**Finding the Mass of a Metre Rule using the Principle of Moments**

In this experiment you will use the principle of moments, together with the idea of the centre of gravity, to find the mass of a metre rule.

##### Theory

The centre of gravity of a body is a point through which the weight of the body acts, or appears to act. A metre rule has a uniform shape and a constant density and so the centre of gravity will be a point exactly in the middle of the rule (at the 50 cm mark).

The principle of moments states that an object is in equilibrium if the sum of all anticlockwise moments about the pivot is equal to the sum of all clockwise moments about the same pivot. If a metre rule is balanced horizontally at any point, this means that the clockwise moments and the anticlockwise moments must be equal. The arrangement for the experiment is shown in the diagram below.

In this situation, the weight *F1* of the masses provides the anticlockwise moment and the weight* F2* of the rule provides the clockwise moment. The weight of the rule acts through the centre of gravity at the middle of the rule. This is shown in the diagram below.

If the rule is balanced, we can apply the principle of moments. This results in the equation

**F1d1 = F2d2**

where d1 is the distance between the hanging mass and the pivot and d2 is the distance between the pivot and the centre of gravity of the rule. This equation can be rewritten as

**m1gd1 = m2gd2**

where *m*1 is the mass hanging from the rule, *m2* is the mass of the metre rule and *g* is the acceleration of free fall.

This can be rearranged to give

**m2 = m1 × (d1 / d2)**

##### Making measurements and observations

1. Set up the stand, boss and clamp so that the bar of the clamp is horizontal and its height above the bench is a few centimetres more than the length of the mass hanger.

2. Hook the thread loop over the zero end of the metre rule.

3. Hang the mass hanger from the bottom of the thread loop underneath the metre rule.

4. Slide the thread loop so that it is at the 1 cm mark of the metre rule.

5. Move the metre rule and the hanging masses so that the metre rule balances horizontally on the bar of the clamp stand. (This may be a bit fiddly, so be patient.)

6. When it is balanced, record *m1, d1* and *d2.*

7. Repeat the experiment for six different values of *m1*.

##### Recording and presenting your data

1. All your measurements should be recorded in a table of results. Your table of results should include a column for *m2*.

2. The values in the column for m2 should be calculated using the equation.

*m2 = m1 × (d1 / d2)*

##### Analysing your data

1. Calculate the average of your values for the mass *m2* of the metre rule.

##### Evaluation

1. Estimate as to the actual uncertainty in each measurement you have taken.

2. Describe any steps you took to reduce experimental errors.

3. Describe any limitations or problems with the method used to find *m2* in this experiment.

4. Suggest ways in which the accuracy of the measurements taken could be improved.