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,
: Measured in radians. Represents the amount of rotation about the axis. - Angular velocity,
: Measured in radians per second. Represents the rate of change of angular displacement. - Angular acceleration,
: Measured in radians per second squared. Represents the rate of change of angular velocity. - Relations between linear and angular quantities:
(Linear speed = radius × angular velocity) (Linear acceleration = radius × angular acceleration)
- Rotational equations of motion:
(Final angular velocity = Initial angular velocity + Angular acceleration × Time) (Final angular displacement = Initial angular displacement + Initial angular velocity × Time + ½ × Angular acceleration × Time²) (Angular acceleration = (Final angular velocity - Initial angular velocity) / Time)
- Rolling motion: Combination of rotational and translational motion.
Dynamics:
- Torque,
: A measure of the force causing rotation. Perpendicular to both the force vector and the displacement vector. - Moment of inertia,
: 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
, where is the moment of inertia about the center of mass, is the mass, and 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
, where and 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
. - Power in rotational motion: Power equals the rate at which work is done in rotation. Rotational power is given by
. - Conservation of angular momentum: The total angular momentum of an isolated system remains constant. Angular momentum is given by
.
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.