From Serendip

## SCIENCE AS EXPLORATION INSTITUTE 2001

Circular Motion

### An Inquiry-Based Teaching Module developed by Matthew Rice, Department of Physics, Bryn Mawr College

I have grouped the experiments into those that I think explore similar physics, but often these distinctions are hard to make and there is much overlap. Also I have ordered the groups, and again, this ordering is quite hard to do. It is hard to know what concept to explore first.

So, I have listed many experiments from which you can pick and choose and decide which order you should tackle them. Also, as there are many experiments presented, I will run this session in a way that is probably not as "Inquiry-Based" as they should be offered to your students. Options one could use here are:

1) Present the student with the equipment and minimal guidance and let students explore.

2) Use the equipment and demonstrate the intended effect, and then discuss the physics. One could then perhaps let the students reaffirm or go beyond on their own.

3) Discuss the physics and then look at an example. If this is the option that you think will work best for a particular experiment you should try to couple it with another experiment that students can then explore on their own, guided by the previous discussion.

I have left much unsaid in what follows and hopefully we will fill in these gaps as we work through this module. These will often be things that will depend on your personal preference. Such as "How will I asses my student's achievements?" or "what everyday phenomenon does this relate too?" I will be happier if we try to address these questions for fewer experiments rather than rush through each experiment just to get them all done in our three hour block.

List of experiments.

A) Centripetal and Centrifugal Force

This group of demonstrations is designed to get straight to the heart of a serious misconception that many people have: that of the fictitious centrifugal force.

Circular motion is not a natural state of motion (Newton I: "A body will tend to move in a straight line at constant speed unless acted upon by an external force.")

To keep a body in circular motion a centripetal (center seeking) force must act on it. Often this force is interpreted as fighting against a fictitious centrifugal (or center fleeing) force, rather than just the force required to enable circular motion. One source of this conceptual error may be that the upward force from the ground on our feet is needed to counter the real force of gravity pulling us down. So people think that the reason that say, a car door pushes them to make them take a sharp turn, is that it is countering a (fictitious) force pulling them out.

Experiments:

Paper plate and marble:

Roll the marble around the inner rim of the paper plate. The banked rim provides the centripetal force.

Cut a quarter out of the plate and try to predict the path of the ball when it leaves the plate. Note that there is no force "pulling" it away from the center.

Tie a dense object (e.g. a washer) to a string and whirl it in a horizontal circle.

Here the string clearly provides the centripetal force.

Let go of, or cut the string and predict the path.

Place pennies on a turntable.

The pennies further from the center do not perform circular motion ("fly off")

The required centripetal force (provided by friction), needs to be greater the

a) faster the spin (proportional to spin rate squared), and

b) the further from the center (proportional to the radius of the circle).

Spin a bucket of water and see the parabolic shape formed by the surface.

The further out from the center the greater the centripetal force needed to keep the water in a circular path. This is provided by the horizontal component of the buoyant force due to the fluid, the other component counters gravity.

BTW, a sailor floating on this surface would say that the surface (near him) was horizontal, but as he sailed away from the center, gravity would "seem" to get stronger. (See the advanced topic, artificial gravity.)

Examples:

Spin dryer, shaking the water off something, or a centrifuge.

Turning a corner in a car.

Compare to braking and accelerating. The layman description of, say braking in a car may be "thrown forward," but one can easily be convinced to simply say "continuing forward." Compare this to turning a corner. One should not say thrown out, but just "continuing on."

Artificial gravity

I feel that this should be discussed only to advanced students or students who have been grappling with the "non-existence" of centrifugal forces for a while. It tends to cloud the issue. But these days, if we were to call it virtual gravity, I think students would be able to keep it in their heads that it is not a real force.

Things to consider:

On a revolving space station what is the direction of the (centripetal) force that must act on a body? What is the perceived direction of gravity?

What really "is" our weight here on Earth. It is the normal reaction of our feet on the ground, or the upward pull of the tendons in our shoulders! This is what we really perceive as our weight. It is, of course, due to the attractive force we feel towards the center of the Earth, but if we were to simply fall, and give ourselves over to this force, we would actually say we were weightless.

Look at the magnitude and the variation of the virtual gravity with radius.

A demonstration here could be a bucket of water whirled in a vertical circle fast enough so that the water does not spill out.

Related topic:

Changing weight at different latitudes here on earth.

B) Stability and the Center of Mass

Here we look at the need to have the center of mass of an object between the supports. I suppose that we should speak of the polygon that is at least a triangle formed by the base of an object, but we will only work in 2-D. 3-D could be an advanced topic. The concept of touques can be introduced before or during this discussion, or torques can not be mentioned at all, effectively alluding to zero net torque, with no mention of this.

Experiments:

A shoe box with a lump of clay, or weight on an edge, or in a corner.

If the clay is placed strategically the box can be placed on a ramp on its outwardly similar faces for drastically different stability results.

If the students are not shown the clay, or weight, nor allowed to touch the box, they will be able to predict the location of the clay by observing enough of the behavior of the box.

Touching toes whilst backed against a wall.

One can not do this as, to stay upright (whilst stationary) you must have your center of mass between your heels and toes.

Related to stability, but more to torques, listed below, is the extra force that your toes must take as you lean forward.

Thinking outside the box: Do a squat.

Standing on balls of feet whilst fronted onto a wall.

Here your center of mass must move over the balls of your feet, but it cannot.

Balancing a spoon, fork and match with a "strategically" placed center of mass.

One can find commercially produced toys or knick-knacks that also balance by having the center of mass below the pivot point to achieve seeming magical stability.

Balance a ruler on two fingers and then slide them together.

They will always end up at the center, as the center must be kept between the two fingers. The finger that is closer to the center of gravity will have to support more weight and will feel more friction and move more slowly. Eventually the other finger will be closer to the center of gravity and the situation will be reversed. Thus the fingers will alternate moving until they meet at the center of gravity.

Now tape a weight somewhere on the top of the ruler. It is now possible to guess approximately where the center of gravity is, but not exactly. Its position can be found, however, by the above method.

Examples:

A high-wire walker with a long pole that flexes, thus putting the center of mass in a stabilizing position below him.

The tower of Pisa. (One might also talk of structural integrity.)

Discus normal reaction, line of action and net torque about the pivot point.

Learn to calculate/predict the location of the center of mass and to test the prediction.

The touching your toes experiment is a good chance to "feel" the "movement" of the normal reaction, and its changing line of action as you bend over and move your center of gravity.

Brick stacking

To stack two bricks for that they extend as far as possible, but don't topple one clearly stacks them with one overhanging 1/2 way. How would you stack 3? 4? How many do you need to get the top brick "clear" of the bottom brick? What is the limit to the overhang you can achieve before toppling is inevitable?

C) Torque

Moving on from stability, we now look at situations where the net torque is zero, leading to no circular motion (strictly no angular acceleration), and those where the net torque is non-zero leading to circular motion.

Even at the most elementary level one can get quantitative and explore the three factors that are central to generating a torque. These are the force applied (studied by varying the weight of objects used), the distance from the pivot point that the force is applied (sometimes called the lever arm) and to a lesser extent, the angle between the force and the lever arm. This last aspect will not be looked at to any great extent in this list of experiments, but if one wants to explore it, a good place to start is the simplest action of opening a door. Clearly a larger force used will increase the chances of opening a door. It is also easy to see that if the force acts far from the pivot (hinges) it’s effect will be greater. Doorknobs are far from the hinges! The final piece to the torque puzzle can be demonstrated by pulling on the doorknob along the line of the door. The door will not move, no matter how hard you pull.

Experiments:

Plank with a pivot and six equal masses.

This is essentially a seesaw, and one may actually be able to use the one in your playground to do this experiment (it needs to have a smooth, close to friction free pivot and one may have to add a little mass to one side or other to equilibrate the system.)

Balance one mass on each side. Note that they are the same distance from the center.

Now place two masses on one side and only one on the other. Note the relative distances.

Place three masses on one side and one at the same distance on the other side. Now place a second mass (somewhere) on the second side. Predict where the final mass should be placed to achieve balance.

Balance two masses on the plank with one end hanging over the edge of the desk. Now tie a string to the mass on the overhang and hang it below the rod from the same point as it was sitting. The rod is still balanced no matter how far away the mass is, as long as the "lever arm" is the same length.

A shoe box with a lump of clay, or weight on an edge, or in a corner.

Same as the box above but now let the student find the location of the mass by holding the box. We instinctively know where the mass is. Our brain is doing the magnitude of the force/lever arm calculation for us! Help the student break this action down and understand what he/she instinctively knows (that if one holds closer to the center of mass one needs to provide a greater force.)

Soda can sprinkler/Hero engine/Feynman problem

Make two small holes on a diameter at the base of a soda can. Bend the metal so that the holes are not radial but close to tangential. Hang the can by the pull-tab so that it hangs vertically. Fill the can with water and see the "Hero engine" begin to rotate due to the net torque.

A question that has caused much discussion (among theorists) is the anti-Hero engine, when the empty can is weighted and sunk into a bucket of water so that it fills. Which was does it rotate?

Examples:

The weight scale at the gym or doctor’s office with a sliding weight (a beam balance)

Your weight acts at a point close to the pivot of the scale. One then moves a much smaller weight away from the pivot point and the distance you move it is proportional to your weight.

Seesaw, or teeter-totter

One can balance a larger person if you sit farther from the pivot.

Counter weight on a construction crane

The weight is moved further out to balance a larger load.

Carrying two shopping bags is easier then one

One can avoid awkward leaning if you carry two similarly weighted bags.

Pan balance

The pans are equal distances from the pivot, and hence if the pans balance then the weights in each are equal. As one is known, the other one is, also.

Note that there is no need to place the masses at the same location in the pans. Why is this?

D) Moment of inertia

Just as a heavy, or inert, body will resist changes in its motion, so too a rotationally inert body will resist changes in its rotational state. Rotational inertia (or moment of inertia) is more interesting than regular inertia (or mass) as there is a distributional component. If the mass is further from the axis of rotation then the moment of inertia will be greater.

Experiments:

Dumb bells with movable weights.

Use a meter stick and some clay. Place two lumps of clay equidistant from, but close to, the center. Grasping the meter stick at the center, note how easily it can be spun. Now move the weights out toward the ends. It is now much harder to spin. The mass is the same, but the distribution is different, it is harder to get the stick spinning (and harder to slow it down). Its moment of inertia is greater.

Balancing a meter stick using a lump of clay as an aid.

One can try this using a lollypop or a hammer. If you place the clay on the ruler it is easier to balance. If you place it at one end and try to balance on the other end it is the easiest of all! If the moment of inertia is higher (mass further from the pivot point) then the stick tilts more slowly, giving you time to adjust.

Similar effects:

Balancing with a long pole.

Balancing on a tight rope is easier than balancing on a slack rope. On a slack rope your pivot point is closer to your waist (you lean one way, and your feet go the other) and so your moment of inertia is lower than if you rotate about your feet, as on a tight rope. (The change in your moment of inertia is about a factor of four.)

Some astute observers will note that in the vertical balancing examples, if the mass is further from the pivot point the moment of inertia may be higher, but so is the torque provided by gravity, that is trying to topple the object. These effects do oppose each other, but the moment of inertia effect wins. This is because the moment of inertia is proportional to the square of the distance, the torque varies only linearly with distance.

Rolling objects with the same shape and weight, but different mass distributions down a slope.

Affix five (or more) masses inside a biscuit tin, around the edges. Now place the same number in the center of a similar can. Roll them down a slope. The can with the higher moment of inertia loses the race. One could also bring in the concepts of potential, kinetic and rotational kinetic energy if one were so inclined (no pun intended.)

Two cans of soup, one solid, one not, can also be an interesting demonstration.

Examples:

You reduce the moment of inertia of your legs (about your hip) by bending them at the knee. They can then be drawn forward with less effort, ready for the next stride. (Also one keeps ones arms bent. BTW, why more your arms at all?)

F) Conservation of Angular Momentum and Changing Angular Momentum by Applying Torques

The total angular momentum of a closed system does not change. A closed rotational system is one that experiences no net external torque. Angular momentum is the product of both angular velocity (spin rate) and moment of inertia. Many of the effects of conservation of angular momentum involve changing the moment of inertia, and hence changing the angular velocity.

Another extremely interesting aspect of angular momentum is its vector nature. If angular momentum is conserved then its direction stays constant, leading to gyroscopic stabilization. A very advanced topic is the precession of a gyroscope.

Experiments:

Make a hole in the cap of a 2 liter drink bottle. Fill the bottle with water, tip it upside down with the hole covered and shake it with a circular motion. Uncover the hole and let it drain. A tornado should form and persist. As the water drains it gives up its angular momentum to its surroundings, mostly to the rest of the water. (If the water is not allowed to drain, the angular momentum of the water will soon dissipate.)

The classic example: The spinning ice-skater

By pulling her arms (and legs) in, a skater can reduce her moment of inertia. If she is spinning, she will thus spin faster, as her angular velocity increases to compensate.

One can do this without the (frictionless) ice if one uses a spinning baseboard or a swivel chair. Grasp two exercise weights and stand, squat or sit on your swiveling device. Hold the weights out and have someone spin you. Now pull the weights in. Your spin rate will increase. Letting the weights out will slow your rate. You should be able to do this a few times before friction slows you to a stop.

What role do the weights play?

It takes energy to pull the weights in. Energy considerations can form the topic of advanced discussions.

More fun with the spinning baseboard

Stand on the disk and throw a ball from your chest. Now throw it forward, whilst holding it to the side. Discuss the observed differences.

"Zero angular momentum" turn.

This is the type of turn that cats can do and astronauts practice. Again on the spinning baseboard, hold your arms out and move them to the right. Pull them in and move then left. Repeat. What is the net effect on your body? Why? (One needs a baseboard that is quite friction free otherwise friction can give spurious results.)

A bicycle wheel on a short axle acts as a good gyroscope. (One may wish to protect the spokes to save potential jammed fingers.)

Hold the spinning gyroscope and try to move it around. Hang it from the end of its (horizontal) axle and observe the classic gyroscopic precession. Try to explain these effects by considering the torques acting (this can be very sophisticated!)

Now hold the gyroscope whilst on the baseboard. One can get a good feel for conservation of angular momentum by moving the gyroscope around or slowing it down.

Spin a heavy object on a string and let the string wrap around a finger.

As the string gets shorter the moment of inertia decreases and the spin rate increases.

A neat trick is to tie your keys to a piece of string with most at one end and one at the other. Hang the keys over a pencil. If both ends start vertically the keys fall to the ground as expected. But if the lone key is held out at an angle gravity acts as a torque, imparting angular momentum and, as the set of keys fall, shortening the lone key's string, conservation of angular momentum means that the key's rotation rate increases. This wraps the key around pencil and friction (which BTW is an exponential function of the amount of wrapping) soon stops the fall.

It is easier to balance a spinning basketball than it is a stationary one.

This is a consequence of the conservation of angular momentum and its vector nature.

Discuss the term "gyroscopically stabilized".

Applications:

Here are a few more everyday examples where rotational motion plays a defining role.

A Bicycle

Steering, pedaling, the construction of the wheels, gearing, balancing (easier whilst in motion), the line of the front forks relative to the contact point of the front wheel, the bend in the front forks.

Amusement park rides

Playground equipment

Eggs

Spin a hard-boiled and a fresh egg. Explain any differences.

If they are spun and then stopped then the fresh egg should "start up" again.

Toys

Yo-yo, light saber

Tools

Screwdriver (two uses), hammer

Kitchen utensils

Coffee cup

Jar opener