Download OBJECTIVES PRE-LECTURE 2-1

17
FORCE
FE2
OBJECTIVES
Aims
This chapter will almost certainly require a re-organisation or replacement of your intuitive views
about the physical world! You should aim to understand the physical concept of force which is
probably quite different from the normal intuitive concept that you have already acquired. You will
also learn the principles, known as Newton's laws of motion, which underpin the scientific idea of
force. You will learn to apply the concept of force and Newton's laws to simple examples.
Minimum learning goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain, interpret and use the terms
mass, vector, component, force, newton, fundamental force, total force [resultant force, net
force], weight [gravitational force], gravitational acceleration [gravitational field], equation
of motion [Newton's second law], Newton's third law, centripetal force, contact force,
tension.
2.
Describe the general system of classification of fundamental forces.
3.
State, explain and apply Newton's third law.
4.
Relate tensions in strings or ropes to the forces exerted by those strings or ropes.
5.
Draw diagrams showing the forces acting on objects.
6.
State the equation of motion and apply it to simple problems by considering force components
in mutually perpendicular directions.
7.
Use force-time graphs to find accelerations, velocities and speeds of objects.
PRE-LECTURE
Introduction
Accelerations are caused by unbalanced forces. This idea was discussed in the last chapter. A
mathematical expression of this result, the equation of motion, is the main topic in this chapter.
Revise questions 1.9, 1.10 and 1.11 in FE1.
2-1
COMPONENTS
The idea of components, which was applied to the vector quantities position, velocity and acceleration
in chapter FE1, can be extended to other kinds of vector quantities such as force.
The value of a vector quantity can be described by specifying its magnitude (number × unit)
and its direction. An alternative, equivalent, description is in terms of three components referred to
three mutually perpendicular directions or coordinate axes. The relation between a vector and its
components is most easily illustrated on two-dimensional paper using two dimensions, with two
reference directions.
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Direction V
a
a sin θ
θ
a cos θ
Figure 2.1
Direction H
Components of a vector
The component of vector a in a direction at an angle θ to the vector's direction is defined to be
equal to a cosθ. In the case illustrated above the component in the direction labelled H is aH = a
cosθ and the component in the direction labelled V is aV = a cos(90° - θ ) or a sinθ . Note that each
component is a scalar quantity (i.e. one which has no direction). The value of a component might
be positive, zero or negative, depending on the value of the angle θ .
2-2
MASS
Mass is a property of an object which is one way of specifying the quantity of matter in it. The
concept of mass can be explained formally in terms of the effect of a total force on a body, using the
equation of motion discussed later in this chapter. Alternatively mass can be determined in terms of
the gravitational force on the object: at a fixed location the gravitational force on different objects
(their weights) are proportional to their masses. The SI unit of mass is the kilogram (symbol kg)
and the unit of weight is the newton (N).
LECTURE
2-3
THE NATURE OF FORCE
Pulls, pushes, compressions, friction, gravity - these are forces that we all are familiar with. All the
forces occurring in nature can be classified into four fundamental types. In decreasing order of
strength these types are:
strong nuclear,
electromagnetic,
weak nuclear,
gravitational.
Nuclear forces
The strong nuclear force holds neutrons and protons together in the nucleus of an atom. It is the
force associated with radioactive alpha decay and nuclear energy, including the atomic bomb and the
hydrogen bomb.
The weak nuclear force holds the neutron together. It is associated with radioactive beta
decay. (Beta decay of carbon-14 can be used for radioactive tracing in biological studies, and for
carbon dating of archaeological items.)
Both types of nuclear force act only over a very short range of distances comparable with the
size of an atomic nucleus, about 10-14 m (10 fm).
Gravity
The gravitational force is the force which causes objects to fall. It is an attractive force between
the Earth and the falling object. This force on a falling object is a special instance of a universal
force which acts between every pair of objects. The fundamental law of gravitational attraction says
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that the magnitude of the force between between two particles, with masses m1 and m2 , at a distance
d apart is
m1 m2
F ∝
.
d2
Note that this force extends to infinite distances and is always attractive. The attractive force
between Earth and Moon, Sun and planets, etc is gravitational.
The weight of an object, on the Earth or on the Moon etc, is the gravitational force acting on it.
At any particular place near the Earth's surface, the weight W of a body is proportional to its mass m:
W = mg
... (2.1).
The constant of proportionality g is the magnitude of the Earth's gravitational field but it is often
loosely called the acceleration due to gravity.
Electromagnetic force
The interactions between atoms and the charged particles within atoms are all electromagnetic.
These forces determine whether a substance will be a solid, liquid, gas or a plasma. They are the
forces involved in chemical reactions and biological processes.
The macroscopic forces described as pulls, pushes, friction etc can be explained, on a
molecular scale, as electromagnetic forces. In general, the force between molecules is strongly
repulsive at short distances but weakly attractive at greater distances.
Force
component
REPULSIVE
Equilibrium
Separation
0
ATTRACTIVE
Figure 2.2 Intermolecular force
The equilibrium separation, where the force is zero, is typically about 10
-10 m (0.1 nm).
On a macroscopic scale the forces on electric charges are electromagnetic, as are those
involved in power generation and electronic communications, computers, and measuring instruments.
Electromagnetic forces extend to infinite distances and can be attractive or repulsive.
2-4
PAIRS OF FORCES
Note that all forces, whether they be fundamental or macroscopic, always occur in pairs. For
example, the gravitational force exerted on the Sun by the Earth is equal in magnitude but opposite in
direction to the gravitational force exerted on the Earth by the Sun.
The electrostatic force exerted on a charged particle, A, by another charged particle charge, B,
is equal in magnitude but opposite in direction to the electrostatic force exerted on B by A. The
same is true for the molecular forces between two molecules
This has macroscopic consequences. Consider a block sitting on a table. The block exerts a
downward force on the table and the table exerts an upward force of equal magnitude on the block.
FE2: Force
These are all examples of Newton's third law of motion.
2-5
THE EQUATION OF MOTION
Forces are responsible, not for motion itself, but for changes in the motion of a body. Change in
motion implies acceleration. The effect of the forces is described by the equation of motion, which
is also known as Newton's second law of motion. The equation relates total force F to acceleration
a. For a body whose mass m does not change:
total force = mass × acceleration.
In symbols this equation of motion is
F = ma
... (2.2).
A word about notation: symbols for vector quantities are printed in bold-face type (e.g. F) while
ordinary italic type is used to indicate the magnitude of a vector (e.g. F means |F|.)
Note the following important points about the equation of motion.
•
Only those forces which act on a body affect the motion of that body.
•
All the forces acting must be included in the total force. (Total force is also called net force or
resultant force.)
•
Directions of forces must be considered in the calculation of the total force. (The procedure
for doing this is given below.)
•
The acceleration has the same direction as the total force.
•
The SI unit of force is the newton, symbol N.
Example with one force acting: ball thrown in the air
A ball is thrown into the air (either vertically or at some angle). If the force exerted on the ball by the air
is negligible, then the total force on the ball is equal to the ball's weight, W = mg downwards. Hence the
acceleration a is equal to g, downwards.
Near the earth's surface, g = 9.8 m.s-2.
Uniform circular motion
Consider an object moving in a circle with constant speed v. The total force in this case is called
centripetal force. It follows from the equation of motion (F = ma ) and the formula for
centripetal acceleration (v2 /R), which is towards the centre of the circle, that
mv2
F = R
and the direction of F is towards the centre of the circle.
Note that centripetal force is not a new kind of force; it is just a fancy name for the radial
component of the total force. In the case of uniform circular motion the radial component is the only
component of the total force.
Calculation of total force
When several forces are acting on an object, they must be combined to give the total force. In
general, the forces will be acting in different directions, so combining them presents a problem.
Fortunately the components of the motion in each of any three mutually perpendicular directions
can be treated separately. This separation of the motion into components is helpful even when there
is only one force acting.
For example, if a ball is thrown into the air at some angle to the horizontal, the vertical and
horizontal components of the motion can be treated separately. Provided that air resistance is
negligible, the vertical motion has an acceleration vertically down, but the horizontal motion has
constant horizontal velocity.
Example: a sliding block
A block is being pulled along a frictionless table. The pulling force is directed upwards at an angle θ to
the horizontal.
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N
P
θ
W
Figure 2.3
Block pulled across a frictionless table
Forces acting on the block:
•
the weight of the block, W, vertically downwards;
•
a contact force, N, exerted by the table on the block, vertically upwards;
•
the pulling force, P.
In this case all the forces act in one plane, the plane of the diagram, so the problem is two-dimensional.
We need only two component directions: vertically up and horizontally to the right are convenient.
Components of the pulling force P:
• horizontal component
PH =
• vertical component
PV
=
P cosθ
P cos (90°-θ )
to the right;
upwards.
Note that the two component forces PH and PV are together equivalent to the pulling force P.
P
PV = Pcos(90° - θ)
is equivalent to
θ
Figure 2.4
PH = Pcos(θ)
Representing a force by its components
The total force
The horizontal component of the total force is FH = PH to the right. So the horizontal component of the
acceleration is equal to
PH
to the right. (Note that components of accelerations are defined in the same
m
way as components of forces.)
The vertical component of the total force FV = N + P V - W, upwards. This must equal zero since the
block does not accelerate vertically. We see therefore that the force the table exerts on the block,
N = W - P V. Note that components, not magnitudes, are added; the component of the weight in the
vertically up direction is -W.
General procedure
Summarizing, the general method is as follows.
(i) Draw in all the forces - make sure that none is omitted.
(ii) Choose three mutually perpendicular directions and calculate the force components along these
directions. (Note that if the motion is confined to a plane as in the example above you need
only two mutually perpendicular directions.)
(iii) Calculate the component of the total force in each direction, by adding the individual
components, and taking account of their signs.
(iv) Calculate the acceleration component in each direction, using the equation of motion.
Note that sometimes you are given the acceleration, in which case you can use the equation of
motion to give the components of the total force.
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POST-LECTURE
2-6
QUESTIONS
Fundamental forces
Q2.1 a ) We have said that the gravitational force is weaker than the electromagnetic force. Is this your experience?
Why might the electromagnetic force appear to be weaker than the gravitational force?
b) Nuclear forces were the last to be discovered. Why was this so if they are the strongest?
c) How would you rate the forces in terms of their significance to biological systems?
Molecular forces
Q2.2 a) Why does a block of metal resist changes in its shape when you attempt to compress it, or when you
attempt to stretch it?
b) Suppose an object is hanging from the end of a fine wire. What holds the wire together? If the load is
increased the wire stretches. How do the forces in the wire change?
c) Compare the force required to pull a brick apart with the force required to lift it up. Which is greater?
What does this suggest about the relative strengths of molecular (electromagnetic) force and gravitational
force?
Weight
Q2.3
A spring balance measures the weight of an object, i.e. the gravitational force on it. The gravitational
force on an object near the surface of the Moon is about one-sixth of that near the surface of the Earth.
a) Suppose a particular object has a mass of 2 kg. According to the spring balance, what would its weight be
on Earth? On the Moon? (An approximate answer will do.)
b) How could you use the spring balance to measure the mass of an object?
Equation of motion
Q2.4 a ) An object on the end of a string is being swung in a horizontal circle at constant speed. What provides the
centripetal force?
If you want the object to swing around at twice the speed, how much greater would the centripetal force
have to be?
Sketch the path the object takes if the string breaks.
b) What provides the centripetal force on the various parts of a revolving wheel?
c) What provides the centripetal force on a satellite orbiting about the earth?
The following dynamics problems, where several forces are involved, can all be attacked using
the method outlined in the lecture.
Q2.5
A block, mass 2.00 kg, is being pulled along a frictionless table. The pulling force is 20.00 N in a
horizontal direction (figure 2.5).
F
= 20.00 N
Figure 2.5
Draw in the weight of the block and the force that the table exerts on the block.
Take components of forces in horizontal and vertical directions.
a)
What is the horizontal acceleration?
b)
What is the total vertical force? (Hint: what is the total vertical acceleration?)
c)
Suppose now that there is a horizontal frictional force of 10.00 N opposing the motion. What is the
horizontal acceleration?
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Q2.6 a )
A car is being driven along a flat road.
What are the forces acting on the car?
What difference does it make if the car is moving at constant velocity or accelerating?
b)
The car is rounding a curve.
What provides the centripetal force?
c)
The car is on a hill with slope θ.
What is the downhill component of its weight?
Q2.7
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A block, mass 2.00 kg, is being pulled along a frictionless table. The pulling force is 20.00 N at an angle
of 30° to the horizontal (figure 2.6).
F = 20.00 N
30°
Figure 2.6
Draw in all the other forces.
a)
b)
What is the total horizontal force? What is the total vertical force?
What is the horizontal acceleration? What is the vertical acceleration? (Will the block be lifted off the
table?)
c)
What would happen if the magnitude of the force were doubled?
d)
Suppose the table were not frictionless, how would this affect your answers?
Q2.8
Two forces are acting on an object (figure 2.7).
4.00 N
y
x
90°
30°
8.00 N
Figure 2.7
If these were the only forces acting, the object would accelerate.
a)
Find the components of the total force in the y ↑ and x → directions.
b)
What additional force or forces would stop the object from accelerating?
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TENSION
When you pull on one end of a rope which is attached to some object at the other end, the rope will
exert a force on that object. Something also happens to the rope itself. To understand this, think
about some place inside the rope and imagine that the rope consists of two parts. Each part exerts a
force on the other part and, according to Newton's third law, those forces have the same size (figure
2.8). This common magnitude of the force between the two parts of the rope is called the tension in
the rope at the particular place considered.
magnitude
Figure 2.8
T
magnitude
T
Internal forces and tension
The force exerted on the object is equal to the tension in the rope at the point where the rope is
joined to it and the force exerted on the rope at the other end is equal to the tension there. In general,
the tension in the rope will be different at different parts of the rope. In some cases where the
applied forces are much larger than the rope's weight, or if the rope is not accelerating, we can make
the useful approximation that the tension is roughly the same throughout the rope. That is the case
in the next problem.
Q2.9
Suppose that your car is stuck in mud at the side of the road. You have the idea that if you tie a rope, one
end to the car and the other end to a tree, and apply a force perpendicular to the rope at the middle, you will
be able to move your car (figure 2.9).
Figure 2.9
a)
How to pull a car out of a bog
Why is this a good idea ?
Hints
• Suppose you have started applying the force to the rope. The rope has taken on the shape shown in
figure 2.10 but the car has not yet started to move. θ is a very small angle.
T
θ
T
P
θ
Figure 2.10
Forces on a point in the rope
• Consider the forces acting on the bend in the rope. The forces at this stage still balance, so take
components to see how the tension compares with the applied force, P. What is the magnitude of the force
on the car? On the tree?
b)
Where might this good idea go wrong ?
FE2: Force
2-8
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AN ACCELERATING SYSTEM
Consider a person in a lift which has a component of acceleration, vertically upwards, of magnitude a
(figure 2.11). The forces on the person are her weight W = mg downwards and a contact force
exerted by the floor of the lift, upwards.
a
mg
N
Figure 2.11
Forces on a person in an accelerating lift
The person's upward acceleration is equal to the lift's acceleration, a, so
N - mg = ma .
In common usage the term weight has a different meaning from the definition given in the
lecture. Commonly, the weight of an object is the force that it exerts on its support, the floor, etc.
In a lift accelerating upwards, the downward force exerted on the floor by the passenger is
equal to N. Since this is greater than its usual value, mg, the passenger's apparent weight is greater
than normal.
Q2.10 a ) Suppose the lift is accelerating downwards. How does N compare with mg now?
b)
When do you experience "weightlessness"?
c)
Why do you get a strange feeling in your stomach?
Interlude: Times
INTERLUDE 2 - THE RANGE OF TIMES IN THE UNIVERSE
age of the universe
time since earliest humans
1 year
1 day
1 minute
period of a sound wave
time for light to travel 1 m
period of an atomic vibration
period of a nuclear vibration
time/seconds
101 8 __
_
age of the Earth
_
15
10 __
_
_
12
10 __
_
_
109 __ human life span
_
_
6
10 __ 1 month
_
_
3
10 __ 1 hour
_
_
1
__ heart beat
_
_
10-3 __
_
_
-6
10 __ period of a radio wave
_
_
-9
10 __
_
_
-12
10 __
period of a molecular vibration
_
_
10-15__
_
_
18
10 __ time for light to cross an atom
_
_
-21
10 __
_
_
time for light to cross a nucleus
-24
10 __
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