rotation

Examples of rotation in the following topics:

• Rotational Kinetic Energy: Work, Energy, and Power

  • The rotational kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy.
  • Rotational kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy .
  • Looking at rotational energy separately around an object's axis of rotation yields the following dependence on the object's moment of inertia:
  • The mechanical work applied during rotation is the torque ($\tau$) times the rotation angle ($\theta$): $W = \tau \theta$.
  • The earth's rotation is a prominent example of rotational kinetic energy.

• Rotational Angle and Angular Velocity

  • The rotational angle is a measure of how far an object rotates, and angular velocity measures how fast it rotates.
  • The amount the object rotates is called the rotational angle and may be measured in either degrees or radians.
  • The speed at which the object rotates is given by the angular velocity, which is the rate of change of the rotational angle with respect to time.
  • The radius of a circle is rotated through an angle $\Delta\theta$.
  • The angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs.

• Conservation of Energy in Rotational Motion

  • Energy is conserved in rotational motion just as in translational motion.
  • The simplest rotational situation is one in which the net force is exerted perpendicular to the radius of a disc and remains perpendicular as the disc starts to rotate.
  • Kinetic energy (K.E.) in rotational motion is related to moment of rotational inertia (I) and angular velocity (ω):
  • The final rotational kinetic energy equals the work done by the torque:
  • This confirms that the work done went into rotational kinetic energy.

• Angular Position, Theta

  • The angle of rotation is a measurement of the amount (the angle) that a figure is rotated about a fixed point— often the center of a circle.
  • When objects rotate about some axis—for example, when the CD (compact disc) rotates about its center—each point in the object follows a circular arc.
  • The rotation angle is the amount of rotation, and is analogous to linear distance.
  • Thus, for one complete revolution the rotation angle is:
  • The radius of a circle is rotated through an angle Δ.

• Rotational Inertia

  • Rotational inertia is the tendency of a rotating object to remain rotating unless a torque is applied to it.
  • There are, in fact, precise rotational analogs to both force and mass.
  • Rotational inertia, as illustrated in , is the resistance of objects to changes in their rotation.
  • In other words, a rotating object will stay rotating and a non-rotating object will stay non-rotating unless acted on by a torque.
  • The quantity mr2 is called the rotational inertia or moment of inertia of a point mass m a distance r from the center of rotation.

• Torque

  • Torque is the force that causes objects to turn or rotate (i.e., the tendency of a force to rotate an object about an axis).
  • Rotation is a special case of angular motion.
  • In the case of rotation, torque is defined with respect to an axis such that vector "r" is constrained as perpendicular to the axis of rotation.
  • In other words, the plane of motion is perpendicular to the axis of rotation.
  • Clearly, the torque in rotation corresponds to force in translation.

• Relationship Between Torque and Angular Acceleration

  • In fact, this equation is Newton's second law applied to a system of particles in rotation about a given axis.
  • It makes no assumptions about constant rotational velocity.
  • The net torque about an axis of rotation is equal to the product of the rotational inertia about that axis and the angular acceleration, as shown in Figure 1 .
  • With rotating objects, we can say that unless an outside torque is applied, a rotating object will stay rotating and an object at rest will not begin rotating.
  • From this we might conclude that just because a rotating object is in translational equilibrium, it is not necessarily in rotational equilibrium.

• The Coriolois Force

  • In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right.
  • The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame.
  • It is proportional to the object's speed in the rotating frame.
  • They allow the application of Newton's laws to a rotating system.
  • This effect is responsible for the rotation of large cyclones.

• Constant Angular Acceleration

  • Constant angular acceleration describes the relationships among angular velocity, angle of rotation, and time.
  • Simply by using our intuition, we can begin to see the interrelatedness of rotational quantities like θ (angle of rotation), ω (angular velocity) and α (angular acceleration).
  • Similarly, the kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
  • By using the relationships a=rα, v=rω, and x=rθ, we derive all the other kinematic equations for rotational motion under constant acceleration:
  • Relate angle of rotation, angular velocity, and angular acceleration to their equivalents in linear kinematics

• Angular Quantities as Vectors

  • The direction of these quantities is inherently difficult to track—a point on a rotating wheel is constantly rotating and changing direction.
  • The axis of rotation of a rotating wheel is the only place that has a fixed direction.
  • Imagine the axis of rotation as a pole through the center of a wheel.
  • This dependency on perspective makes determining the angle of rotation slightly more difficult.
  • Figure (a) shows a disk is rotating counterclockwise when viewed from above.