Rotation of Rigid Bodies and Torque – Study Notes

Rotation of Rigid Bodies and Torque

Introduction to Rotational Motion

Rotational motion describes the movement of objects around a fixed axis. Unlike linear motion, where all points move in the same direction, in rotational motion, all points rotate through the same angle in a given time interval. The position of a rotating object can be specified by the angle θ it makes with a reference axis.

Angular Velocity and Angular Acceleration

Just as velocity and acceleration describe linear motion, angular velocity (ω) and angular acceleration (α) describe rotational motion. These quantities are defined as follows:

• Instantaneous angular velocity: $\omega_z = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$

• Instantaneous angular acceleration: $\alpha_z = \lim_{\Delta t \to 0} \frac{\Delta \omega_z}{\Delta t} = \frac{d\omega_z}{dt}$

For constant angular acceleration, the equations of motion are analogous to those for linear acceleration:

Angular Displacement and Radians

The angle θ is measured in radians, where one radian is the angle subtended by an arc length equal to the radius of the circle. The relationship between arc length s, radius r, and angle θ is:

• $\theta = \frac{s}{r}$

• $s = r\theta$

Linking Linear and Angular Kinematics

For a point at a distance r from the axis of rotation, the linear (tangential) speed v is related to the angular speed ω by:

• $v = r\omega$

The tangential and radial (centripetal) accelerations are given by:

• Tangential acceleration: $a_{tan} = r\alpha$

• Radial (centripetal) acceleration: $a_{rad} = r\omega^2$

Moment of Inertia

The moment of inertia (I) quantifies how mass is distributed relative to the axis of rotation and determines how difficult it is to change an object's rotational motion. It is defined as:

• $I = \sum_i m_i r_i^2$

Objects with mass farther from the axis have a larger moment of inertia and are harder to rotate.

Common Moments of Inertia

Object	Moment of Inertia (I)
Solid sphere	$\frac{2}{5}MR^2$
Thin-walled hollow sphere	$\frac{2}{3}MR^2$
Slender rod (center)	$\frac{1}{12}ML^2$
Slender rod (end)	$\frac{1}{3}ML^2$
Solid cylinder	$\frac{1}{2}MR^2$
Thin-walled hollow cylinder	$MR^2$

Rotational Kinetic Energy

The kinetic energy of a rotating rigid body is given by:

• $K = \frac{1}{2}I\omega^2$

Torque

Torque (τ) is the rotational equivalent of force. It measures the tendency of a force to rotate an object about an axis. The magnitude of torque is given by:

• $\tau = rF\sin\theta$

Where r is the distance from the axis of rotation to the point of application of the force, F is the magnitude of the force, and θ is the angle between the force and the lever arm.

Forces applied farther from the axis or at a greater angle produce more torque. The direction of torque is determined by the right-hand rule.

Torque and Angular Acceleration (Newton's Second Law for Rotation)

Newton's second law for rotation relates net torque to angular acceleration:

• $\sum \tau_z = I\alpha_z$

This equation is the rotational analogue of $\sum F_x = ma_x$ for linear motion.

Worked Example: Calculating Torque

Consider a force of 9.0 N applied tangentially to a wheel of radius 0.120 m. The torque produced is:

• $\tau = rF = 0.120 \times 9.0 = 1.08\ \text{N} \cdot \text{m}$

Summary Table: Linear vs. Rotational Quantities

Linear Motion	Rotational Motion
Displacement: $x$	Angular displacement: $\theta$
Velocity: $v$	Angular velocity: $\omega$
Acceleration: $a$	Angular acceleration: $\alpha$
Mass: $m$	Moment of inertia: $I$
Force: $F$	Torque: $\tau$
Newton's 2nd Law: $F = ma$	Rotational analogue: $\tau = I\alpha$

Additional info: The above notes synthesize and expand upon the provided lecture slides, integrating standard textbook context for clarity and completeness.