Chapter 20 : Rotational Motion

Introduction

Rotational motion is a fundamental aspect of physics that describes the motion of objects as they rotate around an axis. While linear motion involves movement in a straight line, rotational motion involves the circular or angular movement of an object. Understanding rotational motion is crucial in various scientific disciplines, including physics and engineering.

Basics of Rotational Motion

Angular Displacement

Angular displacement, denoted by θ (theta), represents the angle through which an object has rotated. It is measured in radians and is positive for counterclockwise rotation and negative for clockwise rotation. The angular displacement is related to the linear displacement (distance traveled) by the formula:

Angular Displacement (θ)=Linear Displacement (s)/Radius (r)

Angular Velocity

Angular velocity (ω) is the rate of change of angular displacement with respect to time. It is measured in radians per second (rad/s). The relationship between angular velocity, angular displacement, and time is given by:

Angular Velocity (ω)=Change in Angular Displacement (Δθ)/Change in Time (Δt)

Angular Acceleration

Angular acceleration (α) is the rate of change of angular velocity with respect to time. It is measured in radians per second squared (rad/s²). The relationship between angular acceleration, change in angular velocity, and time is expressed by the equation:

Angular Acceleration (α)=Change in Angular Velocity (Δω)/Change in Time (Δt)

Kinematics of Rotational Motion

Rotational Kinematic Equations

Similar to linear motion, rotational motion has kinematic equations that relate angular displacement, angular velocity, angular acceleration, and time. The three fundamental rotational kinematic equations are:

θ=ω₀t+(1/2)αt²

ω=ω₀+αt

θ=ω₀t+1/2αt²

Where:

• θ is the angular displacement,
• ω is the final angular velocity,
• ω₀​ is the initial angular velocity,
• α is the angular acceleration,
• t is the time.

Dynamics of Rotational Motion

Torque

In rotational motion, torque (τ) plays a role similar to force in linear motion. Torque is the measure of the tendency of a force to rotate an object about an axis. Mathematically, torque is defined as the product of force (F) and the perpendicular distance (r) from the axis of rotation to the point where the force is applied:

τ=r⋅F

Moment of Inertia

The moment of inertia (II) is a measure of an object's resistance to changes in its rotation. It depends on both the mass distribution of the object and the axis of rotation. The formula for moment of inertia is given by:

I=∑m_ir_i²

Where:

• m_i​ is the mass of the i^th particle,
• r_i​ is the perpendicular distance of the i^th particle from the axis of rotation,
• The sum is taken over all particles in the object.

Newton's Second Law for Rotation

Newton's second law for rotation relates the net torque applied to an object to its angular acceleration. The equation is:

τ=Iα

Conservation of Angular Momentum

Similar to linear momentum, angular momentum (L) is conserved in the absence of external torques. The conservation of angular momentum is expressed by the equation:

L=Iω

This principle explains phenomena like the rapid rotation of an ice skater pulling in their arms, showcasing the conservation of angular momentum.

Conclusion

Rotational motion is a fascinating and complex topic with applications ranging from the spinning of wheels to the rotation of planets. Understanding the principles of angular displacement, angular velocity, torque, and angular momentum is crucial for physicists, engineers, and anyone interested in comprehending the dynamics of rotating systems. The concepts presented in this chapter provide a foundation for further exploration into advanced topics like rotational energy, oscillations, and more specialized applications in various fields.