Physics

11.7 - Principle of Moments

6 min

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The experiment

Aim

To test the principle of moments by balancing a metre rule with masses hanging on either side of a pivot, and showing that the clockwise moment about the pivot is equal to the anticlockwise moment.

Background

A force on an object that can turn about a pivot has a turning effect, called a moment, which is either clockwise or anticlockwise.
A moment in Nm is $M = F \times d$ , where $F$ is the force in N and $d$ is the perpendicular distance in m from the pivot to the line of the force.
The principle of moments says that when an object is balanced, the clockwise moments about a point add up to the same as the anticlockwise moments.
You can't measure a moment directly, but you can measure each hanging weight and its distance from the pivot.
So in this practical you balance a metre rule with a mass on each side of the pivot, and check that force × distance is the same on both sides.

Variables

Independent variable - The masses, and the distance of the first mass from the pivot.
Dependent variable - The distance at which the second mass balances the rule.
Control variables - The metre rule and the position of the pivot.

Method

Set up the pivot - Support a metre rule at about its 50 cm mark on a pin, a knife-edge, or a loop of twine.
Balance the bare rule - With nothing on it, slide the rule until it stays level, or add a small blob of Plasticine to the higher end. Don't move the pivot after this.
Hang the first mass - Hang a known mass, such as 200 g, on one side with a loop of cotton, and measure its distance $d_{1}$ from the pivot.
Balance it with the second mass - Hang a different mass, such as 400 g, on the other side and slide it until the rule is level. Measure its distance $d_{2}$ .
Repeat - Move the first mass and balance the rule again, for at least five pairs of distances, then use different masses.
Record the results - In a table giving each force, its distance, and force × distance, for both sides.

Safety

Falling masses - Put a tray under the masses, and stand rather than sit, so nothing lands on your feet.

A metre rule balanced on a pivot at the 50 cm mark, a 200 g mass hanging on the left at distance d1 and a 400 g mass nearer the pivot at d2. — The heavier mass sits closer to the pivot, so the two moments still balance.

note 2 of 3

Worked example - Testing the principle of moments

A student balanced a metre rule on a pivot at its 50 cm mark. She hung a 200 g mass at the 20 cm mark, then slid a 400 g mass along the other side until the rule was level again, at the 65 cm mark. In her next row of results, the anticlockwise moment came out as 0.40 Nm and the clockwise moment as 0.44 Nm.

Show that her first results agree with the principle of moments, and say what the mismatch in the next row could mean.

Step 1 - Turn the masses into forces

Use $W = m g$ with $g$ = 10 N/kg.

The 200 g mass is 0.200 kg, so its weight is 0.200 × 10 = 2.0 N.
The 400 g mass is 0.400 kg, so its weight is 0.400 × 10 = 4.0 N.

Step 2 - Find each distance from the pivot

Measure from the pivot at 50 cm, not from the end of the rule, and convert to metres.

The 200 g mass is 50 − 20 = 30 cm = 0.30 m from the pivot.
The 400 g mass is 65 − 50 = 15 cm = 0.15 m from the pivot.

Step 3 - Work out the anticlockwise moment of the 200 g mass

$M = F \times d = 2.0 \times 0.30 = 0.60 Nm$

Step 4 - Work out the clockwise moment of the 400 g mass

$M = F \times d = 4.0 \times 0.15 = 0.60 Nm$

The two moments are equal, so the results agree with the principle of moments.

Step 5 - Decide what the mismatch in the next row means

The difference is 0.44 − 0.40 = 0.04 Nm.

Systematic error - If the clockwise moment is larger in every row, the pivot isn't at the rule's centre of mass, so the rule's own weight adds a turning effect to every reading. Balancing the bare rule first removes this.
Random error - If the difference changes direction from row to row, it comes from reading positions to the nearest millimetre and judging by eye when the rule is level.

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Common exam questions

Question 1

The metre rule has a weight of its own. Why doesn't it appear in the moments calculation?

The bare rule is balanced first, so the pivot is at its centre of mass, where its weight acts. That weight acts at zero distance from the pivot, so its moment is zero.

Question 2

A child weighing 400 N sits 1.5 m from the pivot of a see-saw. How far from the pivot must an adult weighing 600 N sit on the other side to balance it?

The child's moment is 400 × 1.5 = 600 Nm. The adult's moment must also be 600 Nm, so the distance is 600 ÷ 600 = 1.0 m.

Question 3

Why are the masses hung from thin loops of cotton rather than stood on top of the rule?

A thin loop marks one definite point to measure to. A mass on top can slide, and its width makes the distance hard to pin down.

Question 4

The rule is balanced and not moving. What can you say about the upward and downward forces on it?

They're equal. The upward force from the pivot equals the weight of the rule plus the hanging weights.

Question 5

A student records the distances in centimetres. Can they still check the principle of moments?

Yes, as long as both sides are in centimetres. Both moments then come out in N cm, so they can still be compared.

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