Complete Rotational Motion Summary

1. Rotational Kinematics

Rotational kinematics describes the motion of rotating bodies without considering the forces that cause the rotation.

1.1 Angular Displacement

Angular displacement \(\theta\) is the angle through which an object rotates. It is given by:

\[\theta = \frac{s}{r}\]

Where:

\(s\) = arc length
\(r\) = radius of the circular path

1.2 Angular Velocity

The average angular velocity of a body for a finite time interval is given by:

\[\omega_{\text{av}} = \frac{\Delta\theta}{\Delta t} = \frac{\theta_f - \theta_i}{t_f - t_i}\]

The instantaneous angular velocity is defined as:

\[\omega = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}\]

The unit of angular velocity is radian per second (rad/s).

1.3 Relation with Period and Frequency

In terms of period \(T\) and frequency \(f\), the angular velocity is given by:

\[\omega = \frac{2\pi}{T} = 2\pi f\]

1.4 Relation Between Linear and Angular Speed

The relation between linear speed and angular speed is given by:

\[\omega = \frac{v}{r}\] \[v = \omega r\]

Although all particles have the same angular velocity, their speeds increase linearly with distance from the axis of rotation.

1.5 Equations of Rotational Kinematics

For constant angular acceleration, the equations of rotational kinematics are:

\[\omega = \omega_o + \alpha t\] \[\theta = \theta_o + \omega_0 t + \frac{1}{2} \alpha t^2\] \[\omega^2 = \omega_o^2 + 2\alpha (\theta - \theta_o)\]

Where:

\(\omega_o\) = initial angular velocity
\(\omega\) = final angular velocity
\(\theta_o\) = initial angular displacement
\(\theta\) = final angular displacement
\(\alpha\) = angular acceleration
\(t\) = time

2. Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation rate. It depends on the mass distribution relative to the axis of rotation.

2.1 Definition

For a discrete system of particles, the moment of inertia is defined as:

\[I = \sum m_{i} r_i^2\]

Where:

\( m_i \) = mass of the \( i^{th} \) particle
\( r_i \) = perpendicular distance of the \( i^{th} \) particle from the axis of rotation

For a continuous body, the moment of inertia is given by:

\[I = \int r^2 dm\]

2.2 Parallel Axis Theorem

Parallel Axis Theorem

The moment of inertia of a body about an axis is equal to its moment of inertia about a parallel axis through its centre of mass plus the product of the mass of the body and the square of perpendicular distance between the two axes.

\[I = I_{\text{cm}} + md^2\]

Where:

\( I_{\text{cm}} \) = Moment of inertia about the center of mass
\( m \) = Total mass of the body
\( d \) = Perpendicular distance between the two parallel axes

2.3 Perpendicular Axis Theorem

Perpendicular Axis Theorem

For a planar body (lying in the xy-plane), the moment of inertia about the z-axis is equal to the sum of the moments of inertia about the x and y axes.

\[ I_z = I_x + I_y \]

Where:

\( I_x \) = Moment of inertia about the x-axis
\( I_y \) = Moment of inertia about the y-axis
\( I_z \) = Moment of inertia about the z-axis (perpendicular to the plane)

2.4 Common Moments of Inertia

Object	Axis of Rotation	Moment of Inertia
Thin rod of length L	Through center, perpendicular to length	\( \frac{1}{12} ML^2 \)
Thin rod of length L	Through end, perpendicular to length	\( \frac{1}{3} ML^2 \)
Solid sphere of radius R	Through center	\( \frac{2}{5} MR^2 \)
Hollow sphere of radius R	Through center	\( \frac{2}{3} MR^2 \)
Solid cylinder of radius R	Through central axis	\( \frac{1}{2} MR^2 \)
Hollow cylinder of radius R	Through central axis	\( MR^2 \)

3. Torque and Angular Momentum

3.1 Torque

Torque is a measure of the force that can cause an object to rotate about an axis. It is the rotational analog of force.

The torque of a force \( F \) that acts at a distance \( r \) from the origin is defined as:

\[\tau = r F \sin \theta\]

Where \( \theta \) is the angle between the position vector and the force vector.

In vector form:

\[\vec{\tau} = \vec{r} \times \vec{F}\]

3.2 Newton's Second Law for Rotation

Since torque is a rotational analog of force, Newton's second law for rotational motion is given by:

\[\tau = I \alpha\]

Where:

\( \tau \) = net torque
\( I \) = moment of inertia
\( \alpha \) = angular acceleration

3.3 Rotational Work and Energy

The rotational work done by a force about the fixed axis of rotation is defined as:

\[W_{rot} = \int \tau d\theta\]

Where \( \tau \) is the torque produced by the force, and \( d\theta \) is the infinitesimally small angular displacement about the axis.

The rotational kinetic energy of a body about a fixed rotational axis is defined as:

\[K_{rot} = \frac{1}{2} I \omega^2\]

Where \( I \) is the moment of inertia about the axis.

3.4 Work-Energy Theorem for Rotation

In complete analog to the work-energy theorem for translational motion, it can be stated for rotational motion as:

\[W_{rot} = \Delta K_{rot}\]

The net rotational work done by the forces is equal to the change in rotational kinetic energy of the body.

3.5 Rotational Power

In complete analog with linear motion, the instantaneous rotational power is defined as:

\[P_{rot} = \frac{dW_{rot}}{dt} = \tau \cdot \omega\]

3.6 Angular Momentum

Angular momentum about the origin is defined as:

\[\vec{L} = \vec{r} \times \vec{p}\]

The magnitude of angular momentum is given by:

\[L = r p \sin \theta\]

For a rigid body rotating about a fixed axis:

\[L = I \omega\]

3.7 Conservation of Angular Momentum

Conservation of Angular Momentum

If the net external torque on a system is zero, the total angular momentum of the system remains constant.

\[\text{If } \tau_{ext} = 0, \text{ then } \frac{dL}{dt} = 0 \text{ and } L = \text{constant}\]

3.8 Angular Impulse

Angular impulse is defined as:

\[\tau = \int \tau_{ext} dt\]

4. Rolling Motion

Rolling motion is a combination of translational motion of the center of mass and rotational motion about the center of mass.

4.1 Pure Rolling Motion

In pure rolling motion (rolling without slipping), the point of contact between the rolling body and the surface is instantaneously at rest.

The condition for pure rolling is:

\[v_c = \omega R\]

Where:

\( \omega \) = angular velocity of the wheel about its center of mass
\( v_c \) = linear velocity of the center of mass
\( R \) = radius of the rolling object

4.2 Velocity of Points on a Rolling Body

Since rolling is a combination of translation of the center and rotation about the center, the velocity of any point on the rim is the vector sum:

\[\vec{v} = \vec{v_c} + \vec{v'}\]

Where:

\( \vec{v_c} \) = velocity of the center of mass
\( \vec{v'} \) = velocity of the particle with respect to center of mass

4.3 Kinetic Energy of a Rolling Body

The total kinetic energy of a rolling body is given by:

\[K = \frac{1}{2} m v_c^2 + \frac{1}{2} I_c \omega^2\]

Where:

\( m \) = mass of the body
\( v_c \) = velocity of the center of mass
\( I_c \) = moment of inertia about the center of mass
\( \omega \) = angular velocity

In pure rolling motion, \( v_c = \omega R \), so:

\[K = \frac{1}{2} m (\omega R)^2 + \frac{1}{2} I_c \omega^2\] \[K = \frac{1}{2} (I_c + m R^2) \omega^2\]

Key Points:

In pure rolling, the point of contact is at rest relative to the surface
The total kinetic energy is the sum of translational and rotational kinetic energies
For objects with different mass distributions, the fraction of energy in rotational form varies
A solid sphere will roll faster down an incline than a hollow cylinder of the same mass and radius