The principle of moments

​​In a nutshell

The moment of a force is the measure of how much it causes the body to rotate about a specific point. The principle of moments states that when an object is in rotational equilibrium, then the sum of the clockwise moments about a point is equal to the sum of the anticlockwise moments.

Definitions

Equations

Variable definitions

The moment of a force

The moment of a force is a measure of how much an object tends to rotate around a point when a force is applied to it. It can be calculated with the following equation.

​$$moment = Fx$$​​

where $$F$$ is the force being applied and $$x$$ is the perpendicular distance from the line of action of the force to the point of rotation.​

If the force is at an angle, then the perpendicular distance between the force and the point of rotation should be calculated using trigonometry.

Note: The point of rotation can also be referred to as a pivot in an exam!

Example

The force in the diagram above is acting at an angle to the object. The object has a length $$L$$. $$x$$ is the distance between the line of action of the force and the pivot point and is always perpendicular to the force. It can be calculated in the following way using trigonometry.

$$x = L\cos\theta$$

Substitute this with the equation for moment above creates the following equation.

​$$moment = FL\cos\theta$$​​

The principle of moments

The principle of moments states that when a body is in rotational equilibrium, then the sum of the clockwise moments about a point will be the same as the sum of the anticlockwise moments around the same point. This can be used to solve problems using objects in rotational equilibrium.

Note: For an object to be in equilibrium, the principle of moments must apply and the net force on the object must be equal to zero. 

Example

​

Calculate the force $$F$$ in the above diagram. The object is in rotational equilibrium.

The object is in rotational equilibrium, so the sum of the clockwise and anticlockwise moments will be the same.

$$clockwise\ moment = anticlockwise\ moment$$

Next, substitute the values and rearrange.

$$F \times 0.08 = 15 \times 0.2$$​​

$$F = \dfrac{15\times0.2}{0.08}$$​​

$$F = 37.5\,N$$​​

The force has a magnitude of $$\underline{37.5\,N}$$.

​

Example

​

Calculate the forces $$A$$ and $$B$$ in the above diagram. The object is in equilibrium.

First, resolve around one of the unknown forces. Taking the pivot to be at the base of the force $$A$$, the moment caused by that force will be equal to zero, and so only the moment caused by weight and $$B$$ need to be compared.

$$clockwise\ moment = anticlockwise\ moment$$​​

Next, substitute in the values.

$$0.25 \times 20 = 0.35 \times B$$​​

$$B = \dfrac{0.25 \times 20}{0.35}$$​​

$$B = 14.3\,N$$​​

Now that $$B$$ has been calculated, A can be calculated. Since the object is in equilibrium, the sum of the two unknown forces must sum to the weight.

$$A + B= W$$​​

$$A = W - B$$​​

$$A = 5.7\,N$$​​

The unknown force $$A$$ is $$\underline{5.7\,N}$$ and the unknown force $$B$$ is $$\underline{14.3\,N}$$.