Clockwise and anticlockwise moments

Definition

Principle of moments

The principle of moments states that for rotational forces in equilibrium, the sum of the clockwise moments about a pivot equals the sum of the anticlockwise moments about that same pivot.

Every moment turns an object one of two ways about its pivot, either clockwise or anticlockwise.
The direction is found by picturing which way the object would begin to rotate if that force acted on its own.
On a seesaw with a central pivot, a weight on the right-hand side produces a clockwise moment and a weight on the left-hand side produces an anticlockwise moment.
Several forces can turn the object the same way, so a beam may carry two or three clockwise moments at once and they are added together.
When the total turning effect one way exactly matches the total the other way, there is no resultant moment and the object will not start to rotate.
The principle of moments states that for rotational forces in equilibrium, $moments\text{sum of clockwise moments} = \text{sum of anticlockwise moments}$ .
Every moment in that statement must be taken about the same pivot, because a moment has no meaning until the pivot it turns about has been named.
Equilibrium here does not mean that there are no moments, only that the clockwise total and the anticlockwise total are equal in size.
If the two totals are not equal there is a resultant moment, and the object turns in the direction of the larger total.
The pivot itself pushes on the object, but its line of action passes through the pivot, so it contributes no moment to either total.

A seesaw in rotational equilibrium: the weights on either side of the central pivot act at different distances so that the clockwise and anticlockwise moments are equal, while the pivot pushes upwards on the plank.

Using the principle in a calculation

Each moment in the sum is worked out from $F\,d$ , using the perpendicular distance from the pivot to that particular force.
The working follows a fixed order.
1. Mark the pivot and label every force with its distance from it.
2. Decide whether each force turns the object clockwise or anticlockwise.
3. Calculate every clockwise moment and add them together.
4. Calculate every anticlockwise moment and add them together.
5. Set the two totals equal to each other, because the object is in equilibrium.
6. Rearrange to find the unknown force or distance, and give the unit.
When a load is given as a mass, turn it into a weight with $m\,g$ before taking any moments, using the gravitational field strength stated in the question.
The weight of a uniform beam acts at its centre of mass, which is the middle of the beam, so the beam's own weight produces a moment whenever the pivot is not at that point.
A force whose line of action passes through the pivot contributes nothing to either total, so it can be left out of the sum entirely.
A useful check is that the heavier load always sits closer to the pivot, because a larger force needs a smaller distance to give the same moment.

Example

Balancing a seesaw

A child of weight $N300\ \text{N}$ sits $m1.2\ \text{m}$ to the left of the pivot of a seesaw, and a second child of weight $N450\ \text{N}$ sits at an unknown distance $d$ to the right, so that the seesaw balances.
The anticlockwise moment is $m300\ \text{N} \times 1.2\ \text{m} = 360\ \text{N m}$ .
The clockwise moment is $N×d450\ \text{N} \times d$ .
Applying the principle of moments gives $d=360450\,d = 360$ .
Rearranging gives $\dfrac{360}{450} = 0.80\ \text{m}$ .
The heavier child sits $m0.80\ \text{m}$ from the pivot, which is closer in than the lighter child, as expected.

Example

A beam carrying several moments

A beam rests on a central pivot and carries two loads on the left and two on the right.
The two left-hand loads produce anticlockwise moments of $m18\ \text{N m}$ and $m7\ \text{N m}$ .
One right-hand load produces a clockwise moment of $m9\ \text{N m}$ , and the second produces an unknown clockwise moment $M$ .
The anticlockwise total is $25\ \text{N m}$ .
The clockwise total is $9 + M$ .
Setting the totals equal gives $25 = 9 + M$ , so $16\ \text{N m}$ .
Both totals now come to $m25\ \text{N m}$ , which confirms that the beam is in rotational equilibrium.

Common Mistake

Do not compare only one moment on each side when a question gives several forces, because every moment about the pivot belongs in the totals.
Do not take moments about a different point for each force, since the principle only holds when all the moments are about one named pivot.
Do not ignore the weight of a heavy uniform beam, because it acts at the centre of mass and produces a moment whenever the pivot is somewhere else.
Do not describe a balanced beam as having no forces on it, because the pivot pushes upwards with a force equal to the total downward force.

Exam technique

Laying out a moments answer

State the principle of moments in words or symbols before substituting any numbers into it.
Label each moment as clockwise or anticlockwise as you calculate it, so the two totals stay separate.
Keep every distance in metres and every force in newtons, and write $m\text{N m}$ on the final answer.
Convert a mass into a weight before taking moments, because a mass in kilograms cannot be used in $F\,d$ .
If the question asks why something balances, say that the clockwise and anticlockwise moments about the pivot are equal rather than saying that the forces are equal.

Self review

State the principle of moments in full.
Explain why every moment in the sum must be taken about the same pivot.
Explain why a heavier child must sit closer to the pivot of a seesaw in order to balance a lighter one.
Calculate the distance at which a $N200\ \text{N}$ weight balances a $N400\ \text{N}$ weight placed $m0.50\ \text{m}$ from the pivot.
Explain where the weight of a uniform beam acts and when it produces a moment.

Clockwise and anticlockwise moments

Definition

Principle of moments

The principle of moments states that for rotational forces in equilibrium, the sum of the clockwise moments about a pivot equals the sum of the anticlockwise moments about that same pivot.

Every moment turns an object one of two ways about its pivot, either clockwise or anticlockwise.
The direction is found by picturing which way the object would begin to rotate if that force acted on its own.
On a seesaw with a central pivot, a weight on the right-hand side produces a clockwise moment and a weight on the left-hand side produces an anticlockwise moment.
Several forces can turn the object the same way, so a beam may carry two or three clockwise moments at once and they are added together.
When the total turning effect one way exactly matches the total the other way, there is no resultant moment and the object will not start to rotate.
The principle of moments states that for rotational forces in equilibrium, $moments\text{sum of clockwise moments} = \text{sum of anticlockwise moments}$ .
Every moment in that statement must be taken about the same pivot, because a moment has no meaning until the pivot it turns about has been named.
Equilibrium here does not mean that there are no moments, only that the clockwise total and the anticlockwise total are equal in size.
If the two totals are not equal there is a resultant moment, and the object turns in the direction of the larger total.
The pivot itself pushes on the object, but its line of action passes through the pivot, so it contributes no moment to either total.

Using the principle in a calculation

Each moment in the sum is worked out from $F\,d$ , using the perpendicular distance from the pivot to that particular force.
The working follows a fixed order.
1. Mark the pivot and label every force with its distance from it.
2. Decide whether each force turns the object clockwise or anticlockwise.
3. Calculate every clockwise moment and add them together.
4. Calculate every anticlockwise moment and add them together.
5. Set the two totals equal to each other, because the object is in equilibrium.
6. Rearrange to find the unknown force or distance, and give the unit.
When a load is given as a mass, turn it into a weight with $m\,g$ before taking any moments, using the gravitational field strength stated in the question.
The weight of a uniform beam acts at its centre of mass, which is the middle of the beam, so the beam's own weight produces a moment whenever the pivot is not at that point.
A force whose line of action passes through the pivot contributes nothing to either total, so it can be left out of the sum entirely.
A useful check is that the heavier load always sits closer to the pivot, because a larger force needs a smaller distance to give the same moment.

Example

Balancing a seesaw

A child of weight $N300\ \text{N}$ sits $m1.2\ \text{m}$ to the left of the pivot of a seesaw, and a second child of weight $N450\ \text{N}$ sits at an unknown distance $d$ to the right, so that the seesaw balances.
The anticlockwise moment is $m300\ \text{N} \times 1.2\ \text{m} = 360\ \text{N m}$ .
The clockwise moment is $N×d450\ \text{N} \times d$ .
Applying the principle of moments gives $d=360450\,d = 360$ .
Rearranging gives $\dfrac{360}{450} = 0.80\ \text{m}$ .
The heavier child sits $m0.80\ \text{m}$ from the pivot, which is closer in than the lighter child, as expected.

Example

A beam carrying several moments

A beam rests on a central pivot and carries two loads on the left and two on the right.
The two left-hand loads produce anticlockwise moments of $m18\ \text{N m}$ and $m7\ \text{N m}$ .
One right-hand load produces a clockwise moment of $m9\ \text{N m}$ , and the second produces an unknown clockwise moment $M$ .
The anticlockwise total is $25\ \text{N m}$ .
The clockwise total is $9 + M$ .
Setting the totals equal gives $25 = 9 + M$ , so $16\ \text{N m}$ .
Both totals now come to $m25\ \text{N m}$ , which confirms that the beam is in rotational equilibrium.

Common Mistake

Do not compare only one moment on each side when a question gives several forces, because every moment about the pivot belongs in the totals.
Do not take moments about a different point for each force, since the principle only holds when all the moments are about one named pivot.
Do not ignore the weight of a heavy uniform beam, because it acts at the centre of mass and produces a moment whenever the pivot is somewhere else.
Do not describe a balanced beam as having no forces on it, because the pivot pushes upwards with a force equal to the total downward force.

Exam technique

Laying out a moments answer

State the principle of moments in words or symbols before substituting any numbers into it.
Label each moment as clockwise or anticlockwise as you calculate it, so the two totals stay separate.
Keep every distance in metres and every force in newtons, and write $m\text{N m}$ on the final answer.
Convert a mass into a weight before taking moments, because a mass in kilograms cannot be used in $F\,d$ .
If the question asks why something balances, say that the clockwise and anticlockwise moments about the pivot are equal rather than saying that the forces are equal.

Self review

State the principle of moments in full.
Explain why every moment in the sum must be taken about the same pivot.
Explain why a heavier child must sit closer to the pivot of a seesaw in order to balance a lighter one.
Calculate the distance at which a $N200\ \text{N}$ weight balances a $N400\ \text{N}$ weight placed $m0.50\ \text{m}$ from the pivot.
Explain where the weight of a uniform beam acts and when it produces a moment.

9.2.2 The principle of moments

Clockwise and anticlockwise moments

Using the principle in a calculation

Balancing a seesaw

A beam carrying several moments

Laying out a moments answer

Review 9.2.2 The principle of moments by teaching Genie

9.2.2 The principle of moments

Clockwise and anticlockwise moments

Using the principle in a calculation

Balancing a seesaw

A beam carrying several moments

Laying out a moments answer

Review 9.2.2 The principle of moments by teaching Genie

9.2.2 The principle of moments

Clockwise and anticlockwise moments

Using the principle in a calculation

Balancing a seesaw

A beam carrying several moments

Laying out a moments answer

Review 9.2.2 The principle of moments by teaching Genie

9.2.2 The principle of moments Revision Guide

9.2.2 The principle of moments

Clockwise and anticlockwise moments

Using the principle in a calculation

Balancing a seesaw

A beam carrying several moments

Laying out a moments answer

Review 9.2.2 The principle of moments by teaching Genie

9.2.2 The principle of moments Revision Guide