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.