Rotational Motion

Introduction

Rotational motion is a fundamental concept in physics, describing the movement of objects around a fixed axis. This topic includes the study of torque, rigid body equilibrium, and moment of inertia, which are essential for understanding how forces cause objects to rotate and how their mass distribution affects rotational dynamics.

Torque

Definition and Properties

Torque is a measure of the effectiveness of a force in causing an object to rotate about a pivot point or axis. It depends on three main factors:

• Magnitude of the force

• Distance from the pivot (lever arm)

• Direction of the force with respect to the pivot

The mathematical definition of torque () is: where:

• r: Distance from the pivot to the point of force application

• F: Magnitude of the applied force

• \theta: Angle between the force vector and the lever arm

• F_{\perp}: Component of force perpendicular to the lever arm

Units: Newton-meter (N·m)

Examples

• Applying a force at the end of a wrench to loosen a bolt: The longer the wrench (greater r), the greater the torque for the same force.

• Opening a door: Pushing farther from the hinges increases the torque.

Calculating Torque

There are two equivalent methods to calculate torque:

• Method 1:

• Method 2:

The lever arm or moment arm is the perpendicular distance from the axis of rotation to the line of force.

Common Mistakes

• The lever arm must be perpendicular to the force, not just to the object.

Centre of Mass

Definition and Role in Torque Calculations

The centre of mass of an object is the point at which its mass can be considered to be concentrated for the purposes of analyzing translational and rotational motion. For torque calculations involving gravity, the force of gravity is assumed to act at the centre of mass.

• The centre of mass is not always the same as the pivot point.

• For uniform objects, the centre of mass is at the geometric center.

Adding Torques

Net Torque

The net torque acting on a system is the vector sum of all individual torques:

• Torques that cause rotation in opposite directions are assigned opposite signs (e.g., clockwise vs. counterclockwise).

Example Calculation

Consider a bar of length 4 m with three forces applied at different points and angles:

• N at

• N at

• N at

Calculate the torque about the left end:

• (if force acts through the pivot)

Net torque:

Rigid Body Equilibrium

Conditions for Equilibrium

A rigid body is in equilibrium if both translational and rotational equilibrium are satisfied:

• Translational equilibrium:

• Rotational equilibrium:

This means the object does not accelerate linearly or rotate.

Choosing the Pivot Point

• The pivot point can be chosen to simplify calculations, often to eliminate unknown forces from the torque equation.

• For statics problems, any point can be used as the pivot.

Example: Ladder Against a Wall

A ladder of length and mass leans against a frictionless wall at angle . To prevent slipping, the minimum coefficient of friction at the base must be found by setting both and .

• Normal force and friction act at the base.

• Gravity acts at the centre of mass.

• Wall provides a normal force but no friction.

*Additional info: The solution involves balancing torques about the base and using to solve for .*

Moment of Inertia

Definition

The moment of inertia () quantifies how mass is distributed relative to the axis of rotation and determines an object's resistance to changes in rotational motion. where is the mass of the -th particle and is its distance from the axis.

• Units: kg·m2

• Depends on both mass and its distribution relative to the axis.

Moment of Inertia for Common Shapes

*Additional info: The moment of inertia changes if the axis of rotation changes, even for the same object.*

Rotational Kinetic Energy

Formula and Explanation

Rotational kinetic energy is the energy due to the rotation of an object and is given by: where:

• : Moment of inertia

• : Angular velocity

This formula is analogous to the translational kinetic energy .

Summary Table: Key Quantities in Rotational Motion

Applications

• Engineering: Design of rotating machinery, flywheels, and vehicle axles.

• Sports: Analysis of rotational motion in gymnastics, diving, and weightlifting.

• Everyday life: Opening doors, using tools, and playground equipment.