Torque in Rotational Motion - Notes for Quick Revision 📚

Torque in Rotational Motion - Rotational motion is a fundamental topic in physics that deals with objects rotating around an axis. A central concept in rotational motion is torque, which measures the effectiveness of a force in producing rotational acceleration. Torque is essential for understanding systems like levers, gears, engines, and other rotational mechanisms.

This article provides comprehensive notes on torque, its definition, mathematical formulation, factors affecting it, real-world applications, and essential equations.

What is Torque?

Definition: Torque is the rotational equivalent of force. It measures the turning effect of a force applied to an object about an axis.

Formula:

$$\tau =rF\sin \theta$$ Where:

• $\tau$: Torque (measured in Newton-meters, $\text{Nm}$)
• $r$: Perpendicular distance (lever arm) from the axis of rotation to the line of action of the force (in meters).
• $F$: Magnitude of the applied force (in Newtons).
• $\theta$: Angle between the force vector and the lever arm.

Characteristics of Torque

Vector Quantity:

• Torque has both magnitude and direction. Its direction is determined using the right-hand rule.
• If you curl the fingers of your right hand in the direction of rotation caused by the force, your thumb points in the direction of the torque.

Unit of Torque:

• SI Unit: Newton-meter (Nm)
• CGS Unit: dyne-cm

Sign Convention:

• Positive Torque ($+\tau$): When the force causes a counterclockwise rotation.
• Negative Torque ($-\tau$): When the force causes a clockwise rotation.

Factors Affecting Torque

Magnitude of Force ($F$):

• Larger forces produce greater torque, assuming the same lever arm and angle.

Distance from Axis ($r$):

• Torque increases with the perpendicular distance between the axis of rotation and the point of force application.

Angle of Force ($\theta$):

• Maximum torque occurs when $\theta =90^{\circ }$, as $\sin 90^{\circ }=1$.
• No torque is produced if $\theta =0^{\circ }$ or $\theta =180^{\circ }$, as $\sin 0^{\circ }=0$.

Torque in Rotational Motion

1. Torque and Angular Acceleration

Torque is related to angular acceleration ($\alpha$) through the moment of inertia ($I$):

$$\tau = I\alpha$$

Where:

• $I$: Moment of inertia of the object about the axis of rotation.
• $\alpha$: Angular acceleration (rad/s²).

2. Torque and Rotational Equilibrium

An object is in rotational equilibrium if the net torque acting on it is zero:

$$\sum \tau = 0$$

This condition ensures the object remains at rest or rotates at a constant angular velocity.

Applications of Torque

1. Lever Systems

Torque is the basis of lever mechanics. The principle of levers states:

$$F_1r_1 = F_2r_2$$

This relationship allows small forces to lift heavy objects by increasing the lever arm.

2. Gears and Machines

Torque is transmitted between gears, pulleys, and engines to perform work efficiently.

Adjusting gear ratios alters the torque and angular velocity in vehicles and machinery.

3. Engineering Structures

Torque calculations are critical for designing beams, bridges, and rotating components to ensure structural stability.

4. Sports

In sports like golf, baseball, and cricket, applying torque through a bat or club determines the speed and direction of the ball.

5. Everyday Tools

Wrenches and screwdrivers rely on torque to tighten or loosen bolts effectively.

The Relation Between Torque and Work

Torque contributes to rotational work when applied to an object causing angular displacement:

$$W = \tau \theta$$

Where:

• $W$: Work done (in Joules)
• $\tau$: Torque (in Nm)
• $\theta$: Angular displacement (in radians)

Torque in Terms of Angular Momentum

Torque is the time rate of change of angular momentum ($L$):

$$\tau = \frac{dL}{dt}$$

Where:

• $L = I\omega$ (angular momentum).
• $\omega$: Angular velocity.

Difference Between Torque and Force

Torque	Force
Causes rotational motion.	Causes linear motion.
Depends on the lever arm.	Acts directly on the object.
SI Unit: Newton-meter ($\text{Nm}$).	SI Unit: Newton ($\text{N}$).

Examples of Torque in Action

Opening a Door:

Applying force at the edge (farthest from the hinges) maximizes torque and makes it easier to open.

Tightening a Nut with a Wrench:

A longer wrench increases the lever arm, generating more torque for the same force.

Seesaw:

A seesaw balances when the torques on either side are equal.

FAQs About Torque

What is the significance of torque?

Torque determines the rotational effect of a force and is crucial for analyzing rotational systems in mechanics.

Can torque be zero if force is applied?

Yes, if the force is applied at the axis of rotation ($r = 0$) or along the line of action of the lever arm ($\theta = 0^\circ$).

How does torque relate to energy?

Torque does rotational work when it causes angular displacement, contributing to the system's rotational kinetic energy.

What is the difference between torque and moment of force?

Torque and moment of force are often used interchangeably, though "torque" is more commonly used in rotational dynamics.

Torque is a fundamental concept in physics, bridging the gap between linear and rotational motion. Understanding how torque works, its dependence on force, distance, and angles, and its role in real-world applications equips students with the knowledge to solve complex rotational dynamics problems. Use these notes to master torque and its significance in various physical and engineering contexts!

Explore related topics:

• Rotational Motion and Moment of Inertia
• Angular Momentum in Rotational Motion
• Thermodynamics