Moments, levers and gears

​​In a nutshell

A moment is the turning effect from an applied force on an object. A lever is a bar that rotates about a pivot point. A gear transmits the rotational effect from a force.

Equations

Variable definitions

Moments

Definition

A moment is the turning effect from an applied force on an object. This causes the rotation of the object. A moment depends on the size of the force and the distance from the pivot point that the force is applied.

1. Force applied 2. Rotation 3. Pivot point

A pivot is the point around which an object turns.

Example

• A Ferris wheel turns around the centre of its circle. 
• A seesaw turns around the middle of the seesaw - the part that's connected to the ground.

Note: A pivot point is not always at the centre of an object. A spanner turns around a nut.

The size of the moment can be calculated using the following equation.

​$$moment\:of\:a\:force = force\times distance\: \text{(perpendicular to the direction of the force)} \\ \ \\ M = F\times d$$​

Note: $$d$$ is used as the symbol for perpendicular distance. This is different to $$s$$ which is used for parallel distance.​

The maximum turning effect (moment) occurs when the force acts perpendicular to the object.

There are two types of moment: clockwise and anticlockwise. A clockwise moment is when the force applied causes the object to rotate in a clockwise $$\circlearrowright$$ direction. An anticlockwise moment is when the force applied causes the object to rotate in an anticlockwise $$\circlearrowleft$$ direction.

Principle of moments

The principle of moments states that the sum of the clockwise moments is equal to the sum of the anticlockwise moments for an object in equilibrium. This means there will be no overall turning effect, as the object is balanced.

​$$sum \:of\:clockwise\:moments = sum\:of\:anticlockwise\:moments$$​

This equation only applied to objects in equilibrium. If an object is rotating, the sum of the clockwise moments will not be equal to the sum of the anticlockwise moments.

Example

A seesaw is in equilibrium. Alice sits $$2.0\,m$$​ to the right of the pivot point of the seesaw. Bob sits $$4.0\,m$$​ to the left of the pivot point of the seesaw. Alice exerts a force of $$300\,N$$​ perpendicular to the seesaw. What force does Bob exert?

First, write out the quantities and make sure they are in the correct units

$$F_{Alice} = 300\, N, \:d_{Alice}=2 \,m \\ \ \\ F_{Bob} =\:?,\: d_{Bob} = 4\,m$$

Next, write down the equations you need to use

​

$$M=F\times d \newline sum \:of\:clockwise\:moments = sum\:of\:anticlockwise\:moments$$

Next, rearrange equation to make Bob's force the subject

$$F_{Alice}\times d_{Alice} = F_{Bob} \times d_{Bob} \\ \ \\ F_{Bob} = \frac{F_{Alice}\times d_{Alice}}{d_{Bob}}$$​​

Then, substitute the values into the equation

$$F_{Bob} = \frac{300 \times 2}{4} = 150 \,N$$​​

​​

Therefore, the force Bob exerts is $$\underline{150\,N}$$​.

​

Levers

​​Definition

A bar that rotates around a point.

The purpose of a lever is to increase the distance between the pivot and the point at which the force acts. This increases the moment and makes it easier to do work.

Example

A heavy load placed on the end of a lever. A force is applied to the opposite end of the lever. The longer the lever, the greater the moment will be. This means that the heavy load will be easier to lift up. 

1. Applied force 2. Large distance 3. Pivot point 4. Heavy load

Curiosity: Archimedes said "If you give me a lever and a place to stand, I can move the world"!  

Gears

​​Definition

A circular disc able to transmit the rotational effect of a force.

Gears are discs with 'teeth' around the edges. These teeth can interlock with the teeth of other gears.

When one gear turns, the turning effect is also applied to the next gear. The effect causes the gear to turn in the opposite direction to the previous interlocked gear.

In a chain of interlocked gears, their rotational speed depends on the size of a gear. A smaller gear turns faster than a larger one.

A larger gear produces a larger moment. This is because the distance from the pivot (the centre of the disc) is greater than that of a small gear. 

​	1 Small gear 2. Large gear