that mass remains constant.
Forces act to change the momentum of an object as:
dp
F=
(8)
dt
This is how Newton originally wrote his second law. Again, it is for the case that mass
remains constant that this reduces to the more commonly written version:
(9)
F = ma
Consider again a ball that has been thrown. If the ball was thrown in deep space, away from
any other bodies, then it would continue in a straight line with constant momentum. However
a ball thrown on Earth experiences external forces – gravity and friction (air resistance). These
two forces act to change the momentum of the object.
If you consider an approximately isolated system – say a rocket ship in space – with no
external forces there will be no change in momentum. To make a rocket ship work it burns
fuel and ejects it out the back. If the rocket starts off with zero momentum relative to the
surface of the Earth, then the total momentum of the rocket and its contents, including fuel,
continues to be zero unless an external force acts. When the fuel is ejected it has a momentum
away from the rocket ship. Conservation of momentum tells us that the rocket ship moves in
the opposite direction so that the total momentum is still zero. (Water rockets are always a
favourite demonstration of conservation of momentum with students.)
Newton’s third law is itself a statement of conservation of momentum:
Fa on b = !Fb on a
hence from equation (8), in any interaction between objects a and b,
!p a = "!p b
and momentum is conserved.
(10)
(11)
Angular momentum and conservation of angular momentum are linked to the equivalent force
(torque) laws for rotational systems in the same way. Conservation of angular momentum is a
very useful way of solving many problems in mechanics.
Conservation of “stuff”
Charge: Charge is a fundamental property of matter, and is conserved. Charge comes in two
varieties – positive and negative. The total amount of charge in the universe seems to constant,
and not far from zero. Most objects are nett neutral – they have as much positive charge in
them as negative. Atoms have a positively charged nucleus, surrounded by negatively charged
electrons. When you charge something up, charge is not being created, it is being moved from
somewhere else. If you charge up an object, like a cat, by adding electrons, those electrons are
coming from somewhere else. If the cat is becoming negative, then something else is
becoming positive at the same rate.
Conservation of charge is useful for understanding how circuits work. Current is the flow of
charge, the amount of charge moving past a point in a circuit per unit time. Kirchoff’s junction
rule, which says that the sum of all currents coming into a junction must be equal to the sum of
currents going out, is just a statement of conservation of charge. Combine that with Kirchoff’s
loop law, which is a statement of conservation of energy, and you can figure out what’s going
on in just about any circuit.
Mass and Volume: Mass is not in fact a conserved quantity. Mass can be considered to be a
form of potential energy, and we know that given enough energy particles with mass can form,
and that in some circumstances mass can be converted to energy. These are not circumstances
that we commonly encounter, hence for most purposes mass can be treated as a conserved
quantity. Where there are no chemical reactions occurring, and a material is incompressible,
the idea of conservation of volume can be useful. The equation of continuity for fluid flow,
which states that the volume rate of flow which is the product of cross sectional area and flow
velocity is a constant:
Q = Av = constant
(12)
is indispensable for understanding flow of incompressible fluids such as water. Coupled with
Bernoulli’s equation, a statement of conservation of energy, most problems involving flow of
incompressible fluids can be solved.
The following pages give some suggestions for activities that students can do, and some of my
favourite questions that can be solved by applying conservation principles.
Using conservation principles: Some Activities
These activities have been chosen because they use mostly simple everyday objects, or things
that can be easily and cheaply made.
These activity sheets are extracts from the “Workshop Tutorials for Physics: Activities” book
(Wilson et al). Each sheet has a photo or diagram of the equipment, suggested questions to be
asked and a short explanation of the relevant physics.
Balloons
Apparatus
packet of balloons, long ones work particularly well
Action
The students blow up a balloon and then let it go. They should watch the balloon and
note the direction it moves in, and the position of the hole as it flies.
The Physics
When the balloon is released the air inside it rushes out because it is under pressure.
The air comes out the neck of the balloon. For momentum to be conserved the balloon (and
remaining air) must move in the opposite direction. This is what happens, and the balloon
whizzes around the room, moving in the opposite direction to the airflow.
Students at the
Australian Catholic
University ready to
launch a balloon.
Accompanying sheet
Balloons
Blow up a balloon, and do not tie off the neck.
Now let it go. What does the balloon do?
Explain what happens in terms of conservation of momentum.
Bicycle Wheel
Apparatus
smoothly rotating stool, bicycle wheel with handle through centre about which the
wheel can spin (i.e. the handle should not spin with the wheel)
Note: a foot pedal operated motor with a belt attached to a pulley is useful to spin the
wheel to large angular velocities, however this is not necessary. A good velocity can be
obtained by hand, and this is safer with large classes.
Action
The students spin up the wheel and attempt to tilt it. The wheel should be carefully
passed to a student sitting at rest on the rotating stool. This student slowly tilts the wheel to
different angles. They can control the direction and speed at which the stool rotates by
changing the angle of the spinning wheel.
The Physics
A large torque is needed to tilt the spinning wheel. Angular momentum must be
conserved, and a rapidly spinning bicycle wheel has a large angular momentum. The wheel
exerts a large reaction force upon the person attempting to change its angular momentum.
When the person is on the rotating stool there is no strong frictional force holding the person
still, and so they begin to rotate due to the large reaction force, and the total angular
momentum is constant.
Note: the bicycle wheel should not be spun too fast as it
can be dangerous if dropped or held too close to the
body. The student on the stool should be cautioned to
tilt the wheel very slowly. This activity needs close
supervision.
Accompanying sheet
Tilting the bicycle wheel
causes the person to rotate
on the stool.
Bicycle Wheel
Spin up the bicycle wheel.
What do you feel when you try to tilt the wheel?
Carefully hand the wheel to someone sitting on the rotating stool.
What happens when they tilt the wheel? Why?
Caution: hold the spinning wheel carefully away from yourself.
Blowing and Lifting
Apparatus
polystyrene block or light cardboard sheet with rod (nail) through the middle, air
blower, tube
Action
The students lift the block by blowing the air down the tube over the block. The nail or
rod allows them to locate the tube over the block. They should explain why blowing on the
block lifts it, instead of pushing it down.
The Physics
The high velocity air flowing over the upper surface of the block is at lower pressure
than the air below which is at atmospheric pressure, so the block lifts. This is an application of
Bernoulli’s principle.
tube
block
with rod
high velocity,
low pressure
block lifts
Accompanying sheet
Blowing and Lifting
Lift the foam block by blowing down into the hollow tube above it.
How is this possible?
Why isn’t the block blown away from the tube?
Bouncing Balls I
Apparatus
selection of balls; for example super-balls, tennis balls, balls of blu-tac or play –dough
Action
The students experiment with dropping and throwing down the different balls. They
should consider the energy changes as the balls fall, collide with the floor, and then bounce up
again.
The Physics
Balls that lose less energy to non-mechanical forms on impact rise higher than balls
that lose more energy. Balls of blu-tac or play-dough lose a lot of energy to internal frictional
forces as they deform on impact. This energy is dissipated as thermal energy. Balls which
undergo more elastic collisions bounce higher as they lose less kinetic energy.
No ball which is dropped (from rest) will bounce higher than the height it was dropped
from. A ball can bounce higher than the original height if thrown down. These balls start off
with kinetic energy and gravitational potential energy instead of just gravitational potential
energy.
Accompanying sheet
Bouncing Balls I
Drop the balls from the same height.
Why do some balls bounce higher than others?
Can you make any of the balls bounce higher than the original height?
Does this contradict conservation laws?
Bouncing Balls II
Apparatus
two balls, one large and one small – for example a basketball and a tennis ball
Action
The students hold the small ball on top of the large ball and drop them together. They
then repeat this with the large ball on top. They should try to explain their observations in
terms of conservation of momentum.
The Physics
The small ball held over the big ball bounces off higher as some momentum is
transferred from the big ball to the small ball, increasing its velocity. Momentum has been
conserved during the collision and the change in momentum of the small ball is large. The
results are most spectacular when the small ball is held exactly vertically above the large ball
before they are released.
If the balls are switched around the momentum is still conserved, but the transfer of
momentum from the small to the big ball makes little difference to the big ball’s velocity due
to its large mass.
Note: a reasonably clear space is needed for this activity as the small ball will gain a large
amount of momentum and can fly off very fast and a long way.
v
v
v
v
Accompanying sheet
Bouncing Balls II
Hold the little ball on top of the big ball and drop them together.
What happens and why?
Does the same thing happen if you hold the big ball on top and drop them?
v
A Loaded Race
Apparatus
ramp, several balls of different sizes and masses, several cylinders (for example large
and small cans of different masses)
Action
The students try to predict which can or ball will win a given race. They then allow the
objects to roll down the ramp and check their predictions.
Note that a full “chunky soup” can is interesting because it can not be considered a
solid mass, the contents move about changing the behaviour of the can.
The Physics
Neglecting air resistance, all the solid spheres will hit the bottom at the same time.
From energy conservation equations we have mgh = ! mv2 + ! I!2 rearranging for v gives v =
10
gh ~ 1.19 gh for solid spheres. Thus the velocity at the bottom of the ramp is
independent of M and R so all the balls should reach the bottom at the same time.
7
For a solid cylinder v = 4 3 gh ~ 1.15 gh, so generally spheres have a higher speed
than a cylinder and will win the race.
A soup can with contents that slosh about will also take longer than one with more
solid contents.
Accompanying sheet
A Loaded Race
What determines how fast a sphere or cylinder rolls down a hill?
Examine the various objects.
Which will roll fastest?
Newton’s Cradle – Two Steel Balls
Apparatus
Newton’s cradle with only two steel balls
Action
The students swing one of the balls out and observe what happens when it is released
and collides with the other ball. They should try to identify the forces acting on the balls and
identify an action-reaction force pair. This demonstration is used to demonstrate Newton’s
third law on the Newton’s laws sheets, and conservation of momentum of the Momentum
Conservation sheets.
The Physics
When the swung ball collides with the stationary ball is exerts a force on it which
accelerates it and causes it to swing outwards. The stationary ball exerts an equal and opposite
force on the initially moving ball, which decelerates it, causing it to stop. The action-reaction
force pair is the force due to ball A on ball B and the force due to ball B on ball A. Note these
forces have opposite directions, equal magnitudes and act on different objects. At all times
gravity and tension due to the strings act on the balls, but these are not an action-reaction pair.
When the first ball (ball A) swings back and hits the second ball (B) it stops. The
second ball swings out. Momentum is conserved, so the change in momentum of ball A must
be equal in magnitude and opposite in direction to the change in momentum of ball B. We can
write this as !pA = - !pB.
r
dp
r
We also know that F = dt , which is Newton’s second law. Since the momentum
changes of the two balls are equal in magnitude and opposite in sign, the forces acting on them
must also be equal in magnitude and opposite in sign. This is equivalent to Newton’s third law
which states that the force exerted by ball A on ball B must be equal and opposite to the force
exerted by ball B on ball A, FAB = - FAB.
A
B
Accompanying sheet
Newton’s Cradle – Two Steel Balls
Swing one of the balls out and release it.
What happens? Explain your observations.
Is there an action-reaction pair here?
If so, what is it?
Pendulum
Apparatus
simple pendulum, such as a ball on a string suspended from a retort stand
Action
The students swing the pendulum and determine how the kinetic and gravitational
potential energy of the bob vary in time. They identify the positions at which kinetic energy is
a maximum and minimum, and where gravitational potential energy is maximum and
minimum. For the “motion” worksheets the students determine where the velocity and
acceleration are a minimum and maximum.
For the Work, Power and Energy worksheets the students look at energy changes and
determine which forces are doing work.
The Physics
At the lowest point of its motion, kinetic energy is maximum and potential energy is
minimum. This is where the velocity is a maximum. At the highest point of its motion, kinetic
energy is minimum (i.e. zero) and potential energy is maximum. The acceleration is a
maximum at the end points of the swing, and a minimum (zero) in the middle, at the lowest
point.
The forces acting are gravity, tension and friction. Tension does no work because it is
always at right angles to the direction of motion. Both gravity and friction do work, although
the only net work done on the pendulum in a complete swing is done by friction slowing the
pendulum down.
Students at the
Australian Catholic
University observing
energy changes in a
pendulum.
Accompanying sheets
Pendulum
Pendulum
At what position is the kinetic energy maximum?
Where is it minimum?
What forces are acting on the pendulum?
What work is being done on the pendulum?
At what position is the potential energy maximum?
Where is it minimum?
Draw energy bar graphs for the pendulum at
different points in its swing.
What forces are doing the work?.
Pendulum on Trolley
Apparatus
simple pendulum (not too big) mounted on a trolley
Action
The students swing the pendulum bob out while holding the base still. They then let go
of the base and release the bob. They should observe the motion of the pendulum and the
trolley.
The Physics
When the bob is raised and the base held still the total momentum of the pendulumtrolley system is zero. When you release the bob it swings down, gaining momentum. In order
for momentum to be conserved the trolley must move the opposite way, which it does. As the
pendulum swings back and forth the trolley will roll back and forth in the opposite direction,
until friction eventually stops it.
release bob
bob and pendulum move
in opposite directions
Accompanying sheet
Pendulum on Trolley
Hold the trolley still and swing the pendulum bob out.
Now let go of the trolley and release the bob.
What happens to the trolley?
Why does it behave like this?
Rotating Stool
Apparatus
a stool which rotates smoothly,
small hand held weights,
enough floor space for other students to stand well back.
Action
The students take turns sitting on the stool and being spun SLOWLY by each other.
They should start off with their arms drawn in, so that they can brake by extending their arms.
If they start with arms out they can spin too quickly and fall off when they draw their arms in.
The Physics
When the hands are stretched the system has a larger rotational inertia and a smaller
angular velocity. When the hands are pulled inward towards the body the rotational inertia
decreases and hence the angular velocity increases. Angular momentum of the system (person
and weights) is conserved.
Notes:
It is important that other students stay well clear of the chair to avoid mishaps, and students do
not rotate for too long and get dizzy and/or nauseous.
large I,
small ".
small I,
large ".
Accompanying sheet
Rotating Stool
Sit on the stool with equal weights in your hands.
Start rotating with the hands stretched out
and slowly bring your hands towards your chest.
What happens? Why?
What happens when you stretch your arms out again?
Warning: Stand well back from the chair when it is in use!!
Do Not Rotate Too Quickly!!!
Solar Panel and Electric Circuit
Apparatus
small solar panel connected to small motor or light, desk lamp
Action
The students hold the solar panel beneath the desk light and observe the light or motor.
They track the energy changes from the desk light through the circuit. It is helpful to draw a
flow chart showing the energy conversions taking place.
The Physics
Energy as light is converted to electrical energy by the solar cell which is then
converted to kinetic energy by the motor or back to light by the globe. Some energy is also
converted to thermal energy at each step.
A student at the
Australian Catholic
University using the
light energy from a
desk lamp to run a
small fan to cool
himself.
Accompanying sheet
Solar Panel and Electric Circuit
Trace the energy conversions and identify
which ones are "not useful" and which are.
Think of a case where this "not useful" energy may be useful.
Trace energy transformations that occur as water stored in a dam,
supplies energy to a hydro-electric power station,
which supplies energy to turn on an appliance at home.
Tennis Racquet
Apparatus
tennis (or squash) racquet, tennis balls
Action
The students hold the tennis racquet in one hand and drop a ball onto it, observing how
high the ball bounces from the racquet. They then hold the racquet with the handle firmly
pressed against a horizontal surface (such as the floor or desktop). This is quite difficult, and
works best when the handle is stood or sat on. They then drop the ball onto the racquet from
the same height above the racquet as previously, and observe how high it bounces this time.
The Physics
Kinetic energy from the ball is transferred to the racquet strings and through the handle
to the wrist or hand of the person holding the racquet. When the racquet is held loosely it
vibrates, dissipating energy. When it is held firmly so that it cannot vibrate there is less energy
lost from the strings to the handle. Hence more energy is stored in the strings and returned to
the ball, allowing it to bounce higher than when the handle is loosely held.
Students at the University
of Sydney experimenting
with the tennis racquet.
Accompanying sheet
Tennis Racquet
Hold the racquet out horizontally and drop a ball onto it.
How high does the ball bounce?
Hold the handle still by stepping on it.
Drop a ball onto it from the same height as before.
How high does the ball bounce now? Why?
Two Sheets of Paper
Apparatus
two pieces of paper, two retort stands with clips to hold the paper
The sheets of paper should be hung parallel, spaced a few centimeters apart.
Action
The students blow between the two pieces of paper. An extension is holding a single
piece horizontally and blowing along it so that it rises.
The Physics
When you blow between the two pieces of paper the air between them is at a higher
velocity and lower pressure than the air around them and they are pushed together. This is an
application of the Bernoulli effect.
~ 2cm
blow
Accompanying sheet
Two Sheets of Paper
Blow between the sheets of paper.
Explain what you observe.
Where else have you observed this effect?
Using conservation principles: Some Questions
1. You throw a ball vertically into the air and catch it when it returns. Answer parts a and b
first by ignoring air resistance, and then taking it into account.
a. What happens to the ball’s kinetic energy during the flight?
b. What happens to the total energy of the ball?
c. What happens to the energy of the ball as you catch it and it comes to rest?
2. Why is it easier and less painful to catch a fast moving ball by moving your hand back with
the ball, rather than keeping your hand still?
3. Energy bar graphs can be used to represent the fraction of energy in a system in different
forms. The graphs below show how the energy in a system is distributed over time, with time
increasing to the right.
a. Describe and draw a set of diagrams for a situation which could be described by the
following energy bar graphs.
b. Make up your own scenario and swap with another student or group and draw appropriate
energy bar graphs.
4. Astronauts use a strong line to attach themselves to the outside of a space craft when they
go outside. Draw a diagram showing what happens when an astronaut pulls on the line to get
back to the space craft. How does the momentum of the astronaut change? How does the
momentum of the space craft change? What about the astronaut and space craft combined?
5. A box of mass m starts at the top of a slide, which you may treat as frictionless. The box
slides down a vertical distance h and at the bottom encounters a horizontal spring with spring
constant k.
a. By what maximum distance, x, is the spring compressed?
b. Describe the motion of the box after this.
6. Describe the changes in energy that occur as a current moves around a circuit. Explain how
Kirchoff’s loop law can be considered as a statement of conservation of energy.
7. An open cart is moving along with constant velocity on a frictionless track when it begins
to rain.
a. What happens to the velocity of the cart? Explain your answer.
b. The rain stops, and a plug is opened in the bottom of the cart. What happens to the velocity
of the cart now, and why?
8. Two students slide down a water slide, one on the straight slide and the other on the curved
one as shown. Which student is moving faster when they reach the bottom? Explain your
answer.
9. An empty can of dog food and a full can roll down a hill. Which one is moving faster when
it gets to the bottom? Explain your answer.
10. A figure skater is spinning with her arms outstretched. She pulls her arms in to her body
and her angular velocity increases.
a. Explain why this happens.
b. What happens to her rotational kinetic energy when she pulls in her arms? How can you
account for any change in energy?
11. A physicist observes an unstable particle at rest. Suddenly the particle decays into at least
two fragments, perhaps more. Consider the three cases shown below. The paths and velocity
vectors of two of the resulting fragments are shown, but the masses of the fragments are
unknown.
A
C
B
a. In which cases (if any) can the physicist be certain of the existence of at least one other
additional, unobserved particle associated with the decay? Explain your reasoning.
b. For each case, what can be said (if anything) about the relative masses of the fragments?
12. Water flows through the pipe shown below from left to right.
A
B
C
a. Rank the volume rate of flow at the four points A, B, C and D.
b. Rank the velocity of the fluid at the points A, B, C and D. Explain your answer.
c. Rank the pressure in the fluid at points A, B, C and D. Explain your answer.
D
Solutions
1.
a. and b. With no air resistance:
The ball has an initial velocity, vi, and kinetic energy KEi =! mvi2. It also has an initial height
hi =0 and potential energy PEi = mghi = 0. As the ball goes up, its kinetic energy decreases and
is zero at a height of h, its gravitational potential energy increases and is maximum at h. The
reverse happens on the way down, such that the total energy of the ball is constant, i.e. at
every instant PE + KE = total energy. Or, in terms of work rather than potential energy, there
is no change in kinetic energy of the ball between the initial and final positions thus the total
work is zero. During the flight the work done by weight on the way up is W=-mgh and on the
way down is W = mgh.
a. and b. With air resistance:
As the ball goes up, there is work done by air resistance, so the ball’s kinetic energy decreases
and is zero at a height h' which is less than h. On the way down again there is work done by
air resistance. Consequently the final kinetic energy is less than the initial kinetic energy, i.e.
final speed is less than initial speed. The total energy of the ball-earth system is PE + KE +
Wair resistance. Total energy is constant but the ball has less mechanical energy when caught.
Some of its energy has been converted into heat due to work done by air resistance.
c. The energy is converted into heat, sound and motion of the muscles in your hand.
2. We can approach this in terms of either conservation of energy or conservation of
momentum. Your hand must do work on the ball to change its kinetic energy from ! mv2 to 0.
The work done is given by the force times the distance, so if you increase the distance over
which your hand applies the force to stop the ball, the force required is less. If the force on the
ball by your hand is less then the force by the ball on your hand will also be less. In terms of
conservation of momentum, increasing the time over which the momentum is transferred to
your hand decreases the force that must be applied. This is done by moving your hand
backwards with the ball.
3. There are many possible scenarios that could be described by this set of energy bar graphs,
for example someone sliding down a slide and landing in water at the bottom.
4. Astronauts use a strong line to attach themselves to the space craft when they go outside.
When an astronaut pulls on the line to get back to the space craft he moves towards the space
ship and it moves towards him. He applies a force to the ship, via the rope. The ship applies a
reaction force, equal in magnitude but opposite in direction, to the astronaut. The rate of
change of momentum is equal to the net force applied, and so assuming no other forces the
change in momentum of the space ship is equal to that of the astronaut, and they move towards
each other. As no external forces are acting the total change in momentum is zero. Note that
the mass of the ship is much larger than the astronaut’s mass, hence the astronaut accelerates
rapidly towards the ship, while the ship accelerates only slowly towards the astronaut.
5. A box of mass m starts at the top of a slide, which you may treat as frictionless. The box
slides down a vertical distance h and at the bottom encounters a horizontal spring with spring
constant k.
a. We can solve this problem using conservation of energy. The box begins with gravitational
potential energy mgh, which is converted to kinetic energy as it slides down the slide. At the
moment when the spring is compressed to the maximum extent, the box no longer has any
kinetic energy, and all the energy is in the form of elastic potential energy, ! kx2. Setting this
elastic potential energy equal to the change in gravitational potential energy and solving for x
gives: x = (2mgh/k)1/2.
b. The box will oscillate back and forth, sliding back up the slide and down again. If there is
no friction at all it will undergo simple harmonic mtion, each cycle returning to the height
from which it started. If there is any friction, as of course there will be in any real system,
energy will be lost as heat and the oscillations will steadily decrease until the box comes to a
stop.
6. As charge moves around a circuit, once a steady current is flowing, it gains potential energy
when it passes through sources of emf. As it passes through resistors or resistive elements it
loses this potential energy in the form of thermal energy or heat due to the resistance (this is a
lot like friction). When it gets back to the same point in the circuit that it started from, it must
have the same potential energy and if the current flow is steady it will have the same kinetic
energy. Hence the energy gained plus the energy lost is zero around a loop. This is Kirchoff’s
loop law.
(I find a good gravitational analogy that can be demonstrated in class is students moving
around a tiered room in circuits – whatever gravitational potential energy they gain going up
stairs, they lose coming back down to their starting point.)
7. An open cart is moving along with constant velocity on a frictionless track when it begins to
rain.
a. There are two ways to answer this question; using forces and Newton’s third law, or using
conservation of momentum. They are, of course, equivalent. The rain falls into the cart, where
it is accelerated by a force applied by the cart to the rain. The rain therefore applies an equal
an opposite force to the cart, slowing it down. If we consider the rain and cart as a system, the
rain gains momentum in the horizontal direction, hence by conservation of momentum the cart
must lose momentum and slow down.
b. Pulling the plug will not change the speed of the cart. The water flows out with the same
horizontal velocity as the cart. Since no external horizontal forces are acting, the horizontal
component of the momentum of both water and roller coaster are conserved, and the
horizontal component of the roller coaster’s velocity does not change.
v
v
v
v
v
v
8. If we ignore friction, both students start at the same height, hence with the same
gravitational potential energy, mgh, which is converted to kinetic energy as they slide down.
At the bottom each will have kinetic energy ! mv2 = mgh (by conservation of energy), hence
each has a velocity given by v2 = 2gh so they must have the same speed at the bottom.
9. As the cans roll down the hill their gravitational potential energy is converted into kinetic
energy. This kinetic energy is divided between translational ( ! mv2 ) and rotational ( ! I!2).
The empty can has a greater fraction of its mass at larger radius, hence it has a greater moment
of inertia for its mass and so a greater fraction of its energy goes into rotational energy. Hence
it has a smaller translational kinetic energy and takes longer to get to the bottom of the hill.
(Note that the objects do not have to have the same mass or radius – these will cancel out.)
10. As a figure skater pulls her arms in towards her body her angular velocity increases.
a. As she pulls her arms in towards her body she decreases her moment of inertia by moving
some of the rotating mass closer to the axis of rotation. By conservation of angular
momentum, in the absence of external forces a decrease in momentum of inertia must result in
an increase in angular velocity.
b. Her angular momentum is conserved, so L = I! = constant. Her rotational kinetic energy is
given by KE = ! I!2 = ! L!. Hence while L is constant, her rotational kinetic energy has
increased. By conservation of energy this extra energy must have come from somewhere. She
must do work to pull her arms in, thus changing her kinetic energy. The energy comes from
potential energy (chemical energy) in her muscles.
11. A particle, initially at rest decays into at least two fragments, perhaps more.
In all cases the initial momentum is zero, pi = 0, so the final momentum must also be zero.
In diagram A the velocities are in opposite directions but are different in magnitude. This
means that either the masses of the particles are different and the lower one is heavier, or that
there is another particle involved. See diagram opposite.
There must be another particle in B. The total momentum as shown in this diagram is not zero,
as the two velocity vectors are in different directions. Regardless of the masses of the two
particles their momenta cannot add to zero. Hence there must be at least one other particle
present.
In diagram C if the masses are equal then momentum is conserved. If the masses are not equal
then a third particle is needed to satisfy conservation of momentum
A
B
small
mass
or
big
mass
C
equal
masses
equal
masses
third,
unobserved
particle
12. Water flows through the pipe from left to right.
a. Water is incompressible, and as there is no water either entering or leaving between points
A and D the volume flow rate must be same at all points, just as current must be the same at
all points along an arm of a circuit. This is called the principle of continuity- A#v = constant,
and it is a statement of conservation of mass and hence volume for an incompressible fluid.
b. As the water flows from A to B the area increases, hence to maintain continuity v must
decrease, therefore vA > vB. As the water then flows downhill to point C it will gain energy,
however the area has not changed so we know, because of continuity, that the velocity has not
changed, vB = vC. When the water flows from C to D the area increases again, so the velocity
will decrease again, vC> vD. The ranking is therefore vA > vB = vC> vD.
c. When the water flows from A to B there is no change in gravitational potential energy,
however the velocity has decreased which means that the kinetic energy of the water has
decreased. By conservation of energy we know that if kinetic energy decreases, some other
form of energy must increase. If we look at Bernoulli’s equation, "gh+ ! "v2 + P = constant,
(which is a statement of conservation of energy density), we can see that the pressure must
have increased, PB > PA. When the water flows downhill from B to C it loses gravitational
potential energy, but the velocity and hence kinetic energy does not change. The pressure must
again increase in going from B to C, PC > PB. Finally, as the water flows from C to D the
velocity decreases again and the pressure must once more increase, PD > PC. So the final
pressure ranking will be PA < PB < PC < PD
Session 2: Forces
Forces and Energy
We are now looking at the top right hand section of the concept map from the previous
session. A somewhat more detailed map for mechanics is shown below.
There is a relationship between force and energy – the amount of energy transferred to an
object (or taken from it) by a force acting on the object is the force times the displacement of
the object in the direction of the applied force. This is usually called work, and written
W = " F • dx = Fx cos !
(1)
where we can use the right hand side expression when force and displacement are constant and
at an angle $ to each other. In the simple case of a single force acting on a body initially at rest
the displacement will be in the same direction as the force, so the energy transferred (or the
work done) is just W = Fx. When a net force acts on an object it does work on it, which results
in a change in the objects kinetic energy, the change in kinetic energy is the work done. This is
the work-energy theorem from session 1:
W = " F .dx = !KE
(2)
from which, by applying conservation of energy, we cam also write:
! dU
F=
dx
(3)
Note that the dot product in equation 1 is very important. The relative directions of force and
displacement tell you whether there is work being done at all, and whether it is positive or
negative.
A car that is slowing down due to friction does so because it experiences a net force. The
direction of this force is opposite to the direction of displacement. This means that the force
does negative work, and decreases the car’s velocity and hence its kinetic energy.
What happens when a force acts in a direction perpendicular to the direction of motion? This
is what happens in circular motion, for example the orbit of a satellite around the Earth. The
satellite is held in orbit by the force of gravity. If there was no gravitational force the satellite
would simply fly off at a tangent. The direction of the gravitational force acting on
the satellite is directly in towards the Earth, which is at right angles towards the path of the
satellite. This force causes an acceleration, and the acceleration is in the same direction as the
force – towards the Earth. But the force is at right angles to the path, and hence no work is
done on the satellite. This makes sense when you consider work as the change in kinetic
energy, the satellite has not changed speed (but it has changed velocity), so there is no change
in kinetic energy, and no work is done.
The solid curved path is the path taken by the satellite. The dotted straight line shows the path
if there was no inwardly directed (centripetal) force.
Newton’s First and Second Laws
Newton’s laws link force to motion, by telling us how an object will accelerate when subjected
to a force. We have already discussed Newton’s first law – an object at rest will remain at rest,
and an object in motion will continue with constant velocity unless acted on by an external
force. This is a statement of conservation of momentum – unless you apply an external, net
force, the momentum of the system is conserved. In this statement of Newton’s first law we
are assuming that mass is constant.
It is always possible to define your system such that there is no net external force (an isolated
system), although this may mean defining your system as the whole universe. Often it is
enough to define your system as a collection of objects that are interacting with each other, but
nothing external. In this case the total momentum is conserved, while the objects may change
their velocity, the average velocity of the system as a whole remains constant. This means that
the centre of mass of an isolated system has a constant momentum. Sometimes this can be
used to solve otherwise messy problems.
Newton quantitatively defined force with his second law :
dp
F=
= ma
(4)
dt
This tells us that the bigger the force, the greater the acceleration, but that the bigger the mass,
the smaller the acceleration for a given force. It is important to remember that both force and
acceleration are vectors – and that the net force is always in the same direction as the
acceleration and the change in momentum.
When an object, such as a car, is stationary, a force must be applied to the car to make it move.
Once it is in motion, it will continue with constant velocity unless a force is applied to it. This
seems to contrary to our common understanding – if we didn’t have to apply a force to keep
the car moving, why does it slow and stop if we take our foot off the accelerator? Of course
there is a force acting to slow down the car – friction. The car is rolling, and momentum is lost
due to friction acting between the wheels and axles, and air resistance acting on the car body.
All the petrol that you burn while driving at a constant speed goes to overcome the forces of
friction acting to slow down the car.
It is important to note that total or nett force is in the direction of acceleration, not necessarily
in the same direction as the motion. Friction acting to slow down a car, or the gravitational
force acting to slow down a ball which has been thrown into the air acts in the opposite
direction to the velocity of the object. Many students have a great deal of difficulty with this
idea, and when asked to draw the forces acting on a ball which has been thrown upwards will
draw a force in the direction of motion. This is an indication of an Aristotelian view or similar,
in which it is assumed that there is some “impetus” required to keep an object moving.
Newton’s Third law
Newton’s third law says that for every force there is an equal and opposite force, often stated
as for every action there is an equal and opposite reaction. This is written as:
Fa on b = !Fb on a
(5)
and tells you that if object a exerts a force on object b, then object b exerts a an equal force but
in the opposite direction on object a. The really important thing to remember here is that the
forces act on different bodies. If we go back to the way Newton wrote his second law:
dp
F=
(6)
dt
we can see that Newton’s third law is telling us that whatever the change in momentum of a is,
the change in momentum of b is equal and opposite – i.e. the net change in momentum is zero
and total momentum is conserved. It is easier to remember the key ideas of Newton’s third law
when considered as a statement of conservation of energy.
Many students find Newton’s third law difficult to apply correctly, even though they can recite
it easily enough. A typical question is how can a horse pull a cart if the cart exerts an equal
and opposite force on the horse. The answer is simple. The force that the horse exerts on the
cart causes the cart to accelerate. The force acting on the horse due to the cart will cause the
horse to accelerate in the other direction – but this has no effect on the motion of the cart
(unless they collide). So the cart is accelerated forward. If the horse is also to accelerate
forward there must be a nett forward force acting on the horse, which means there must be
another force acting on the horse greater than the force applied by the cart. This other force is
the frictional force of the ground on the horse’s hooves.
The four forces
There are only four known forces, and in fact it now seems that there are only really three.
These are the gravitational force, the electromagnetic force and the strong and weak nuclear
forces, although there is now evidence that the electromagnetic force and the weak nuclear
force are aspects of the same force.
The Gravitational Force: The gravitational force is the force that acts between all bodies with
mass. It has the form
mm
(7)
F = G 12 2
r
The force is proportional to the masses of the bodies and inversely proportional to the square
of the distance between the bodies. The constant G is the universal gravitational constant. It
is possible to derive Kepler’s laws, which describe the behaviour of orbiting bodies, such as
the planets around the sun or satellites around the Earth, from Newton’s law of gravitation.
Gravity is a central force, such that it acts towards the center of mass of the objects.
Gravity is responsible for the “weight force” that we feel on Earth. If we plug the numbers into
the gravitational force we find that on the surface of the Earth, which has a mass of 6# 1024 kg
and a radius of 6,400 km, the force is equal to F = m. 9.8 m.s-2. Using Newton’s second law, F
= ma, we can see that the acceleration of an object on or very close to the surface of the Earth
is 9.8 m.s-2. You may wish to point out to students that in using this approximation they are
treating the gravitational field as constant, and hence of the form due to a large flat plain of
mass – in other word using F = mg is equivalent to assuming the Earth is flat (at least locally).
The Electromagnetic Force: The electromagnetic force is the force that acts between
stationary or moving electrically charged bodies. Apart from gravitational forces, virtually all
the forces that we experience are electromagnetic in nature. The forces that bind atoms
together are electromagnetic in nature, and frictional forces and contact forces are also
electromagnetic.
Stationary charges exert a force on each other called the Coulomb force, given by:
qq
(8)
F = k 1 22
r
where the q are the charges, r is their separation and k is a constant.
You have probably noticed that this force looks very similar to the gravitational force. It is
proportional to the product of the charges, and inversely proportional to the square of the
distance between them. There are a couple of differences – this force is repulsive for similarly
charged objects, and gravity is always attractive. Mass only comes in one type – charge comes
in two – positive and negative.
A charge has an electric field associated with it – which is just a way of describing force at a
distance. The field is the force that would act on a positive test charge at a given point. When a
charge moves its electric field moves with it.
A moving charge creates a magnetic field. Hence sources of magnetic fields are currents and
magnetic materials which have moving charges (generally electrons) within them. Many
particles such as electrons have an associated magnetic field even when they are not moving.
To explain how this is possible we ascribe a property called “spin” to these particles, which
accounts for their magnetic field. (Note that we don’t really believe electrons are literally
spinning, its just a way of labeling the property that gives them a magnetic field.)
A moving magnetic field, such as that due to a moving bar magnet, creates an electric field.
This is a beautiful symmetry – a changing electric field creates a magnetic field, and vice
versa. It is also incredibly useful, apart from solar energy, all our electricity supplies are based
on this effect. A generator uses a changing magnetic field in a coil of wire, by moving either
magnets or the coil, to produce an electric field. This electric field supplies a force (an
electromotive force) on the electrons in the wire coil, which makes them move – a current. A
generator converts kinetic energy into electrical potential energy. We can also do it the other
way around – a motor converts electrical energy into kinetic energy using the opposite
process.
The Nuclear Forces
Apart from hydrogen, which has only one proton, all nuclei are made up of multiple protons
and neutrons. Protons are positively charged and hence should repel each other. Yet most
natural nuclei are stable. If the only forces acting were the gravitational force and the
electromagnetic force, nuclei would fly apart because the attractive gravitational force is much
weaker than the repulsive Coulomb force. Hence there must be another force acting, which is
strong enough over small distances to keep the nuclei together – this is called the strong
nuclear force. It does not obey a 1/r2 rule, like the electromagnetic and gravitational forces, but
drops off much faster.
Friction and the normal force
We know from Rutherford’s experiments bombarding atoms with alpha particles that atoms
are mostly space. So why is it that you can rest your hand on the desk, or push the buttons on a
keyboard, without your fingers going through? This is because the electrons in the atoms of
your fingers interact with the electrons in other matter, preventing them from simply all
joining together or flowing through each other.
The normal force and the frictional force are two components of this contact force. The
frictional force acts in the direction parallel to the interface of the two surfaces that are in
contact. The normal force acts perpendicular (or normal, as the name suggests) to the
interface. Both forces take a value equal to the applied force, up to some maximum value,
beyond which there is a rapid change, usually a decrease, in the contact force. This happens
when enough force is applied perpendicular to a surface such that the surface breaks, or
enough force is applied parallel such that there is sliding of the surfaces relative to each other.
Have you ever tried to push a heavy piece of furniture along the floor and found your feet
sliding out from you instead? This is because the force of friction between your feet and the
floor is smaller than the force that the piece of furniture is exerting on you (which from
Newton’s third law is equal in size but opposite in direction to the force that you are exerting
on the furniture). We often think of friction as a nuisance, something that wastes energy. But
without friction we wouldn’t even be able to walk – our feet would just slide out from under
us and we wouldn’t go anywhere. When we walk, we move forwards because the ground
exerts a frictional force on us. The only way to start moving, or change velocity, is for an
external force to act on us – we cannot change our own state of motion. To be able to move
about we are constantly applying Newton’s third law – we push on the ground, it pushes back
on us, and it is this force of the ground on us that moves us forwards. This force is a
combination of the frictional and normal forces. It is important to remember that frictional
forces act between surfaces to oppose relative motion of the surfaces. Friction does not oppose
motion per se, in fact it enables motion by allowing us to apply a force.
Sometimes forces are described as conservative or non-conservative forces. Conservative
forces are those that completely convert potential energy into kinetic energy or vice versa.
Non-conservative forces convert potential energy or kinetic energy into thermal energy – in
particular frictional forces. Thermal energy is a form of kinetic energy – it is the kinetic
energy due to movement of all the particles in an object or substance. The sort of bulk kinetic
energy we usually talk about, where a whole object is moving with a given velocity, can be
completely converted into thermal energy. But thermal energy cannot be completely converted
back into potential energy or kinetic energy.
Hence frictional forces are called non-conservative – even though the total energy is still
conserved, it’s just not all available to use anymore. This is what the concept of entropy is
really about – while energy is conserved, whenever friction acts some of that energy is “lost”
to thermal energy. Entropy is a measure of the disorder of a system – when all the particles
move together with the same kinetic energy this is a very ordered system. When they all have
their own random kinetic energy this is a very disordered system, with a higher entropy.
Entropy may decrease locally, but overall, the entropy of the universe is always increasing.
Free body diagrams
Within a few weeks of starting any course, when I ask the class “How shall we start this
problem” I get a chorus of “draw a diagram” shouted back. The first step in solving any
problem is to draw a diagram and put all the information on it that you can. More often than
not a series of diagrams showing the evolution in time of the problem, or a set of force
diagrams – free body diagrams – showing the forces acting on each body separately, are
needed.
A free body diagram allows you to quickly see all the forces acting on an object and then
apply Newton’s second law to relate the total force to the acceleration. It is a simplified
diagram that shows the body as a blob (a small square, circle, whatever) with all the forces
acting on it. Only show forces acting on that body and only show them once. It may be
tempting when analyzing an object on an inclined plane to draw both gravity and its
components parallel and perpendicular to the plane. If you do this you are saying that there are
three forces due to gravity acting, and you are also likely to confuse yourself and end up with
a very complicated diagram. Draw each force once, in the direction it is acting. Then draw a
second free body diagram with each force shown broken into its components (but not the sum
of them again). A good diagram contains most of the physics of the problem, and usually there
should be several diagrams, often of different types.
Having a diagram showing the physical set up of the problem also allows you to quickly
identify any Newton’s second law force pairs, and then put the appropriate one on your free
body diagram. But beware! Remember that only ONE of the pair goes on a given free body
diagram. A free body diagram shows only the forces acting on the body, not the forces it
exerts on other objects. (This is also a quick check to see if a pair of forces is an actionreaction pair – if you can draw them on a single free body diagram, they cannot be.)
Forces: Some Activities
These activities have been chosen because they use mostly simple everyday objects, or things
that can be easily and cheaply made.
These activity sheets are extracts from the “Workshop Tutorials for Physics: Activities” book
(Wilson et al). Each sheet has a photo or diagram of the equipment, suggested questions to be
asked and a short explanation of the relevant physics.
See also Newton’s Cradle: 2 steel balls in the section above, and many of the other activities.
Accelerating on a Ramp
Apparatus
wind up toy car, ramp
Action
The students allow the car to roll down the ramp (without winding it up). They draw a
free body diagram showing the forces acting on the trolley.
They then wind the car up so that it can accelerate up the ramp for at least a short
distance. They should draw a free body diagram showing the forces on the car as it climbs the
slope.
The Physics
The forces acting on the toy car as it rolls down are gravity, friction and the normal
force. The component of gravity along the direction of the slope is greater than any frictional
forces and gives an acceleration down the ramp.
When the toy car is wound up and set to climb the ramp its wheels exert a force on the
ground. The ground exerts an equal and opposite force, due to friction, on the cars wheels
which push the car. As long as the coefficient of friction is great enough that slipping does not
occur, and the force on the wheels is greater than the component of gravity down the hill, the
car will accelerate.
accelerating up:
accelerating down:
N = mgcos$
Ffriction due to ground
N = mgcos$
Ffriction
mgsin$
+ Ffriction due to air
mgsin$
mgscos$
mgscos$
$
$
Accompanying sheet
Accelerating on a Ramp
Place the car at the top of the ramp and release it.
What happens? Why?
Draw a free body diagram of the car.
How does a car accelerate up a hill?
Draw a free body diagram showing the forces acting to accelerate a car up a hill.
Acceleration due to Gravity
Apparatus
tennis ball
Action
The students throw the ball directly up into the air and catch it again. They should
watch to see how the ball’s velocity changes, and consider the direction of acceleration of the
ball during its flight.
The Physics
Once the ball has left the student’s hand there is only the force of gravity acting on it
(neglecting air resistance). Hence the only acceleration is that due to gravity, and the
acceleration is always downwards. The velocity of the ball decreases as it goes up, becomes
zero, and then negative as it falls back down. Hence the speed (magnitude of velocity)
decreases and then increases again, but the velocity only decreases. This exercise helps
students recognise that acceleration is not always in the direction of velocity (or
displacement). Many students find this a difficult concept.
x
v
a
t
g
t
Accompanying sheet
Acceleration due to Gravity
Throw the ball straight up into the air,
and catch it when it comes back down again.
Describe what happens to the velocity and acceleration of the ball.
Sketch the acceleration as a function of time.
Sketch the ball’s velocity and displacement as a function of time.
t
Accelerometer
Apparatus
accelerometer on wheels, see diagram below
The accelerometer is a narrow Perspex tank partly filled with a coloured fluid, e.g. water with
a little food colouring, with a string attached to one end so it can be pulled along. A good size
is around 20 cm long by 15 cm high.
Action
The students pull the accelerometer at constant speed, or accelerate forwards and backwards.
The Physics
The surface of the fluid in the accelerometer should be fairly flat and horizontal at
constant speed, as there is no net force acting on the fluid, it looks just as it would if it were
standing still.
When accelerated forwards, the fluid’s surface will make an angle to the horizontal.
The direction of the slope of the fluid shows you the direction of the acceleration. The fluid
surface is at an angle because the net force on the fluid is no longer zero. The fluid collects at
the back of the accelerometer when it accelerates. In general the fluid will “point” like an
arrow in the direction of acceleration.
Note: this device is also handy for a relativity activity as it is impossible to tell whether you
are moving at constant speed or standing still. It is also handy for circular motion – when
placed on a spinning turntable the fluid collects at the two ends of the accelerometer, with the
fluid pointing inwards, showing that the acceleration is towards the centre of the motion.
constant v
accelerating
Accompanying sheet
Accelerometer
Pull the accelerometer at a constant speed.
What does it show? Why?
Now accelerate it forwards. What do you observe?
Allow the accelerometer to roll. What does it show as it slows down?
a
Boxes on a Trolley
Apparatus
three identical boxes filled to have different masses and sealed up, large trolley which
will hold all three boxes in a row across the trolley with a regular surface
Action
The students examine the boxes to see which is heaviest and which is lightest. They
then predict which will fall off the back of a trolley when it is accelerated. They should
consider which direction the trolley accelerates, and predict which way the boxes will fall off.
They then place the three boxes on the trolley and accelerate it to check their predictions. They
should repeat the experiment a few times, changing the box positions, as small differences in
surfaces will make one box or another fall off first.
The Physics
The boxes will all fall off together. Assuming the mass of the
boxes is not so great as to squash the surface of the box or trolley, the
acceleration due to friction will not depend on the mass of the box. The
maximum possible acceleration due to friction (which is the net force
holding the boxes to the trolley and hence accelerating it) is the
maximum acceleration is amax = µN / m = µmg / m = µg. The
acceleration does not depend on mass, only on µ, which should be the
same for all boxes. Note that the net force acting is in the direction of
movement, which is forwards, and this is the frictional force. Many
students have difficulty with the idea that friction is causing the motion
of the box.
Note that it may be easier to decelerate the trolley by stopping
it suddenly with a foot rather than pulling on it. In this case the boxes
slide the other way.
Accompanying sheet
A student at the University
of Sydney pulling the
trolley with boxes.
Boxes on a Trolley
Examine the three boxes.
Which will fall off the trolley first when you accelerate it? Why?
Place the boxes in a row across the back of the trolley.
What happens when you accelerate it?
What force is accelerating the boxes? In what direction is this force acting?
Which box falls off first?
Gaining Weight
Apparatus
two bathroom scales
Action
A student stands on one of the scales. They then see if they can change the reading
without touching anything else. They should try to hold the new reading. They then try to
change the reading while holding on to a friend. They should note the direction of the force
that they exert on their friend, and the direction of the reaction force exerted on them. They
should also note the direction of change of the reading on the scale.
With one foot on each scale the students can experiment with shifting their weight
distribution. They should attempt to draw free body diagrams showing the forces acting on
themselves and on the scales.
The Physics
When the student is not touching anything else the only external forces acting on them
are gravity and the normal force. They can change the reading on the scale momentarily by
bouncing up and down, but they cannot change and hold the reading. When the students can
experience an additional external force, by pushing up or down on a friends arm, they can
change and hold the reading.
If they push down on the other student they experience a
reaction force upwards, which decreases the force acting on the
scales and it reads a lower weight. The forces acting on the
student are now gravity, the normal force due to the scales, and
the external force due to the other student. By pushing upwards
against the other student they can increase the weight shown on
the scale.
With one foot on each scale they can vary the proportion
of their weight on each scale, but the sum of the two readings
should be constant.
The student decreases his
weight (apparently) by
pushing down on the table.
Accompanying sheet
Gaining Weight
Stand on a scale.
Can you change the reading without touching anything else?
Can you hold the new reading?
Can you change and hold the new reading while hanging onto a friend?
What direction do you apply the force in to increase the reading?
What about to decrease it?
Draw a free body diagram showing the forces acting on you and on the scale.
Smooth Variable Ramp
Apparatus
smooth variable ramp, small trolley or toy car, spring balance, protractor
Action
The students attach the trolley to the spring balance and place the trolley on the ramp.
They then adjust the angle of the ramp and observe how the reading on the balance changes.
They should draw a free body diagram showing the forces parallel and perpendicular to the
ramp, and comment on the relative sizes of these forces.
The Physics
The forces acting on the trolley are the normal force, gravity, the force due to the
spring balance and friction. Decomposing these into components along the ramp and
perpendicular to the ramp gives:
Forces perpendicular to ramp: N and mgcos$, which are equal.
Forces along the ramp: mgsin$ , the force due to the spring balance and friction.
The force due to the spring balance plus friction will be equal to mgsin$, the friction should be
very small so the spring balance should read (approximately) mgsin$. This will increase as the
angle increases, and will be equal to mg when the ramp is vertical.
F spring balance
(+ Ffric)
N = mgcos$
mgsin$
mgcos$
$
Accompanying sheet
Smooth Variable Ramp
Draw a free body diagram showing the forces acting on the trolley.
What are the components of the forces
acting parallel and perpendicular to the ramp?
Is the force on the trolley from the spring balance equal to mgsin$?
What happens to the force needed to keep the trolley stationary on the ramp
as the inclination of the ramp is increased?
Forces: Some Questions
1. Draw force diagrams (free body diagrams) for the situations described below. In which of
the situations is the net force on the car zero?
a. A car cruising at a constant speed of 100 km.h-1 on a long straight stretch of a highway.
b. A car going around a curve of radius 20 m at a constant speed of 60 km.h-1.
c. A car accelerating uniformly at 1.3 m.s-2 on a long straight stretch of a highway.
2. A steady force is applied to two boxes, one of mass m the other of mass 2m. The force is
applied to each box for the same time.
a. Which box has the greater momentum after the force has been applied?
b. Which box has the greater kinetic energy after the force has been applied?
c. Which box travels further in this time?
3. Which of the following pairs are action-reaction pairs, and which are not? Explain your
answers.
a. A donkey pulls forward on a cart, accelerating it; the cart pulls backwards on the donkey.
b. A donkey pulls forward on a cart without moving it, the cart pulls back on the donkey.
c. A donkey pulls forward on a cart without moving it, the Earth exerts and equal and
opposite force on the cart.
d. The Earth pulls down on the cart; the ground pushes up on the cart with an equal and
opposite force.
4. A helium nucleus has two protons and two neutrons. It is orbited by two electrons. What is
the ratio of the magnitude of the force on one electron due to the nucleus to the force acting on
the nucleus due to the electron?
5. A passenger sitting in the rear of a bus claims that he was injured when the driver slammed
on the brakes causing a suitcase to come flying toward the passenger from the front of the bus.
Can this occur when the bus is initially travelling forwards?
What if the bus is travelling backwards?
6. A box is placed on the back of a truck and the truck drives away (forwards). The coefficient
of friction between the surface of the truck and the box is µ.
a. Draw a sketch showing the box on the truck.
Add each piece of information, from the questions below, to your sketch.
b. What is the direction of motion of the box relative to the ground?
c. Under what circumstances will the box not move away with the truck?
d. Identify each force, including frictional force, acting on the box.
e. What is the direction of the net force acting on the box?
f. What is the direction of acceleration of the box?
g. What force causes the acceleration of the box?
7. A box is placed on the back of a truck. The coefficient of static friction between the surface
of the truck’s tray and the box is µs . What is the maximum acceleration of the truck before
the box starts to slide? You will need to draw a free-body diagram.
8. We often think of the frictional force as an annoyance - something that wastes energy and
needs to be overcome to get things moving. In fact without friction, we wouldn’t even be able
to walk around!
a. Explain why this is the case.
b. Draw a diagram showing the forces acting on a foot as it steps off the ground and steps
back down again. Show the direction of the frictional force in both cases.
c. What would happen if there was no friction between your feet and the ground? Use your
diagrams to help explain your answer.
9. In New York city cats fall out of apartment windows at a rate of around one per day. Many
of these cat’s fall from windows several floors up. Many of the cats survive because they
reach terminal velocity within a few floors and are able to prepare for landing.
a. What is terminal velocity, and what happens when a cat reaches it?
b. Describe what happens to the gravitational potential energy of the cat/Earth system as the
cat falls. Sketch a graph showing the different forms of energy as a function of time as the cat
falls. Mark on your graph the time at which the cat reaches terminal velocity.
c. If air resistance didn’t increase with increasing speed, would cats ever reach a terminal
velocity? Draw a diagram showing the forces acting on the cat before and after it reaches
terminal velocity. The relative sizes of the arrows representing the forces should indicate the
relative magnitude of the forces.
Solutions:
1. Force on a car in different situations.
a. A car cruising at constant speed in a straight line is not accelerating, and hence experiences
no net force. All the forces acting on it, for example air resistance and the frictional force of
the ground, add to zero.
N
friction- air
resistance.
friction- push of
ground on car.
W
b. A car going around a corner at constant speed is accelerating, and hence experiences a net force
in towards the centre of the curve.
N
friction- air
resistance.
W
centripetal forcefriction- push of ground on car
(perpendicular to v).
friction- push of
ground on car
(parallel to v).
c. A car accelerating in a straight line experiences a net force. In this case, air resistance is less
than the frictional force of the ground on the car.
Diagrams:
friction- air
resistance.
N
friction- push of
ground on car.
W
2. A steady force is applied to two boxes, one of mass m the other of mass 2m. The force is
applied to each box for the same time.
a. If the same force is applied for the same length of time, then the change in momentum is the
same for each box.
b. The box with the smaller mass will have twice the change in velocity as the larger box,
hence as kinetic energy is proportional to v2, not just v, it will have the greater kinetic energy.
c. More work must have been done on the lighter box, as work is force times displacement,
this must mean the smaller box has traveled further in this time.
3. An action - reaction force pair is of the form FAB and FBA where FAB is the force due to B
acting on A, and FAB = - FBA.
a. Donkey and cart - this is an action reaction pair.
b. Donkey and cart - this is an action reaction pair.
c. Donkey, cart, the Earth- there are three objects so this is not an action reaction force pair.
d. Earth, cart; the ground – again there are three objects, and both forces are acting on the
cart, so this cannot be an action-reaction pair.
4. By Newton’s second law, or by writing out the equation (Coulomb’s law) the forces must
be equal in magnitude but opposite in direction. Hence the ratio is 1, or more strictly speaking,
-1 allowing for the directions.
5. The case will only go backwards if the bus is accelerating forwards. A suitcase cannot fly
backwards if the bus is moving forwards at constant speed or braking. If there is not enough
friction to slow the case along with the bus then as the bus slows the case will continue to
move forwards. At constant speed there is no net force on the bus or case, and the case will not
move relative to the bus. Hence the passengers claim cannot be true if the bus was going
forwards and braking.
A suitcase may fly towards the rear if a reversing bus decelerates. When the driver slams on
the brakes the suitcase will continue to move backward, unless the force of friction between
the case and the bus is enough to accelerate it along with the bus.
constant velocity
constant velocity
braking
6.
a. See diagram below.
braking
direction of
motion
b. The box (the system) will move along with the truck, which is to the left unless the truck is
reversing.
c. If the tray and box are very smooth the box will slide off as the truck moves away, i.e. if
there is not enough friction.
d. The forces acting on the system (the box) are the weight force, mg, the normal force, N, and
the frictional force of the truck’s tray on the box.
Ffric
mg
N
e. The net force is the frictional force, which is to the left. The box is accelerating to the left,
hence the net force must be to the left.
f. The acceleration is in the direction of the net force.
g. The only force acting in the horizontal direction is friction, this is the force which
accelerates the box.
7. In the vertical direction: mg = -N or mg + N = 0.
In the horizontal direction: Ffric = net force = ma.
Using Ffric % µN = ma, the maximum acceleration is amax = µN / m = µmg / m = µg.
(Same free body diagram as in question 6 above.)
8. Without friction, we wouldn’t even be able to walk around!
a. When we walk on the ground we use our muscles to push down and back on the ground.
The ground then pushes back on us (Newton’s third law – for every force there is a reaction
force) and this propels us forward.
b. See diagram opposite. When stepping off the ground we exert a force, FFG, on the ground
with our foot, directed down and back. There is a reaction force from the ground, FGF, which
has a component due to friction which prevents the foot sliding backwards on the surface. This
force is mostly exerted around the front of the foot. When the foot comes down again it exerts
a downwards and forwards for on the ground, the ground exerts an upwards and backwards
force on the foot.
foot goes down
foot comes up
FGF
FFG
friction
FGF
friction
FFG
b. If there was no friction (as on ice or an oily patch on concrete pavement), then we could
not apply a backward-directed push to the ground – our foot would just slide over it. Hence the
ground would not be able to push us forward.
9. Falling cats reach terminal velocity within a few floors.
a. Terminal velocity is the name given to the constant velocity which occurs when a falling
body has no nett force acting on it. This is the situation when the drag force upwards equals
the gravitational force downwards. Thus the falling cat will be falling vertically at constant
speed and zero acceleration.
b. The gravitational potential energy (PE) decreases as the cat falls closer to earth. The cat
accelerates gaining kinetic energy (KE). If the cat is moving at terminal speed then the
decrease in gravitational potential energy will appear as an increase in thermal energy (TE) in
the atmosphere due to the drag (frictional ) forces. See graph below.
Energy
PE
KE
TE
vterminal
time
impact
c. See diagrams below. Initially the gravitational or weight force on the cat, mg, is greater then
the frictional force, Ff, due to air resistance. The net force is downwards so the cat accelerates
downwards. When the cat reaches terminal velocity the air resistance is equal to the
gravitational force and the cat no longer accelerates but falls at a constant (terminal) velocity.
This would not happen if air resistance did not increase with velocity
Ff
Ff
mg
mg