### Rotational Motion About A Fixed Axis Kinematics And Dynamics

**Concepts to remember for Rotational Motion About a Fixed Axis-Kinematics and Dynamics:**

**Kinematics:**

**Angular displacement, $\theta$**: Measured in radians. Represents the amount of rotation about the axis.**Angular velocity, $\omega$**: Measured in radians per second. Represents the rate of change of angular displacement.**Angular acceleration, $\alpha$**: Measured in radians per second squared. Represents the rate of change of angular velocity.**Relations between linear and angular quantities**:- $v = r\omega$ (Linear speed = radius × angular velocity)
- $a = r\alpha$ (Linear acceleration = radius × angular acceleration)

**Rotational equations of motion**:- $\omega_f = \omega_i + \alpha t$ (Final angular velocity = Initial angular velocity + Angular acceleration × Time)
- $\theta_f = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$ (Final angular displacement = Initial angular displacement + Initial angular velocity × Time + ½ × Angular acceleration × Time²)
- $\alpha = \frac{\omega_f - \omega_i}{t}$ (Angular acceleration = (Final angular velocity - Initial angular velocity) / Time)

**Rolling motion**: Combination of rotational and translational motion.

**Dynamics:**

**Torque, $\tau$**: A measure of the force causing rotation. Perpendicular to both the force vector and the displacement vector.**Moment of inertia, $I$**: A measure of an object’s resistance to angular acceleration. Depends on the object’s mass distribution.**Parallel axis theorem**: The moment of inertia about an axis parallel to the axis through the center of mass is given by $I = I_{CM} + Md^2$, where $I_{CM}$ is the moment of inertia about the center of mass, $M$ is the mass, and $d$ is the distance between the axes.**Perpendicular axis theorem**: The moment of inertia about an axis perpendicular to two other perpendicular axes is given by $I = I_x + I_y$, where $I_x$ and $I_y$ are the moments of inertia about the two other perpendicular axes.**Work and energy in rotational motion**: Work done on an object in rotation equals the change in its rotational kinetic energy. Rotational kinetic energy is given by $K_r = \frac{1}{2}I\omega^2$.**Power in rotational motion**: Power equals the rate at which work is done in rotation. Rotational power is given by $P = \tau \omega$.**Conservation of angular momentum**: The total angular momentum of an isolated system remains constant. Angular momentum is given by $L = I\omega$.

**Applications:**

**Simple pendulum**: A mass suspended from a fixed point by a string. Used to study periodic motion and simple harmonic motion.**Compound pendulum**: A rigid body suspended from a fixed point such that it can rotate freely. Used to study rotational dynamics of rigid bodies.**Rotational dynamics of rigid bodies**: Study of the motion of objects that rotate as a whole, without deformation. Includes s like torque, moment of inertia, and conservation of angular momentum.**Gyroscopes**: Devices used to maintain orientation in space due to their resistance to changes in angular momentum.**Centrifugal force**: An apparent force that arises in a rotating frame of reference, such as a spinning washing machine.