Torque and Angular Acceleration

Overview

Torque produces angular acceleration in rotational motion in the same way that force produces linear acceleration in translational motion.

Newton’s second law for linear motion is:

F = ma

Its rotational form is:

τ = Iα

Learning Outcomes

Students should be able to:

• Define torque and angular acceleration.
• Explain the connection between torque and angular acceleration.
• Derive τ = Iα for a point mass.
• Compare F = ma with τ = Iα.
• Calculate torque, moment of inertia or angular acceleration.
• Determine the direction and sign of angular acceleration.

1. Torque

Torque is the turning effect produced by a force about an axis.

Its vector form is:

τ⃗ = r⃗ × F⃗

Its magnitude is:

τ = r F sin θ

Where:

• τ = torque
• r = perpendicular distance from the axis
• F = applied force
• θ = angle between r⃗ and F⃗

When the force is tangential:

θ = 90°

Therefore:

τ = r F sin 90° τ = r F

2. Angular Acceleration

Angular acceleration is the rate at which angular velocity changes.

α = Δω / Δt

Where:

• α = angular acceleration
• Δω = change in angular velocity
• Δt = time interval

Its SI unit is:

rad s⁻²

Its dimensions are:

[α] = [T⁻²]

3. Direction of Torque

Torque is a vector quantity. Its direction is found using the right-hand rule:

• Curl the fingers of your right hand in the direction in which the force tends to rotate the object.
• Your thumb gives the direction of torque.
• Angular acceleration acts in the direction of the net torque.

A common sign convention is:

• Anticlockwise torque: positive
• Clockwise torque: negative

The opposite convention may also be used, but it must remain consistent throughout the calculation.

Intermediate Level

4. Derivation of τ = Iα

Consider a point mass m rotating at a distance r from an axis. A tangential force F acts on it.

Torque is:

τ = r F sin θ

For tangential force:

θ = 90°

Therefore:

τ = r F

According to Newton’s second law:

F = m a_t

Tangential acceleration and angular acceleration are related by:

a_t = r α

Substituting:

F = m (r α) F = m r α

Now substitute this value of force into τ = r F:

τ = r (m r α) τ = m r² α

For a point mass:

I = m r²

Therefore:

τ = I α

This is the rotational form of Newton’s second law.

Torque and Angular Acceleration

Searchable formulas: τ = r F sin θ · τ = r F · τ_net = Iα · α = τ_net / I

5. Meaning of τ = Iα

From:

τ = I α

Angular acceleration is:

α = τI

Therefore:

α ∝ τ

For a constant moment of inertia, greater net torque produces greater angular acceleration.

Also:

α ∝ 1I

For the same torque, an object with greater moment of inertia has smaller angular acceleration.

6. Net Torque

When several torques act on an object, their signs and directions must be considered.

τ_net = Σ τ

The complete rotational law is:

τ_net = I α

For example, if anticlockwise is positive:

τ_net = τ_anticlockwise − τ_clockwise

Only the net torque determines angular acceleration.

7. Linear and Rotational Motion Comparison

Linear and Rotational Motion Comparison

Important Comparison

Mass represents resistance to linear acceleration.

Moment of inertia represents resistance to angular acceleration.

Force changes linear motion.

Torque changes rotational motion.

8. Special Conditions

Force passing through the axis

If the line of action of force passes through the axis:

r_⊥ = 0

Therefore:

τ = 0

Such a force cannot produce angular acceleration about that axis.

Radial force

For a radial force:

θ = 0°

Therefore:

τ = r F sin 0° τ = 0

Tangential force

For a tangential force:

θ = 90°

Therefore:

τ_max = r F

A tangential force produces the maximum torque for a given r and F.

Zero net torque

If:

τ_net = 0

Then:

I α = 0

For a non-zero moment of inertia:

α = 0

The object either remains at rest or continues rotating with constant angular velocity.

9. SI Unit and Dimensions of Torque

From:

τ = I α

The unit of moment of inertia is:

kg m²

The unit of angular acceleration is:

rad s⁻²

Since the radian is dimensionless:

[τ] = kg m² s⁻²

Therefore:

SI unit of torque = N m

Dimensional formula:

[τ] = [ML²T⁻²]

Torque and energy have the same dimensions, but they are different physical quantities. Torque is measured in N m, while energy is written in joules.

Key Formula Summary

Key Formula Summary

Solved Examples

Example 1: Angular Acceleration of a Wheel

A wheel has a moment of inertia of 2.5 kg m². A net torque of 10 N m acts on it. Calculate its angular acceleration.

Given:

I = 2.5 kg m² τ = 10 N m

Formula:

α = τI

Solution:

α = 102.5 α = 4 rad s⁻²

Final answer:

α = 4 rad s⁻²

Example 2: Torque Produced by a Tangential Force

A tangential force of 15 N acts at the rim of a wheel of radius 0.40 m. Calculate the torque.

Given:

F = 15 N r = 0.40 m θ = 90°

Formula:

τ = r F

Solution:

τ = (0.40)(15) τ = 6 N m

Final answer:

τ = 6 N m

Example 3: Required Torque

A rotating body has moment of inertia 0.80 kg m². Find the torque required to produce an angular acceleration of 12 rad s⁻².

Given:

I = 0.80 kg m² α = 12 rad s⁻²

Formula:

τ = I α

Solution:

τ = (0.80)(12) τ = 9.6 N m

Final answer:

τ = 9.6 N m

Example 4: Calculating Moment of Inertia

A torque of 18 N m produces an angular acceleration of 6 rad s⁻². Find the moment of inertia.

Formula:

I = τα

Solution:

I = 186 I = 3 kg m²

Final answer:

I = 3 kg m²

Example 5: Braking Torque

A wheel of moment of inertia 3 kg m² experiences a braking torque of −12 N m. Calculate its angular acceleration.

Formula:

α = τI

Solution:

α = −123 α = −4 rad s⁻²

Final answer:

α = −4 rad s⁻²

The negative sign indicates that the angular acceleration acts opposite to the original rotation.

Real-Life Applications

• A door rotates more easily when pushed far from its hinges.
• A mechanic applies torque using a wrench.
• Bicycle pedals produce torque around the crank axis.
• Electric motors produce torque to rotate machinery.
• Brakes apply torque opposite to the direction of rotation.
• A light wheel accelerates more easily than a heavy wheel when equal torques act.

Important Exam Facts

• Torque is the rotational effect of force.
• Angular acceleration is produced by net torque.
• The main formula is τ_net = Iα.
• Moment of inertia plays the same role in rotation that mass plays in linear motion.
• A tangential force produces maximum torque.
• A radial force produces zero torque.
• Angular acceleration acts in the direction of net torque.
• A larger moment of inertia produces smaller angular acceleration for the same torque.
• Always calculate net torque before applying τ = Iα.

Common Mistakes

• Using force directly instead of calculating torque.
• Forgetting the factor sin θ.
• Using total distance instead of perpendicular distance.
• Using one torque when several torques act.
• Ignoring clockwise and anticlockwise signs.
• Confusing angular acceleration with centripetal acceleration.
• Writing the unit of torque as joules instead of N m.
• Applying τ = Iα using the wrong rotation axis.

MCQs with Answers

Short-Question Preparation

Short-Question Preparation

Long-Question Preparation

A complete derivation should include:

• Definition of torque.
• Equation τ = r F sin θ.
• Tangential-force condition θ = 90°.
• Simplification to τ = r F.
• Newton’s second law F = m a_t.
• Relation a_t = r α.
• Substitution F = m r α.
• Substitution into the torque equation.
• Result τ = m r² α.
• Identification I = m r².
• Final result τ = Iα.
• Explanation of each symbol.
• SI units.
• Dimensional formula of torque.
• Comparison with F = ma.
• Effect of increasing torque.
• Effect of increasing moment of inertia.
• Direction of torque and angular acceleration.

Sticky Notes

Sticky Notes

• Core formula: τ_net = Iα
• Memory comparison: Force changes linear motion; torque changes rotational motion.
• Maximum torque: Apply force perpendicular to the radius.
• Direction rule: Angular acceleration follows the direction of net torque.
• Exam warning: Use net torque, not merely one of the acting torques.

Ask-the-Tutor Modes

Students may select:

• Explain τ = Iα in simpler words.
• Show the derivation one step per line.
• Compare linear and rotational motion.
• Explain clockwise and anticlockwise signs.
• Check a numerical solution and its units.

60-Second Revision

τ = r F sin θ at = r α I = m r² τnet = Iα α = τ_net / I Torque produces angular acceleration. Tangential force produces maximum torque. Radial force produces zero torque. Greater torque means greater angular acceleration. Greater moment of inertia means smaller angular acceleration.