# Moments

## Moments Revision

**Moments**

When a** force** acts on an object at a distance away from a **pivot** point, a **turning effec**t is caused. These turning effects are known as **moments**. **Moments** cause an object to rotate about a **pivot** point. Understanding moments is crucial as their application is used in daily life.

**Calculating Moments**

A **moment** is a **turning force** and the SI unit for a **momen**t is \text{Nm} (Newton metre). A **moment** can be calculated using the following equation:

\textcolor{aa57ff}{M = F \times d}

- \textcolor{aa57ff}{M} is the
**moment**in Newton-metres \left(\text{Nm}\right) - \textcolor{aa57ff}{F} is the
**force**in Newtons \left(\text{N}\right) - \textcolor{aa57ff}{d} is the
**perpendicular distance**from the pivot to the mass in metres \left(\text{m}\right)

In this example on the right, the **distance** is easily measured from the centre of the** pivot** point to the centre of the force acting on it.

In this example we have not considered the **mass** of the beam holding the object but we will look at this in other examples.

Sometimes we find examples where the rotating beam is at an angle to the horizontal, just as in the example on the right.

This time we are going to need to use some trigonometry to calculate the perpendicular distance from the pivot to the object:

\textcolor{aa57ff}{\text{Moment} = F \times d \cos θ}

**Moments** are all around us. We often consider the idea of moments without realising it. For example, if we are trying to release a tight bolt, we may pick a longer spanner. Or when opening a door, the door handle is on the far side of the door, furthest away from the pivot point to allow it to be opened easily.

In general, the greater the **distance** from the pivot, the greater the **moment** or turning effect as moments and the distance from the pivot are directly proportional.

**Example:** Calculate the clockwise moment produced in the diagram on the right.

**[3 marks]**

\text{Perpendicular distance} = d \cos θ = \boldsymbol{1.8 \cos 35 = 1.474} \: \textbf{m}

\text{Moment} = \text{force} \times \text{perpendicular distance} = \boldsymbol{1100 \times 1.474} = 1621.4 \: \text{Nm}

\text{Moment} = \boldsymbol{1600} \: \textbf{Nm} \left(2 \: \text{sf} \right)

**The Principle of Moments**

Often **moments** are experienced in opposite directions. Picture the scenario below.

The object on the left is causing an **anti-clockwise** turning effect whilst the object on the right is causing a **clockwise** turning effect.

If these two **moments** are equal, we can say that the moments are in **equilibrium**.

If the moments are in equilibrium, **no rotation** will occur.

In our example the moments are in **equilibrium**. We can prove this:

\text{Anti-clockwise moment} = 2F \times 0.5d = Fd

\text{Clockwise moment} = F \times d = Fd

Therefore the moments are in equilibrium, because \text{Anti-clockwise moment} = \text{Clockwise moment} = Fd

**Example: **Two people are on opposite sides of a seesaw. Person A is 2 \: \text{m} away from the pivot and weighs 300 \: \text{N} . Person B sits 1.2 \: \text{m} away from the pivot. What must person B weigh if they are in equilibrium?

**[3 marks]**

\text{Anti-clockwise moment} = \text{Clockwise moment}

F_A \times d_A = F_B \times d_B

\boldsymbol{300 \times 2 = F_B \times 1.2}

So the force due to person B (in other words, their weight) is:

F_B = \boldsymbol{\dfrac{300 \times 2}{1.2}}

F_B = \boldsymbol{500} \: \textbf{N}

**Couples**

A **couple** is a pair of forces, acting in opposite directions which together cause a** turning effect**.

For the forces to be a couple, the forces must be **coplanar** and **equal** and **opposite** in magnitude. For** forces** to be **coplanar**, they must be acting in the same plane (but may be acting opposite in direction).

This pair of** forces** can cause a rotation without the need of a** pivot**. As the forces are equal and opposite, the** resultant force** is zero. Therefore there is no net movement other than the **rotation** of the object.

**Centre of Mass**

The **centre of mass** of an object is the point at which all of the object’s mass could be considered to act. For our calculations, this allows us to consider objects as point masses where the full mass of the object acts in one position.

For an object of **uniform density** (has the same density throughout), its centre of mass is at the centre of the object.

If the object is **symmetrical** this can easily be found. If you take more than one line of symmetry, the centre of mass will be at the point where the lines of symmetry intersect.

The **centre of mass** of this regular hexagon can be seen where all of its lines of **symmetry** meet.

The centre of mass for an **Irregular shaped** object is trickier to identify:

- To find the centre of mass, you take the object and hang it from a fixed point next to a
**plum line**. - Next, draw a straight line where the plumb line touches the object.
- Repeat this by hanging the object from another point. The
**centre of mass**is where the lines**intersect**.

**Stability**

For objects that can change shape, like people, the **centre of mass** will move depending on their position. As a standing person bends their knees, their **centre of mass **becomes closer to the ground. The closer the **centre of mass** is to the ground, the more stable you are.

In the diagram on the right, both objects above are currently stable. This is because their **centre of mass** (indicated by the arrows) are acting above the base of the object. However, we would consider the right hand object to be more stable as it would take more effort to move the object enough so that it’s centre of mass moved outside of its base:

Both objects are now unstable as their **centre of mass** is no longer above their base. However, it has taken a much greater movement to cause the left object to become unstable when compared to the right object. This is because of the **large base** of the right hand object.

## Moments Example Questions

**Question 1: Calculate the resultant moment from the diagram below:**

**[3 marks]**

**Question 2: Two people are on opposite sides of a seesaw. Person A is 1 \: \text{m} away from the pivot and weighs 500\: \text{N} . Person B weighs 700\: \text{N} . How far from the pivot would person B need to sit to ensure they are in equilibrium?**

**[3 marks]**

**Question 3:** Using your knowledge of the centre of mass, explain why would a forklift truck be better carrying its load close to the ground, rather than higher up in the air.

**[2 marks]**

As the mass of the load is lowered towards the ground, the centre of mass of the load and the truck moves **towards the ground**. This **improves the stability of the load and forklift truck**.