# Torque & Equilibrium

12PHYS - Mechanics

2019

# Mahi Tuatahi

\begin{aligned} F=ma \end{aligned}
1. State what each letter stands for
2. Give the units for each letter
3. Rearrange the equation for $$m$$ and $$a$$
4. Derive the SI units for F (not Newtons)

For a car of mass 1500kg which is accelerating at $$3.7ms^{-2}$$:

1. What net force is needed to maintain this acceleration?
\begin{aligned} & && \text{Knowns} \newline & && \text{Unknowns} \newline & && \text{Formula} \newline & && \text{Sub and Solve} \end{aligned}
1. If the engine is producing $$6000N$$ of thrust, what is the difference and what happened to it?

# Torque ($$\tau$$)

Torque can be thought of as the turning effect around a pivot. Torque is sometimes known as moment or leverage.

\begin{aligned} \tau &= Fd_{\bot} \newline torque &= Newtons \times metres \newline torque &= \text{Newton meters (Nm)} \end{aligned}
• $$F =$$ force in Newtons
• $$d_{\bot} =$$ perpendicular distance of force from pivot

# Torque ($$\tau$$)

• A small force at a small distance produces a small torque,
• the same small force at a larger distance produces a larger torque.

# Pātai 1

A force of $$9N$$ acting up at a distance of $$10cm$$ is needed to lift the top off a bottle of soft drink. Start by drawing a rough diagram. Calculate the torque applied.

\begin{aligned} & && \text{Knowns} \newline & && \text{Unknowns} \newline & && \text{Formula} \newline & && \text{Sub and Solve} \end{aligned}

# Pātai 1: Whakatika

A force of $$9N$$ acting up at a distance of $$10cm$$ is needed to lift the top off a bottle of soft drink. Calculate the torque applied.

\begin{aligned} & \tau = Fd_{\bot} \newline & \tau = 9 \times 0.1 \newline & \tau = 0.9 \text{Nm anticlockwise} \newline \end{aligned}

# Pātai 2

Calculate the torque applied if the lever is stretched to $$75cm$$.

\begin{aligned} & && \text{Knowns} \newline & && \text{Unknowns} \newline & && \text{Formula} \newline & && \text{Sub and Solve} \end{aligned}

# Pātai 2: Whakatika

Calculate the torque applied if the lever is stretched to $$75cm$$.

\begin{aligned} & \tau = Fd_{\bot} \newline & \tau = 9 \times 0.75 \newline & \tau = 6.75 \text{Nm anticlockwise} \newline \end{aligned}

# Pātai 3

Calculate the torque applied if the lever is compressed to $$1cm$$.

\begin{aligned} & && \text{Knowns} \newline & && \text{Unknowns} \newline & && \text{Formula} \newline & && \text{Sub and Solve} \end{aligned}

# Pātai 3: Whakatika

Calculate the torque applied if the lever is compressed to $$1cm$$.

\begin{aligned} & \tau = Fd_{\bot} \newline & \tau = 9 \times 0.01 \newline & \tau = 0.09 \text{Nm anticlockwise} \newline \end{aligned}

# Question 4: Does torque have a direction?

Yes, and you must always state which direction it is acting in.

Clockwise or Anticlockwise

# But, What Is Equilibrium?

Newton’s First Law tells us equilibrium is when an object is at rest or moving uniformly.

For this to occur we need two things:

1. Sum of all forces to be 0
2. Sum of all torques to be 0

# Okay, So Where Do We Use It?

Building bridges, setting up scaffolding, see-saws and more!

# Question 1

$$m_{1}=2kg$$, $$d_{1}=15cm$$, $$m_{2}=1kg$$, $$d_{2}=30cm$$

1. Calculate the clockwise and anticlockwise torques
2. Are they in balance?

# Question 2

$$m_{1}=7kg$$, $$d_{1}=65cm$$, $$m_{2}=13kg$$, $$d_{2}=35cm$$

1. Calculate the clockwise and anticlockwise torques
2. Are they in balance?

# Question 3

The rock has mass $$1100kg$$ and is at distance $$50cm$$ from the pivot. If Ash exerts $$70N$$ of downward force at a distance of $$8m$$ from the pivot can he move the rock?

Archimedes once said: “Give me a place to stand and I will move the world”

Question: Assuming the mass of the Earth is $$5.972\times 10^{24} kg$$ at a distance of 1km from the pivot and Archimedes’ mass is $$75kg$$, how long would his lever have to be?

# Mahi Tuatahi

1. Calculate the clockwise torque
2. Calculate the anticlockwise torque
3. Is it balanced?

# Torque & Equilibrium

The plank may not be massless. You may need to take it into account.

• The mass of the plank acts through its center of gravity
• Because the plank is uniform, this is the middle of the plank

# How To Solve A Torque Problem

1. Draw and label all forces on a diagram
2. Draw and label the distances between all forces and the pivot
3. Calculate all clockwise torque
4. Calculate all anticlockwise torque
5. Balance torques & forces

# Pātai

• $$d_{1}=30cm$$, $$d_{2}=70cm$$, $$m_{1}=900g$$, $$m_{2}=300g$$, see-saw mass = $$100g$$.
• Calculate the total anticlockwise moment
• Calculate the total clockwise moment
• Is it balanced?

# Pātai: Case Study

• Assume the system is in equilibrium ($$\tau_{clockwise} = \tau_{anticlockwise}$$)
• $$d_{1}=0.5m$$, $$d_{2}=1.5m$$, $$F_{1}=2.5N$$, see-saw mass = $$0.5kg$$, $$F_{2}=?$$.
• Draw the weight force of the see-saw on your diagram
• Find the unknown force, $$F_{2}$$

## Whakawai / Practise

Textbook: Force, Equilibrium and Motion - Q7, 8, 10, 11, 12