Race the jam jars down a slope to see which one is faster.

## In a Nutshell

Show people that rotational motion is a bit weird - you have to account for the distribution of mass not just the overall properties.

## Setup

Not much to do. Place the ramp on the floor (if using a board then use a spare box or find something else to hold the back end up). If using an opened t-shirt box to catch the jars, place at the bottom of the ramp, or if using foam, place about 10cm past the end of the ramp (taping it down helps if it's small).

## The Experiment

Gain attention by telling people they'll be racing jars down a hill. Explain what's in the jars.

First, bring out the drop test objects and ask kids which one is heavier (they can feel this for themselves) and ask which one will hit the ground first if they're dropped from the same height at the same time. Both should hit at the same time (demonstrate a couple of times, from different heights if necessary). Get them to tell you what force pulls things down. The heavier something is, the stronger the gravity. BUT, the heavier something is, the harder it is to get it to move. These effects cancel each other out so all objects fall under gravity at the same rate.

Explain that we are ignoring air resistance (or get them to tell you that). Use Newton’s second law to explain how mass doesn’t affect acceleration under gravity. It seems odd that the inertial mass term in Newton’s second law happens to be the gravitational mass. Einstein used this empirical fact in his ‘Equivalence Principle’ where he equates the gravitational force experienced by an observer to the same force that would be experienced if the observer were accelerating. This then lead to his Theory of General Relativity: The currently used theory that describes the effect of gravity.

These effects can also be understood if the falling masses are described by their energies. As the mass falls, gravitational potential energy is converted into kinetic energy, but again (use equations to show) the speed at which they fall is the same.

Now ask them to think about what will happen on the ramp - if gravity doesn't care how 'heavy' things are, should the weight of the jars make any difference? Will they all reach the bottom at the same time? Feel free to prompt some more (why do you think X, what might cause Y).

Let them pick up the jar of jam and the empty jar to get a feel for the weights, and then race! A pole or stick acts as a good release mechanism. The jar of jam should beat the empty jar.

It looks like the heavier jar wins but this makes no sense since we've already seen gravity doesn't care about weight. What is the difference between falling and rolling? When falling (or sliding on the slope), the jar stays one way round (in one orientation) whilst moving. When rolling, as well as moving down the slope, the jar is also spinning. This is what changes things. Gravity needs to put in extra effort to get the jars spinning as well as going down.

When two objects fall through the air under gravity we only need to consider their centre of mass motion. However, when the jars roll down the hill we also need to consider their rotational motion. The jars have linear kinetic energy due to the movement of their centre of mass and rotational kinetic energy from their rotation.

Does a roundabout just start spinning on its own without any pushing? Sixth form students should be familiar with F=ma, can introduce them to G=I x angular acceleration. G is the torque, the equivalent of force for rotational motion, students may have come across this before in the context of pivots. I is the moment of inertia. This is the equivalent of mass for rotational motion and is a measure of how easy it is to rotate an object. For linear motion, a larger mass makes it harder to move something. Here, a larger moment of inertia makes it harder to spin something. The difference between mass and I is that I is related to both mass and the distribution of the mass. Referring back to the roundabout, is it easier to make it spin fast if a person is sat in the middle or at the edge? (You can also try to connect this to the 'Spinning Chair' experiment if it is also out.) The larger the mass of an object, the harder it is to move and the larger the moment of inertia the harder it is to get the object to rotate. Additionally, the further away this mass is from the axis of rotation, the harder it is to move and the higher the moment of inertia. Link the two equations together and explain that we can treat angular motion is a very similar way to linear motion.

When different masses of the same shape are dropped and fall through the air they hit the ground at the same time. The larger the mass of the object the the larger the force due to gravity, but also a larger force is required to move it. Can think about how easy/hard it is to pick things off the ground that weigh different amounts. Can also relate back to F=ma. In this case F=mg, so a=g. Acceleration is constant.

Now instead of just thinking about F=ma, need to think about G=I*angular acceleration. The torque, G is proportional to mass. Moment of inertia is also proportional to mass, but with different prefactors depending on the distribution of mass within the object. The means that accelerations for the empty jar and the jar of jam are different because the distribution of mass is different.

As rotating need to think about moment of inertia. The jar of jam has a larger mass than the empty jar and it also has a larger moment of inertia. However, because the jam is distributed uniformly throughout the jar, the mass increases by a larger fraction than the moment of inertia compared to the empty jar. The empty jar has almost all of its mass concentrated in a very thin layer of glass a long way from the axis of rotation whereas the one full of jam has a lot of mass quite close to the middle. Objects with more mass closer to the axis of rotation are easier to rotate than objects with more mass closer to the edges of the object. This means that relatively speaking gravity finds it easier to rotate the jam jar. This means that the jam jar travels faster down the slope. If instead we compared an empty jam jar made of glass and an empty jam jar of the same shape but made of lead, the jars would reach the bottom of the slope at the same time. The important point isn’t that the jam jar has extra mass but the fact that the extra jam mass is distributed throughout the jar.

(could be worth mentioning, to reinforce the idea that total mass is irrelevant, that if there were no friction the jars would all slide (not roll!) down the hill in the same amount of time).

Next try the jar of jam vs the jar of water. The jar of water wins, why? What’s the difference between the water and the jam? Answer: Jam is sticky, but water isn’t. When the jar of the water rolls down, the water doesn’t have to spin, only the jar, whereas in the jar of jam both the jar and jam are spinning. This means the jar of water can go faster.

The difference between water and jam is that water is a liquid and jam is solid-like. Because of this the jam is stuck to the outside of the jar and rotates with the jar when it rolls down the slope. The water however, isn’t ‘stuck’ to the side of the jar so only sloshes slightly as the jar rolls. This costs less energy, so more energy goes into the linear kinetic energy of the jar down the slope. Overall energy must be conserved and both jars start with the same energy. (Neglecting the difference in mass between water and jam). So the water jar travels faster and wins the race.

The interesting case of the half-full jar and the jam jar can now be the ultimate race. The jam jar has the extra weight, but the half-full water jar doesn’t have to spin as much, so you can get the kids to bet on which will win. This one is actually quite close, so you’ll have to see which wins on the day!

In conclusion for linear motion the distribution of mass is irrelevant, only the total, whereas for rotational motion it's not just the mass that matters, its position is important as well.

**An energy argument**

What force is it that causes something to spin in the first place? (Friction acts at the point of contact of the jars). All of the problems can be explained by considering the energy of the jars. When the objects were dropped before, gravitational potential energy was converted into kinetic energy, but with spinning added, we need to incorporate a new energy term for the rotation of jam about the centre of the jar - rotational kinetic energy. The amount of rotational energy that you need to put in depends on where the mass is, so the empty jar has the worst distribution of mass and rolls slowest. For the full jar of water the water isn’t rotating so the rotational energy is only for the glass, meaning it can roll quicker.

We can write the rotational kinetic energy down in an equation similar to the linear kinetic energy:

LKE = (½)mv^2

RKE = (½)Iw^2

w is how quickly the jar is spinning. I is called the moment of inertia, and increases as mass gets further away from the centre, so more energy will be required. Actual definition is sum of mass * (distance from rotation axis)^2. E.g. figure skating: if a skater brings their arms in they've reduced their MOI but to keep the same rotational energy the same they spin faster"). Can demo this in the playground - several people stand at the edge of a roundabout spinning fairly slowly and walk in to the middle. The roundabout should speed up and if they walk back out it should get slower again.

The wikipedia page on 'moment of inertia' has more information and a nice gif animation of (front to back) a solid cylinder, cylindrical shell, ball and spherical shell racing down a slope. http://en.wikipedia.org/wiki/Moment_of_inertia#Scalar_moment_of_inertia_...