A.4.1 Torque and rotational motion

Torque and Its Role in Rotation

When you push a door near its hinges, it barely moves.
But if you push at the edge, it swings open easily.

This difference is due to torque, the rotational equivalent of force.

What is Torque?

Definition

Torque

Torque is the rotational equivalent of force. It measures the ability of a force to cause an object to rotate.

It depends on three factors:

Magnitude of the Force ( $F$ )
Distance from the Axis ( $r$ ): This is the lever arm, the perpendicular distance from the axis to the line of action of the force.
Angle( $θ$ ): The angle between the force and the lever arm.

The formula for torque ( $τ$ ) is:

$τ = F r \sin θ$

Note

Units: Torque is measured in newton-meters (Nm).
Although this is dimensionally equivalent to a joule, torque is never expressed in joules because it is not a form of energy.

Schematic diagram of torque acting on a door.

How Torque Works

Maximum Torque:
- Occurs when the force is perpendicular to the lever arm ( $θ = 90^{\circ}$ ), making $\sin θ = 1$ .
  - In this case, $τ = F r$ .
Zero Torque:
- If the force acts along the line of the lever arm ( $θ = 0^{\circ}$ or $180^{\circ}$ ), then $\sin θ = 0$ , and the torque is zero.

Example

Imagine a wrench turning a bolt.
Applying a force of 50 N at a distance of 0.3 m from the bolt, perpendicular to the wrench, the torque is:

$τ = 50 N \times 0.3 m \times \sin 90^{\circ} = 15 Nm$

Tip

To maximize torque, apply the force as far from the axis as possible and perpendicular to the lever arm.

Equilibrium Conditions: Translational and Rotational

For an object to be in equilibrium, it must satisfy two conditions:

Translational Equilibrium: The net force acting on the object is zero.
Rotational Equilibrium: The net torque acting on the object is zero.

Translational Equilibrium

Definition

Translation equilibrium

An object is in translational equilibrium when the sum of all the external forces acting on the object equals zero.

In other words, the object remains at rest or moves with constant velocity.

Mathematically, this is expressed as:

$\sum \vec{F} = 0$

Rotational Equilibrium

Definition

Rotational equilibrium

Rotational equilibrium is defined as the state of movement where angular acceleration is zero: total torque is zero.

In other words, in rotational equilibrium, the object does not rotate or rotates at a constant angular velocity.

This is expressed as:

$\sum τ = 0$

Tip

When solving equilibrium problems, choose an axis of rotation that simplifies calculations by eliminating torques from forces passing through the axis.

Example question

Applying torque

Consider a seesaw with a child weighing 300 N sitting 1.5 m from the pivot. To balance the seesaw, another child weighing 200 N must sit on the opposite side.

How far should the second child sit?

Solution

Calculate Torque for the First Child: $τ_{1} = 300 N \times 1.5 m = 450 Nm$
Set Torques Equal for Equilibrium: $τ_{2} = 200 N \times d = 450 Nm$
Solve for $d$ : $d = \frac{450 Nm}{200 N} = 2.25 m$

The second child should sit 2.25 m from the pivot.

Rotational Kinematics: Angular Displacement, Velocity, and Acceleration

Rotational motion is described using angular quantities, analogous to linear motion.

Angular Displacement ( $θ$ )

Definition

Angular displacement

Angular displacement is the angle through which an object moves on a circular path. It is measured in radians.

Note

1 radian is the angle subtended by an arc equal in length to the radius of the circle.

Angular Velocity ( $ω$ )

Definition

Angular velocity

Angular velocity ( $ω$ ) is the rate of change of angular displacement. It is measured in radians per second (rad/s).

It is defined as:

$ω = \frac{Δ θ}{Δ t}$

Note

The unit of angular velocity is radians per second (rad/s).

Angular Acceleration ( $α$ )

Definition

Angular acceleration

Angular acceleration is the rate of change of angular velocity.

It is defined as:

$α = \frac{Δ ω}{Δ t}$

Note

The unit of angular acceleration is radians per second squared (rad/s²).

Analogy

Think of angular quantities as the rotational counterparts to linear quantities:

Position( $s$ ) ↔Angular Displacement( $θ$ )
Velocity( $v$ ) ↔Angular Velocity( $ω$ )
Acceleration( $a$ ) ↔Angular Acceleration( $α$ )

Schematic drawing showing the components of rotational kinematics.

Equations of Rotational Motion

The equations of rotational motion mirror those of linear motion, with angular quantities replacing linear ones.

Rotational Motion Equations

$θ = ω_{i} t + \frac{1}{2} α t^{2}$
$ω = ω_{i} + α t$
$ω^{2} = ω_{i}^{2} + 2 α θ$

Tip

Always express angles in radians when using these equations.

Example question

Applying angular quantities

A wheel starts from rest and accelerates at $2 {rad/s}^{2}$ for $5 s$ . How much angular displacement does it undergo?

Solution

Given: $ω_{i} = 0$ , $α = 2 {rad/s}^{2}$ , $t = 5 s$
Use: $θ = ω_{i} t + \frac{1}{2} α t^{2}$ $θ = 0 + \frac{1}{2} \times 2 \times 5^{2} = 25 rad$

Bridging Linear and Rotational Motion

Rotational motion is deeply connected to linear motion. For an object rotating in a circle of radius $r$ :

Linear Velocity: $v = ω r$
Linear Acceleration: $a = α r$

Common Mistake

Students often confuse angular and linear quantities.
Remember, angular velocity ( $ω$ ) is the same for all points on a rotating object, but linear velocity ( $v$ ) varies with distance from the axis.

Reflection and Broader Implications

Self review

What are the conditions for an object to be in equilibrium?
How do angular velocity and linear velocity relate for a point on a rotating object?
Can you derive the rotational motion equations from their linear counterparts?

Theory of Knowledge

How does our understanding of rotational motion influence engineering and design? Consider applications like wind turbines or gyroscopes.

You've read 2/2 free chapters this week.

Upgrade to PLUS or PRO to unlock all notes, for every subject.

A.4.1 Torque and rotational motion

Torque and Its Role in Rotation

What is Torque?

How Torque Works

Equilibrium Conditions: Translational and Rotational

Translational Equilibrium

Rotational Equilibrium

Applying torque

Rotational Kinematics: Angular Displacement, Velocity, and Acceleration

Angular Displacement (θ)

Angular Velocity (ω)

Angular Acceleration (α)

Equations of Rotational Motion

Rotational Motion Equations

Applying angular quantities

Bridging Linear and Rotational Motion

Reflection and Broader Implications

You've read 2/2 free chapters this week.

Introduction to Torque

Angular Displacement ( $θ$ )

Angular Velocity ( $ω$ )

Angular Acceleration ( $α$ )