Download Teaching Forces and Motion with Confidence I.O.P day @ Rugby

Teaching Forces and
Motion with Confidence
I.O.P day
@
Rugby School
2016
By
Dr Andy Davies (Head of Physics, Rugby School)
Dr David Lo (Senior Physics Technician, Rugby School)
David Wan (Technical Manager, Rugby School)
Gabor Toth (Physics Technician, Rugby School)
Beetle Car with Light Gate
Spring-driven cars are pulled back different distances (10, 20, 30, 40, 50 cm) and released through a light gate,
programmed to measure speed using a mask on the car. Plot a graph of speed of car against pull-back distance.
Discuss the reasons for the shape of graph, and whether it would rise indefinitely. Also bring out the energy
changes involved in the car’s motion.
Apply gentle pressure to the car and drag backwards a certain distance (“pull-back distance”), energy is
stored in a spring mechanism inside the toy. When the car is then released, this potential energy is
converted to kinetic energy.
When the 10cm mask on the toy car breaks the light beam, timing starts, when the light is reinstated, the
timing stops. The QED can be programmed to calculate the time for you.
Use <S>ELECT and <E>NTER/DISPLAY buttons to choose:
Time Interval
Readings = 1
Press GO when ready
Beetle Car with Ticker
Timer
The displacement of a vehicle is the distance moved in a particular direction.
Use ticker tape to record the motion of a toy car and plot a graph of its displacement against time.
Students usually work in groups of two and each of them needs to obtain their own ticker tape of the motion.
1. Attach approximately 1.0 metre of ticker tape to the roof of the car. Label this end
of the tape “Start”.
2. Give the car a quantity of potential energy by pulling it backwards a certain
distance. One of you must use a distance of 15cm.
Your partner should use a pullback distance of 30cm.
You will then compare your final displacement-time graph to see the differences
in
the motion
3. Switch on the ticker timer just before releasing the car.
4. Divide the resulting tape into 5 gap sections (0.1 second) by drawing pencil lines as
shown below. You should stop after the tenth strip.
Start
...... . . . . . . . . . .
0
5
10
.
15
5 Gaps
. .
.
.
20
.
.
.
.
.
Beetle Car with ticker Timer continued…
5. Use the resulting tape to complete the table below of total displacement against time.
TOTAL DISPLACEMENT
( CENTIMETRE)
TIME
(SECONDS)
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
6. Now plot the graph of displacement against time, labelling the graph as below:
Displacement (cm)
Time (seconds)
7. Draw your best fit line. What does your graph indicate about the speed of the car- was it constant, increasing
or decreasing?
___________________________________________________________________________
8. Compare your graph with your partner’s
a) Describe the differences.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
b) Try to explain the differences.
___________________________________________________________________________
___________________________________________________________________________
Using Ticker Tape to
Measure g
The ticker timer is clamped to operate vertically above the bench. Attach a tickertape to a mass (e.g. 200 g) and
let it fall (e.g. 2 m) to a crash-mat. The resulting tape can be divided into 5-dot sections (0.1 s) and the length of
each section measured, giving a graph of velocity against time. About 6 velocities should be shown. The gradient
of the best-fit line then gives g.
Pasco Velocity Tubes
Constant Velocity Tubes
Constant Velocity Tubes effectively introduce the relationship between graphs and motion. Students
can use a meter stick and a stopwatch to plot the position of the bubble as a function of time as it
moves up the tube. Each tube has an obstruction that serves as the initial position for the bubble.
Pupils can plot distance-time graphs and calculate the speeds from the slopes of the graphs.
Method 1: Use elastic bands to mark known
distances along the velocity tubes. Record the times
when the bubble reaches these known distances.
Method 2: Record the distances and set time
intervals.
Discuss factors affecting the speed. The speed is
constant for most of the journey – forces acting are
weight and drag (down) and upthrust (up). When
balanced, resultant force is zero and the bubble
travels at terminal velocity.
Friction Compensation
F = ma is the second law of motion proposed by Sir Isaac Newton.
This formula lets us work out the forces at work on objects
by multiplying the mass of the object by the acceleration of the object
1. Friction Compensation – in this experiment the aim is to show pupils how we can compensate
for the effects of friction when performing conservation of momentum experiments. I usually ask
the students to find out the % of momentum conserved with and without friction compensation.
To friction compensate the track, the readings from
the two light gates must read the same when the
cart passes.
Adjust the level of the track by
turning the foot
First set the QED to Speed
2 Readings
10cm mask
Using the two red buttons on the QED, press S–
Speed
D – Select number
S - Reading=2
D – Mask size
S – Size in cm =10(press 10 times)
Press the yellow button it will have “Waiting...”
then gently push the cart along the track.” Display”
will appear on the screen, press the D, this will give
you your first reading, press again for your second
reading.
Measuring Accelerations
The track has already been Friction compensated for you,
First set the QED to S - Acceleration
D – Select number
S – Reading= 1
D – Mask size
S – Size in cm =5(press 5 times)
Press the yellow button, it will have “Waiting…”
Measure the acceleration with 50g pulling down (0.5N). Put the remaining three 50g masses on the
cart. Then do the same with 100g pulling down (1.0N), each time ensuring the total mass of the whole
system is the same.
Continue until you’ve measured all 4 accelerations, recording your results.
Light gate clamped so that the mask cuts the beam twice.
Mass – stays the same, as the mass is only moved from the cart to the mass hanger, and not added/subtracted.
Force – Changes, as more mass is added to the hanger and taken off the cart.
Acceleration – is measured using the lightgate.
2. F=ma with a constant mass system
F=ma with a constant mass system continued…
The whole system which is accelerating is the cart plus the string and mass hanging off the end. Find the
total mass of the cart plus string and masses.
Total mass of the system = __________________ grams
= __________________ kilograms
You are keeping the mass of the whole system the same. The pulling force is altered by changing the mass
hanging off the end of the string by removing individual masses from the trolley.
Remember that you are keeping the mass of the whole system the same.
Mass Hanging Weight of Hanging Mass (Pulling Force)
(g)
(N)
Acceleration of System (m/s2)
50
100
150
200
What happens to the acceleration when the weight (pulling force) increases?
_________________________________________________________________
_________________________________________________________________
________________________________
Parachutes and falling
Spheres
Drop a ball bearing into tube of oil (glycerol) and observe the constant speed fall using markers down the tube.
The ball rapidly reaches terminal velocity, when the forces of weight (down) and drag and upthrust (up) are
equally balanced. Note drag is the only force that varies with speed. Sketch a graph of its motion form rest, and
use it to explain initial acceleration and terminal velocity in terms of the forces acting.
With plastic parachutes, various factors such as area of canopy, weight of load and method of attachment may
be investigated. Best results are obtained from a 2-floor drop, e.g. in a stairwell or from a balcony.
Neodymium magnets are attached to the
base for the removal of the ball bearings
after use, Please feel free to help yourself
to tissues provided.
Another way to show a similar principle is
to have a hairdryer face down onto a
balance with a variac attached.
Stability
Different blocks of wood are provided, with different areas of base platforms. Each has a plumb-line attached at
the centre of gravity. This shows the line of action of the block’s weight. The block is tilted on a hinged platform
until it topples over. The angle of tilt can be measured with a protractor – note the block topples when the
weight acts over the corner of the base of the block.
A stable object has a wide base and low centre of gravity – as in a racing car.
Other stable and unstable objects can be shown and discussed: wobbly man, balancing parrot, wine bottle in
holder, butterfly, etc.
Pulley System
Pulleys are mechanisms using wheel and rope to lift heavy objects onto tall heights.
Try lifting the weight by pulling each Newton metre (carefully) and note the effect required and the distance
moved.
They change the direction of an applied force and they can even reduce the force needed to lift a weight.
.
Pulley systems are commonly used in construction
A SIMPLE BLOCK AND TACKLE
In order to lift the 10N weight W you have to apply a force of
F on the rope equal to the weight W. The rope is now under
a tension T equal to the force F. To lift this weight a distance
of H=1m you will have to pull in a length L= H=0.5m of the
rope.The mechanical advantage M is one: M=W/F=1.
Pulley System Continued…
ADDING A PULLEY
In order to lift the 10N weight W you have to apply a force
of F on the rope. Like in the simple case of the block and
tackle the rope is under a tension T equal to the applied
force F. But in this case the weight is supported by twice the
tension: 2T=W. The force F you have to apply then is only
half of the weight W.
By having the second pulley you have decreased the force
needed to lift the weight. The mechanical advantage M is
now two: M=W/F=2T/T=2.However, order to lift the weight
a distance of H=0.5m you will have to pull in a length L = H
x M =1m of the rope. You gained by having to apply a
smaller force, but had to compensate by having to pull a
longer length of rope.
MORE PULLEYS
Here the mechanical advantage is M=3
and here it is M=4.
Conclusion: The mechanical advantage M is equal to number of ropes present at the weight end. The
force needed to raise the weight is W/M. In order to lift the weight a distance H you will have to pull a
corresponding longer length of rope L = H x M.
Inertia
Newton’s First Law of motion states that "An object at rest stays at rest and an object in motion stays in
motion with the same speed and in the same direction unless acted upon by an unbalanced force."
Objects tend to "keep on doing what they're doing." In fact, it is the natural tendency of objects to resist
changes in their state of motion. This tendency to resist changes in their state of motion is described as
inertia.
Inertia: the resistance an object has to a change in its state of motion.
When the cardboard is jerked quickly,
the coin will fall into the glass.
Explanation:


The inertia of the coin resists the
change of its initial state, which is
stationary.
As a result, the coin does not move
with the cardboard and falls into the
glass because of gravity
Inertia continued…
Two large tins have been hung from the roof beam, one filled with sand, and the other is
empty, which is more difficult to be moved. The same tin is also more difficult to be stopped
from swinging.
Explanation:


Object with more mass offers a greater resistance to change from its state of motion.
Object with larger mass has larger inertia to resist the attempt to change the state of
motion
Momentum
A safely fixed air-gun fires a pellet into a target truck, e.g. on Pasco track.
As the truck moves, its velocity is measured by light gate and mask – ensure the track is friction-compensated
first. Measure the mass of (several) pellets, and the truck. From conservation of momentum the velocity of the
pellet is calculated.
Loading port
Lever
To load the Air rifle with a pellet, first pull the lever down until you hear it click and
replace the lever, place the pellet into the loading port at the top of the rifle and push
it down, you are now ready to fire the rifle.
The pellet will imbed itself in the blu-tac at the end of the pasco trolley
You will find your pellets here
Safety glasses must be worn
Monkey and Hunter
The Monkey and the Hunter is a physics question so old that it has a heritage listing. As far as we know,
it is purely imaginary. A monkey hangs from a tree. A hunter aims a rifle at him and fires. At the instant
that the gun fires, the monkey lets go of the branch and begins to fall, thinking that he will thus fall
below the trajectory of the bullet. (Monkeys don't study physics.) What happens?
Monkey
and Hunter
Thought experiments abound in the world of physics, often revealing some of the most confusing
aspects of quantum mechanics, relativity, and thermodynamics. But the “Monkey and the Hunter”
thought experiment tackles a relatively well-understood topic: gravity. Imagine a hunter in the jungle
who has just spotted a monkey grasping a tree branch above him. To avoid more troubling moral
questions, let’s assume that the hunter must capture the monkey to treat it for a disease before
releasing it back into the wild. The hunter also knows that the monkey will reflexively drop from the
branch immediately after the hunter pulls the trigger. Immediately after the bullet exits the barrel, the
monkey will begin his free fall toward the ground.
Con
tinu
ed
…
However, the hunter doesn't know exactly how fast the tranquilizer bullet will travel after leaving
the gun — he just knows it'll move fast.
With all of this in mind, where should the hunter aim? Where would you aim? Three choices arise:
1. Aim above the monkey
2. Aim directly at the monkey
3. Aim below the monkey
Where should the hunter aim? Above, below or straight on?
Intuitively, you might think that the hunter needs to aim below the monkey due to how fast the
bullet moves. If he shoots at or above the monkey, won’t the bullet whiz over its head as the
monkey falls toward the ground?
But that's wrong.
The correct answer is 2: Aim directly at the monkey.
Once the bullet exits the gun, there’s only one force acting on it with any significance: gravity (and
some negligible air resistance in this case). Likewise, only gravity will act upon the monkey after he
loosens his grasp. The constant acceleration due to gravity affects both the monkey and the
tranquilizer in the same way. Consequently, the bullet will fall a little bit below the initial aim
trajectory .By the time the bullet travels the horizontal distance to reach the branch, the monkey
will have fallen the same amount as the bullet in the vertical direction, leading to impact.
The speed of the bullet doesn’t even matter in this case. A faster bullet will hit the monkey at a
higher height while a slower bullet will simply meet the monkey closer to the ground. Because
gravity imparts the same acceleration on both objects, their vertical positions will be inextricably
linked if the hunter aims directly at the monkey.
Atwood Machine
An Atwood machine uses a cable drawn over a pulley to connect two or more masses. One of the
masses acts as a counterbalance or counterweight to reduce acceleration because of gravity. Elevators
in multi-level buildings are examples of Atwood machines. The counterweight in an elevator is typically
the mass of the elevator plus about half of the mass of the allowable load.
Equipment list:
Pascoe trolley and cart
String
100g masses and
hanger
Mass Balance to at
least 2dp
Set square
Blu tac
Long stand
Pulley
Syllabus Lines
Free body diagram
Vectors
Components of Force
Trigonometry
Atwood machine continued…
Introduction to components: for use in gravity as lesson
•
•
•
•
•
•
•
•
•
•
•
Measure the mass of the Pasco car =500g
Measure mass of the hanging masses =300g
Students realise that the car is more massive
Convert masses to newton’s
Pasco ramp is set up at Ɵ˂ 360
Place car on ramp and hang masses over pulley wheel, but DO NOT LET
GO.
Students predict that the car will fall, and that the hanging mass will rise,
but obviously they would be wrong!
Students are then ask to explain why( they think that the car will fall
down the slope, they only compare weights)
LET GO!! and see the car rise
Draw a free body diagram
Calculate the component of the car weight down the slope mg Ɵ˂ 3N.
Alien mass
This is a version of the wig-wag demonstration to illustrate measuring mass in space.
An oscillator (hacksaw blade) is calibrated by adding 10g masses in turn, to the 10g wing nuts and bolt
and measuring the time period T (e.g. using 10 oscillations). Plot graph of mass added vs. time period.
This is the calibration curve. Then add the model alien to the oscillator, and measure T. The mass of the
alien can be found from the calibration graph. A similar method is used in space to monitor an
astronaut’s mass, using an oscillating seat.
Factors Affecting Solid
Friction
Friction is the force resisting the relative motion of solid surfaces and material elements sliding against each
other. There are several types of friction, this experiment will relate to Dry friction. Dry friction is subdivided into
static friction between non-moving surfaces, and kinetic friction between moving surfaces.
Equipment:
5 different sized blocks
Compression N meters
Balance
100g masses
Loaded wooden boards are pushed along the bench using a compression N meter, (see picture below) and the
dynamic friction measured. Different loads can be added (1 – 10 N), different surfaces can be investigated (wood,
rubber, sandpaper etc.) and different areas of board used while keeping the load constant for a fair test.
This practical could be used to relate friction of vehicle brakes and tyres, grip of shoes, clamps, etc.
Rotational Dynamics
This is a very simple trick that works a lot better than you think it would. It relies on you being very
quick to catch a tin as it rolls UP a slope, before it settles and the kids can then see how it obviously
works.
•
•
•
•
•
•
•
•
•
Tape a heavy mass to the inside wall of a large cylindrical tin, like a sweet or biscuit tin.
You need to remember exactly where you taped it, so perhaps tape it on the seam of the tin wall,
this way you can tell where it is from the outside when it is sealed.
Set up a ramp with a surface that has a high level of friction (we use a rubber webbing mat)
Ask the kids what we know about gravity, they will say basic things like it makes things fall.
Hold the tin on the ramp such that the heavy mass inside is beyond the tin’s centre towards the top
of the ramp.
Let the tin go and it will roll up the ramp, BUT BE QUICK TO CATCH IT as if you are stopping it
continuing to roll.
Ask the kids to shout out why that happened
o options here for a QI style powerpoint where you have the typical answers on slides you
can access by pressing the slide number then ENTER
o They will say,
 String
 Magnet
 Hamster
 Motor
 Attraction
Open the tin to reveal the weight inside
This leads you on to centre of mass lesson.
Aquapods
AquaPod Bottle Rocket Launcher
The AquaPod is a futuristic, one piece design plastic bottle launcher made of highly durable ABS plastic.
The AquaPod requires no assembly and launches ordinary plastic 2-liter soda bottles up in the air. The
AquaPod can launch a bottle up to 30 metres in to the air, and is the only launcher that has a built in
safety-valve that prevent over-pressurization which releases pressure at 345kPa to help keep everyone
safe from over-pressurising the entire system.
To launch
Partly fill a 2 litre plastic bottle with water and secure it upside down over the white launch tube. Using
any bicycle pump, pressurise the AquaPod through the valve stem until the check valve inside the front leg
releases pressure and water. Stand back and give the 4.5 metre release-string a short, quick tug to launch
the bottle high into the air.
Experiments:
To run meaningful experiments you can:
• Use different launch pressures - requires a pressure gauge (not supplied) to monitor pressure
• Vary the amount of water used in the bottle
Specifications:
• Dimensions: 330mm x 195mm x 341mm
• Minimum age recommendation: 14+ Years
• Bottle launch height: up to 30 metres
• Bottle Size: 2 litre (not supplied)
The AquaPod Bottle Rocket Launcher was used as a 2015 PH5 Case Study for the WJEC syllabus (refer to
Case Study sheets).
Aqua pods continued…
The AquaPod is a good demonstration to explore Newton’s third
law and observe firsthand how every action has an equal and
opposite reaction: the bottle pushes some its water downward
and the water responds by pushing upward on the bottle,
propelling the bottle upward. In that respect, the water-bottle
rocket is like any other rocket. All a rocket needs is fuel and
energy. Pushing the fuel backward is what propels the rocket
forward-action and reaction. Energy is what allows the rocket to
push that fuel backward. In many rockets, the fuel and the energy
source are the same thing. Chemical reactions in the fuel release
energy and this energy allows the rocket to push the fuel backward.
However, the water-bottle rocket uses two separate materials as fuel and energy source. The fuel is water
and the energy source is compressed air. Having water as the fuel makes sense because water is dense and
provides lots of inertia for the rocket to push against as it throws water backward out its tail. Having the
compressed air as fuel is a good idea because it has little weight for the amount of energy it stores and
doesn't load down the rocket.
At launch, most of the water-bottle rocket's mass is water. And with air packed tightly inside, the rocket
has lots of energy. When you finally let water start streaming out of the bottle, the compressed air pushes
downward hard on the water and the water pushes upward hard on the compressed air. The air conveys
this upward force to the entire bottle and up it goes.
There is a copy of the 2015 WJEC A2 Case Study available on the desk for you to read, which
uses data obtained from the launch of an Aquapod.
Buggy and Cones (Distance –
Time graph)
A course has been set up a in an open space marked by plastic cones, measured distances apart,
One pupil drives an electric buggy round the course, while other pupils with stopwatches at each cone time the
progress from the start. Plot a graph of distance vs. total time elapsed to determine speed at various points,
from the gradient of the graph. v = Δd/Δt.
Discuss reasons for the speed to vary, errors in measurement, etc.
Leading on to average vs. instantaneous speeds, speed cameras etc.
Each cone is 1 metre apart
Circular motion
The aim of the experiment is to verify the relation between the centripetal force acting on a body
(moving in a circular path at constant speed) and the angular speed of the body.
The apparatus is shown below. Stop-watches and boxes of know masses, M, are also available. The
rubber bung will be made to follow a horizontal, circular path at approximately constant speed.
From the diagram, it is clear that the actual radius of the path followed by the bung, r, is not equal to .
You will notice during the experiment that the radius of the path is not constant. As part of your
"conclusion/evaluation" you should show that (perhaps surprisingly) this is not a problem. Hint: the force
causing the acceleration is the horizontal component of the tension in the string…
Discussion:
A force which pulls an object towards the center of a circle is called a centripetal force. How much
centripetal force needs to be exerted to cause an object to move in a circle? Your experience should tell
you that the amount of centripetal force that you need to exert depends on
a. the mass of the object you are whirling - heavier objects require more force,
b. how fast you are whirling it - going faster requires more force, and
c. the radius of the circle.
The textbook says:
mv 2
FC =
= mrω 2
r
Circular Motion Continued…
however can we verify the quantitative relationship between centripetal force and mass, speed, and radius?
This question can't be answered all at once, since a scientific experiment is designed to vary one quantity
(holding all others constant) and measure its effect on one other quantity. The most difficult quantity (of
mass, speed, and radius) to hold constant from trial to trial in an experiment is the speed of the object, so it
is easiest to study the effect of speed on centripetal force, since it is relatively easy to hold the mass of the
object and the radius of the circle constant.
You will use an apparatus similar to the one pictured above to measure the effect of speed on centripetal
force. You can hold the mass constant during a set of trials by always whirling the same object. You can keep
the radius of the circle constant (with a little practice) by keeping the upper clip a fixed distance below the
glass tube while whirling the object.
Procedure:
1. Place a small number of weights or washers (be sure that all of the washers you use are the same
size.) on the bottom clip of the apparatus. This part of the apparatus hangs straight down, and the
weight of the washers supplies the centripetal force.
2. Practice whirling the stopper (or ball) until you can keep the top clip a short distance below the
bottom of the glass tube while the stopper whirls. IMPORTANT! If the clip touches the bottom of
the glass tube, the weights are no longer supplying the centripetal force! If the clip rises or falls
appreciably as the stopper whirls, the radius of the circle is changing. Practice!
3. Use a stopwatch to measure the time taken for a reasonable number of revolutions (20 - 30 perhaps).
Record your data.
4. Change the number of washers on the bottom clip (centripetal force) and repeat steps 3 and 4. Repeat
for several different weights. Record the data.
5. Change the position of the top clip to change the radius of the circle. Repeat the experiment for this
radius. Be sure to indicate where the radius changes in your data table.
6. If you have time, you might try to determine the relationship between mass and centripetal force. In
order to do this, you need to keep both the radius of the circle and the speed constant while you vary
the mass and the centripetal force. You can design your own data table for this. You could also
investigate the relationship between the radius and the centripetal force.
Circular Motion Continued…
Results:
1. Calculate the period of revolution, T (the time to go around once) for each trial. Show a sample
calculation.
2. Calculate the linear speed, v, of the stopper for each trial. Include a sample calculation. (Note:
3. Theoretically, the centripetal force should be directly proportional to the square of the speed. To
check this, add a column to your data table for v2. Construct a graph of centripetal force versus
v2. Remember that it is customary to put the quantity you change (force, in this case) on the
horizontal axis, and the quantity that changes by itself (speed) on the vertical axis. Be sure that you
pick the largest convenient scale for your graph and draw the best smooth curve through your data
points.
Flic-Flac Projectile
Launcher
This device launches two ball-bearings simultaneously: one horizontally, one vertically. Note they both take the
same time to fall to the floor, showing the independence of vertical and horizontal motion. Calculations using s =
½ a t2 where a = g can yield the time of fall, then the horizontal velocity can be found knowing the horizontal
distance travelled at constant velocity. Further calculation may determine the speed needed to orbit the earth,
or the g field at the radius of the Moon (which Newton used to show gravity followed an inverse square law).
Square the bar until you hear it click, Put the ball bearings onto each of the holes on the bar, press the
black round button to release the ball bearings.
Please ensure that there are no persons in front of you before you release the ball bearings
Use the arrangement to show how the launch height h1, and the height of the vertical drop h2 affect
the horizontal distance travelled.
Ski Jump
Use the apparatus to investigate whether force vectors combine like displacement vectors.
Forces in Equilibrium
Flying Pig
You have heard the phrase "when pigs fly"? Well, some toy maker decided to capitalize on the flying pig
idea, and actually built a flying pig. The flying pig is a toy pig with wings and a propeller. You hook it to
the ceiling or something, and it flies in a circle. So, let's solve a flying pig problem.
Equipment list:
Flying pig toy
Stopclock
Metre rule
String
2 x retort stands with bar across the middle
Draw and label the forces.
We actually only have 2 forces
here. Gravity downwards, and
tension along the rope.
Choose a coordinate system
The same system as we have been using, "z" is up,
"-r" is towards the centre of the circle, and "θ" is
in the direction of motion, as shown
This looks a bit less complicated
Apply Newton's second law.
The tension needs to be broken into its components. So I have drawn the
components of the tension on the diagram.
Flying pig continued …
Since Tz is the side adjacent to φ,
and it is in the positive "z" direction. Since Tr is the side opposite φ,
but it is in the negative "r" direction. In component form we write this
We also know that the force of gravity is in the negative "z" direction. There is no acceleration in
the "z" direction. There is the radial acceleration in the negative "r" direction. There are no forces
or accelerations in the "θ" direction. So when we apply Newton's second law we get
Substitute known forces and accelerations. So we know that
However, we don't know either vT or r. You may remember we can calculate vT with the following
equation
where t is the period (the amount of time it takes the pig to fly around once). So, we make that
substitution and get
We still need an equation for r. Why is r not the length of the string? Remember, r is the distance
from the object (in this case the pig) to the centre of the circle. If the pig was on a table, then r
would be the same as the string length. However, since the pig is flying, the distance from the pig
to the centre of the circle is something less than the length of the string. So what is r? Consider
this picture.
From this picture, you can see that
since it is the side opposite φ. Making this substitution we get
So, substituting these into the Newton's second law equations we get:
Okay, so this doesn't look simple any longer.
Flying pig continued …
Solve the algebra. At the end of the last step we had 2 equations. We know , l and t. We don't know
m, T or φ. We appear not to have enough equations. But we will trudge ahead anyway. We want to find
φ, so we start by solving for T. Notice that both the negative and the term sinφ will cancel from the
second equation. So, that equation is already solved for T.
We can substitute that into the first equation to get:
Add
to both sides.
Now, multiply both sides by t2, divide both sides by 4mπ2l, and cancel the m.
Finally, take the arccosine of each side. (This is also called the inverse cosine or cos-1, see the Math
Appendix for more on that.)
And, lucky us, this doesn't depend on the mass, so we are okay.
Flying pig continued …
Plug in numbers.
We know the period, t = 2.25 s,
equation gives us
= 9.8 m/s and the length is 1.30 m. So plugging these into our
(I know I said in the very beginning of this quarter that we would be using radians this quarter, but since
this angle is not θ and we are not finding arc length, I thought I'd use degrees since you know
approximately how large of an angle 15° is, but I don't expect you to know how large of an angle 0.26
rad is.)
Projectile Launchers
Launching a projectile horizontally at varying initial velocities affects its time-of-flight. And determine if
and how the horizontal and vertical components of motion are related. Horizontal and vertical motion
are independent of one another. The acceleration due to gravity is a constant regardless of what
horizontal velocity an object has. Thus, the time for an object to freefall from a given height is the
same no matter what speed it is moving at in the horizontal direction
Students use the projectile launcher to
launch a ball horizontally three times,
increasing initial velocity on each launch.
They will clearly discover that time-offlight does not vary.
Pull back the lever at the back of the gun
until you hear it click, place the plastic ball
in the hole at the front of the gun, Aim the
launcher gun to the angle of your choice,
using the protractor on the side, and
shoot.
The Pasco Projectile launcher has also
been made available for you to compare if
you wish.