7.1.2 Rotational Motion Explained

Key Concepts

1. Definition of Rotational Motion

Rotational motion is the movement of an object around a fixed axis, where every point in the object moves in a circular path around the axis.

2. Angular Displacement

Angular displacement is the angle through which an object rotates about a fixed axis. It is measured in radians (rad) or degrees (°).

3. Angular Velocity

Angular velocity is the rate of change of angular displacement with respect to time. It is represented by the symbol ω (omega) and is measured in radians per second (rad/s).

4. Angular Acceleration

Angular acceleration is the rate of change of angular velocity with respect to time. It is represented by the symbol α (alpha) and is measured in radians per second squared (rad/s²).

5. Torque

Torque is the rotational equivalent of force. It is the tendency of a force to rotate an object about an axis. Torque is represented by the symbol τ (tau) and is measured in newton-meters (N·m).

6. Moment of Inertia

Moment of inertia is the measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution of the object and the axis of rotation. It is represented by the symbol I and is measured in kilogram-meter squared (kg·m²).

7. Rotational Kinetic Energy

Rotational kinetic energy is the kinetic energy due to the rotation of an object. It is given by the formula K = ½ Iω², where I is the moment of inertia and ω is the angular velocity.

Detailed Explanation

Definition of Rotational Motion

Rotational motion occurs when an object moves around a fixed axis, such as a spinning top or a rotating wheel. Every point on the object follows a circular path around the axis.

Angular Displacement

Angular displacement (θ) is the angle through which an object rotates. For example, if a wheel rotates 360°, its angular displacement is 2π radians (360° = 2π rad).

Angular Velocity

Angular velocity (ω) is the rate at which an object changes its angular displacement. For example, if a wheel completes one full rotation (2π rad) in 2 seconds, its angular velocity is ω = 2π/2 = π rad/s.

Angular Acceleration

Angular acceleration (α) is the rate at which an object changes its angular velocity. For example, if a wheel's angular velocity increases from 0 to 2π rad/s in 4 seconds, its angular acceleration is α = (2π - 0)/4 = π/2 rad/s².

Torque

Torque (τ) is the force that causes rotational motion. It depends on the magnitude of the force, the distance from the axis of rotation, and the angle between the force and the radius. For example, applying a force of 10 N at a distance of 2 m from the axis with a 90° angle results in a torque of τ = 10 N × 2 m × sin(90°) = 20 N·m.

Moment of Inertia

Moment of inertia (I) is a measure of an object's resistance to rotational motion. It depends on the mass distribution and the axis of rotation. For example, a solid cylinder rotating about its central axis has a moment of inertia given by I = ½ MR², where M is the mass and R is the radius.

Rotational Kinetic Energy

Rotational kinetic energy (K) is the energy an object possesses due to its rotation. For example, a rotating wheel with a moment of inertia of 2 kg·m² and an angular velocity of 5 rad/s has a rotational kinetic energy of K = ½ × 2 kg·m² × (5 rad/s)² = 25 J.

Examples and Analogies

Example: Rotational Motion of a Bicycle Wheel

When you pedal a bicycle, the wheels rotate around their central axis. The angular displacement of the wheel increases as you pedal, and the angular velocity depends on how fast you pedal.

Analogy: Rotational Motion as a Spinning Top

Think of rotational motion as a spinning top. The top rotates around its central axis, and the faster it spins, the higher its angular velocity.

Example: Angular Displacement of a Clock Hand

A clock hand rotates around its axis. For example, the minute hand of a clock rotates 360° every hour, so its angular displacement is 2π radians per hour.

Analogy: Angular Velocity as a Car Wheel

Consider a car wheel rotating as the car moves. The faster the car moves, the higher the angular velocity of the wheel.

Example: Angular Acceleration of a Flywheel

A flywheel in a machine accelerates from rest to a certain angular velocity. If it reaches 10 rad/s in 5 seconds, its angular acceleration is α = (10 - 0)/5 = 2 rad/s².

Analogy: Angular Acceleration as a Spinning Figure Skater

Think of a figure skater spinning. When the skater pulls in their arms, their angular velocity increases, indicating angular acceleration.

Example: Torque in a Lever

A lever uses torque to lift a heavy object. If you apply a force of 50 N at a distance of 0.5 m from the fulcrum, the torque is τ = 50 N × 0.5 m = 25 N·m.

Analogy: Torque as a Door

Consider opening a door. The farther you apply the force from the hinge, the easier it is to open the door, demonstrating torque.

Example: Moment of Inertia of a Rotating Disk

A disk rotating about its central axis has a moment of inertia given by I = ½ MR². For a disk with mass 2 kg and radius 0.5 m, I = ½ × 2 kg × (0.5 m)² = 0.25 kg·m².

Analogy: Moment of Inertia as a Merry-Go-Round

Think of a merry-go-round. The more massive it is and the farther the mass is from the center, the harder it is to start or stop its rotation, demonstrating moment of inertia.

Example: Rotational Kinetic Energy of a Spinning Wheel

A spinning wheel with a moment of inertia of 1 kg·m² and an angular velocity of 10 rad/s has a rotational kinetic energy of K = ½ × 1 kg·m² × (10 rad/s)² = 50 J.

Analogy: Rotational Kinetic Energy as a Windmill

Consider a windmill. The faster the blades spin due to the wind, the more rotational kinetic energy they possess.