# Comparison of two kilogram masses

##### Demonstration

Simple demonstrations to illustrate a discussion of gravitational and inertial mass.

#### Apparatus and materials

kilogram masses, 2, made of same substance and with same shape

kilogram mass made of a different material (e.g. aluminium or lead)

dynamics trolley (a second dynamics trolley OPTIONAL)

equal arm balance

inertia balance (‘wig wag’)

stopclock, or other timing device

elastic cord, for accelerating trolley OPTIONAL

runway OPTIONAL

long weak spring or rubber thread OPTIONAL

#### Health & Safety and Technical notes

Read our standard health & safety guidance

Label one of the 1 kg masses ‘standard kilogram mass’.

#### Procedure

See Teaching notes for related discussion at each step.

**a** Produce the ‘standard kilogram mass’ and the other ‘kilogram’ of the same material.

**b** Place the two masses on two pans of the equal arm balance.

**c** Place the ‘standard kilogram mass’ on the dynamic trolley and accelerate it with a ‘standard force’, using the elastic cord. Time how long it takes for trolley to travel a measured distance.

**d** Repeat step **c** with the other ‘kilogram’ of the same material, comparing its motion when accelerated by the standard force.

**e** Using an inertial balance, compare the frequency at which each mass oscillates.

**f** Repeat steps **b**, **d** and **e** to compare what happens when the mass is made of a different material

**g** OPTIONAL Place two trolleys far apart on a level runway, with a weak spring or rubber thread stretched between them. Release so that the trolley accelerate towards each other, travelling equal distances before collision.

**h** Repeat **g** after placing a 1 kg mass on one of the trolleys. The distances they travel after being released will now be unequal.

####

Teaching notes

**1** In step **a**, begin the discussion like this: *Each of these objects has the same mass, 1 kg. What, precisely, does that mean?* Since they look the same and are made of the same material, they are likely to have the equal numbers of atoms.

Ask: *How can we be sure that every object labelled ‘1 kg’ really does have a mass of 1 kilogram?*Every measured mass must be compared with the standard mass kept at Sevres in France. Since the mass kept at Sevres cannot be moved, each country keeps its own standard mass, which has been tested against the mass kept at Sevres.

Ask:* How exactly might you compare two masses, to check that they are the same? *The everyday way of comparing masses is to ‘weigh’ them. An equal arm balance is based on the idea that, if the weights are equal then the masses are equal.

W =* mg* (*W* is weight, *m* is mass and *g *is the gravitational field strength)

so *W*_{1} = *W*_{2} implies *m*_{1} = *m*_{2}

This idea underpins the comparison in step **b**. With more able groups, you might want to make explicit reference to ‘gravitational mass’, a measure of the way that an object behaves in a gravitational field.

**2 **Ask: *Do these two masses have the same inertia?* Compare the two masses by trying to accelerate them, using a trolley (steps **c **and **d **should produce the same result) or in an inertia balance (see the experiment The inertia balance or 'wig-wag'). In step **e** the motion of both masses should be the same.

These experiments show that that both object have the same ‘inertial mass’, or ‘unaccelerability’.

**3 **Steps **g** and **h** provide another demonstration of the fact that different masses have different amounts of inertia. The more massive trolley accelerates more slowly than the lighter one.

**4 **Inertial mass is a measure of the amount of stuff to be accelerated. Gravitational mass is a measure of the amount of stuff to be affected by the gravitational field. Claiming that these are the same thing requires an enormous imaginative jump.

We have no reason to suppose that a large chunk of aluminium and a small chunk of lead, shown by trolley experiments to have the same inertia, will also be pulled to Earth with equal forces. However, if you do simultaneously allow two masses with equal inertia to fall from rest, they reach the ground at the same time. This shows that their accelerations are equal, providing evidence that they do have equal forces acting on them (equal weights).

[Repeat the experiment with unequal masses and you get the same result: equal times, indicating equal accelerations. Here the forces are different; what is the same is the ratio of weight *W* to mass *m*. Acceleration *a* = *F/m* and *W*_{1}*/m*_{1} = *W*_{2}*/m*_{2}.]

The fact that all masses fall with the same acceleration due to gravity in a vacuum, and that inertial and gravitational masses are equivalent, is the basis of Einstein’s theory of general relativity. Einstein’s explanation was developed three centuries after Galileo worked out that large and small bodies fall with the same acceleration.