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, θ: Measured in radians. Represents the amount of rotation about the axis.
  • Angular velocity, ω: Measured in radians per second. Represents the rate of change of angular displacement.
  • Angular acceleration, α: Measured in radians per second squared. Represents the rate of change of angular velocity.
  • Relations between linear and angular quantities:
    • v=rω (Linear speed = radius × angular velocity)
    • a=rα (Linear acceleration = radius × angular acceleration)
  • Rotational equations of motion:
    • ωf=ωi+αt (Final angular velocity = Initial angular velocity + Angular acceleration × Time)
    • θf=θi+ωit+12αt2 (Final angular displacement = Initial angular displacement + Initial angular velocity × Time + ½ × Angular acceleration × Time²)
    • α=ωfωit (Angular acceleration = (Final angular velocity - Initial angular velocity) / Time)
  • Rolling motion: Combination of rotational and translational motion.

Dynamics:

  • Torque, τ: 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=ICM+Md2, where ICM 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=Ix+Iy, where Ix and Iy 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 Kr=12Iω2.
  • Power in rotational motion: Power equals the rate at which work is done in rotation. Rotational power is given by P=τω.
  • Conservation of angular momentum: The total angular momentum of an isolated system remains constant. Angular momentum is given by L=Iω.

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.


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