PPT Activityrotational Motion

PPT ActivityRotational Motion

Developer Notes

I've called all of the rotational analogs for linear motion "rotational x". They are sometimes called "angular x". I find that confusing, and I think that rotation is a clearer description of what we're talking about here, so I've gone against convention.

Goals

Students should understand rotational speed and how that translates to linear speed.
Students should understand that mass and radius affect rotational inertia.
Students should understand torque.
Students should know that rotational motion has analogs for linear motion.
Students should understand conservation of rotational momentum.

Concepts & Skills Introduced

Area / Concept
physics / rotational velocity
physics / rotational inertia
physics / torque
physics / rotational momentum
physics / conservation of rotational momentum

Time Required

min

Warm-up Question

Presentation

Time

Time is the same for both rotational and linear motion.

Distance/speed

This should start with introducing the students to the difference between rotational speed and linear speed. Linear speed is length/time. Rotational speed is revolutions/time. There are a couple of good activities. One is just to have them roll a paper cup. It will roll in a circle. Why? Because one end is bigger. But what does that mean? It means that both ends of the cup revolve in the same amount of time, but that the bigger end goes farther because it is further around.

Another activity: Get a turntable or lazy susan, put two objects on it, one at the outside edge, and one about halfway in. Get the table turning and ask which is going faster. Both revolve at the same speed, but the outside one covers more distance in the same amount of time.

Another interesting activity (Hewitt) is to tape the narrow ends of two cups together and then try to roll them down a pair of rails (meter sticks). The cups will not make it down, but will roll off to the side. Then take the cups apart, tape the wide ends together, and roll them down the rails again. They should self correct and roll down the rails with no problem. This again is due to the difference between rotational speed and linear speed. When the narrow ends are taped together, any error will be amplified. When the wide ends are taped together, any error will be damped - if the cups veer to the right, the diameter on the right will increase, so that side will go faster (linear) and catch up with the left side.

Rotational velocity is expressed in radians/second (rad/s), but it is probably better expressed in revolutions/minute (rpm), or revolutions/second (rps) to lessen the confusion. The symbol for rotational velocity is  (omega).

Do some computation exercises with comparing rotational speed and linear speed for different size wheels.

Acceleration

Like linear acceleration (∆v/t), rotational acceleration is the change in rotational velocity per second (/t). So, the units for rotational acceleration are rev/s2 (rad/s2). The symbol used is  (alpha). And, just like linear acceleration, rotational acceleration requires a force! For example, if you want to accelerate (speed up) a tetherball, you apply a force (hit it) to make it go in a circle faster. (Be a little careful here, because as the rope gets shorter, the ball goes faster due to conservation of rotational momentum, too.)

Remember F = ma. In linear motion, acceleration is directly proportional to force, and inversely proportional to mass. But are force and mass in the rotational world the same as in the linear world? Yes and no.

Mass/rotational inertia

Let's look at mass first. What is the first thing we learned about mass? It has inertia. And that means that it is hard to speed up, slow down, or turn (accelerate). The more mass, the harder it is to change. So what make a rotating object harder to change? More mass will make it harder to change. And, at the same rotational velocity, the farther the mass is from the axis, the harder it will be to change. Think about a flywheel, a potter's wheel, or a playground merry-go-round. If the mass is all in the middle, it will be easier to get going. On the merry-go-round, if all the people are on the inside, it is easier to get going - if they're on the outside, it is harder to get going.

Oscillating dumbbells
Balancing meter stick
Falling meter sticks
Cylinders on ramp
Tops (why don't they fall over when they're spinning?)
Gyroscope? (better as a demo)
Tightrope?
Bike wheel doesn’t tip over when spinning.

POE with two differently weighted identical looking cylinders rolling down a hill.

Force/torque

If mass and acceleration exist in the rotational world, what about force? Yup, there is force, too. And just like F = ma in linear motion, rotational force = rotational inertia  rotational acceleration. Rotational force is called torque, and the symbol is  (tau).  = I. Look at the units, kgm2/s2. It is Nm. Forcedistance. Linear force times the distance from the axis of rotation.

Compare to lug nuts on car wheels. Opening a door - how far from the hinge do you push? Why are steering wheels on buses big?

This can be demonstrated pretty well with stations.

Torque feeler
Wrench/ jack screw w/ scale?
Screw into wood, screw driver vs. wrench
? Spool with string - which way will it roll? Depending on angle.

? POE with off-weighted cylinder rolling uphill.

Rotational momentum

This continues the parallel with linear motion. If something is spinning, it wants to keep spinning. The more rotational inertia it has, the more it wants to keep spinning. Rotational momentum is like mv. It is I. In units it is kgm2/s. t = ∆(I).

Conservation of rotational momentum

Like linear momentum, rotational momentum is conserved. Look at figure skaters. It's the velocity that is a vector and is conserved.

Assessment

Writing Prompts

Relevance

Summary

Exercises

Challenge/ extension

dyn act rot 030221 dk01.doc1Printed: 10/27/20189:24AM

PPT ActivityRotational Motion

Background / History

Almost all of the motion we have looked at so far in this course is linear - in a straight line. There are similar concepts for things that rotate, or spin. Just like there is linear velocity, there is rotational velocity. And the same goes for acceleration, mass, force, momentum, and conservation of momentum. A table is included at the end of this section which details the similarities. You don't need to know all of that information, but it is a good summary.

Rotational velocity is how fast something spins, how many revolutions it makes in a period of time. Depending on how far an object is from the axis (center of spin), it will go a different speed. Objects closer to the middle go slower, and objects farther from the middle go faster. The ratio is theirdistance from the middle times 2π, or 2πr. If an object spins once per second, it's rotational speed is 1 revolution per second. If a part of the object is 1 m from the axis, it's linear speed is 2π m/s. Rotational velocity, (omega), is like linear velocity, .=revolutions/sec.

Rotational acceleration is the change in rotational velocity per second, or ∆/s. Linear acceleration is ∆/s. Rotational acceleration, (alpha), is like linear acceleration, a. = ∆/s

Rotational inertia is like mass. Mass has inertia, meaning it doesn't want to change speed or direction. For spinning things, more mass will make them harder to change, but how far that mass is from the axis affects it, too. Remember that rotational velocity is how fast something spins around. It is harder to get a long object spinning than it is a short one. Rotational inertia (I) is the sum of the mass times distance from the axis, I = ∑mr2. This gets hard to calculate for many objects. For something simple, like a pendulum or a ring, ∑mr2 is just mr2.

Rotational force is like linear force. Remember Fnet = ma. If an object accelerates, there must be a force. How far away the force is from the axis makes a difference. If you push closer to the axis, you need to have more force in order to have the same effect - farther from the axis makes it easier. For rotating objects, force is called torque,  (tau). = I. also =Fr.Torque is measured in Nm.

LinearF = maForce = mass  acceleration

Rotational = ITorque = rotational inertia  rotational acceleration

And on to rotational momentum. Like linear momentum, the more rotational inertia something has, and the faster it is going, the more rotational momentum it has.

Linearmvmomentum = mass  velocity

RotationalIrotational momentum = rotational inertia  rotational velocity

And rotational momentum is conserved, just like linear momentum is. Think of an ice skater spinning with her arms out. When she brings her arms in, she goes faster. Her rotational inertia (I) has been reduced because the radius is reduced, so her rotational velocity () must increase to keep the total rotational momentum the same.

Out of all of this, you need to remember that the parts of linear motion have analogs in rotational motion. You especially should know the linear and rotational speed; torque; rotational inertia; and conservation of rotational momentum.

Problem

Investigate rotational inertia.

Materials

Procedure

Rotate through the stations. Make notes at each station about what you observe.

Oscillating dumbbells - Which is harder to start and stop rotating?
Balancing meter sticks - Which is easier to balance on your finger?
Falling meter sticks - Which one falls faster?
Cylinders rolling down a ramp - Which one rolls down the ramp faster?
Tightrope and pole? Outside?

Summary

Exercises

Two children are on a merry-go-round. One is twice as far from the middle as the other one. How do their rotational speeds compare? How about their linear speeds?
If you roll a paper cup, it will go in a circle. Explain why in terms of rotational velocity and linear velocity.
If a wheel with a radius of 1 m is rolls at 1 revolution per second, how far does it travel each second?
If a wheel with a radius of 0.5 m is rolling down the road and at 3 m/s, what is its rotational speed?
My bicycle has a back tire with a diameter of 0.5 m, and a front wheel with a diameter of 0.4 m. If I ride at 10 m/s, how fast is each one turning?
If the front gear on your bike has 44 teeth, the rear gear has 11 teeth, and the rear wheel has a diameter of 0.6 m, how far will the bicycletravel for each 1 revolution of the pedals?
Tight-rope walkers carry long poles to help them balance. What do the long poles provide to keep them from falling?
Which is harder to spin at 1 rev/s,
a ball on a 0.1 m string, or the same ball on a 5 m string? Why?
What if ball on the 0.1 m string was a bowling ball and the ball on the 5 m string was a tennis ball? Why?
If you ignore air resistance, any two objects will fall with the same acceleration. If you ignore air resistance and friction, ball bearings of different sizes will roll down a ramp with the same acceleration. Why will a can of cranberry sauce roll down a ramp faster when it is full than when it is empty?
Calculate the rotational inertia of a 2 kg ring with a radius of 0.1 m, spinning at 3 rev/s.
Which way does a yo-yo spin on the way up - the same as the way down, or the opposite?
Why is it easier to turn a screw with a wrench instead of a screwdriver?
Try opening a door by pushing on it right near the hinges. Is it easier or harder than pushing on it near the knob? Why?
If you apply more torque to your bicycle by pushing harder on the pedals, what is the result?
The engine oil drain plug on a Mazda Miata should be torqued (tightened) to 36 N-m. If your wrench is 0.18 m long, how much force must you apply to get the correct torque?
When divers flip, they often tuck, but when they un-tuck, the seem to stop spinning. Do they really stop spinning? Explain.
When stunt motorcyclists make big jumps, they rev up or slow down the speed of their back wheel to control the angle of the motorcycle in the air. What principle does this represent?
What principle is it that explains why tops don't fall over when they're spinning?
Write the rotational analogs and the symbols for each of the following linear quantities:
distance
velocity
acceleration
inertia
force
momentum

Vocabulary

Rotational distance: Rotational distance is measured in radians (rad). There are 2π radians in one complete circle. A radian has no units, it is a proportion: radians  radius = arc length.
Rotational velocity (): Radians/sec. How fast something is rotating.
Rotational acceleration (): ∆/s. Radians/sec2. How fast the rotational velocity is changing.
Rotational inertia (I): The property of an object that measures its resistance to a change in rotation. It tends to keep rotating unless acted on by an outside torque. ∑mr2
Torque (rotational force) (): A force that tends to cause rotation.  = Fr
Rotational momentum: Rotational inertia times rotational velocity. I.
Conservation of rotational momentum: The total rotational momentum before an event is equal to the total rotational momentum after the event.

Linear / Rotational
Quantity / Symbol / In Other Symbols / Units / Quantity / Symbol / In Other Symbols / Units
time / t / - / s / time / - / - / s
distance / d / - / m / radian* / rev* / - / -
mass / m / - / kg / rotational inertia / I / ∑mr2 / kgm2
speed / s / d/t / m/s / rotational speed / - / rev*/s / -
velocity / v / d/t / m/s / rotational velocity /  / rev*/s / -
acceleration / a / ∆v/t / m/s2 / rotational acceleration /  / rev*/s2 / -
force / F / ma / N
kgm/s2 / torque /  / I / kgm2/s2
momentum / P / mv / kgm/s / rotational momentum / - / I / kgm2/s
conservation of momentum / - / - / - / conservation of rotational momentum / - / - / -

Challenge/ extension

* Technically, these quantities are represented in radians. A radian is like a revolution. There are 2π radians in one complete revolution. The handy thing about a radian is that it makes the conversion from revolutions to linear distance easy. If you know the number of radians, just multiply by the radius, and you have the length of the arc.

radians  radius = arc length

Both radius and arc length are measured in meters, so a radian has no units, it is just a proportion.

radians = arc m / radius m

For example, a whole circle has 2π radians. If the circle has a radius of 1 m, then there are 2π m all the way around the circle.

dyn act rot 030221 dk01.doc1Printed: 10/27/20189:24AM