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22.1: Rotational Variables
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The angular position of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference. The angular velocity of a rotating body about a fixed axis is defined as ω(rad/s), the rotational rate of the body in radians per second. If the system’s angular velocity is not constant, then the system has an angular acceleration. The instantaneous angular acceleration is the time derivative of angular velocity.
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22.2: Rotation with Constant Angular Acceleration
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The kinematics of rotational motion describes the relationships among rotation angle, angular velocity and acceleration, and time. For constant angular acceleration, the angular velocity varies linearly, so the average angular velocity is 1/2 the initial plus final angular velocity over a given time period. A graphical analysis involves finding the area under an angular velocity-vs.-time or angular acceleration-vs.-time graph to get the change in angular displacement and velocity, respectively.
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22.3: Relating Angular and Translational Quantities
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The linear kinematic equation have the rotational counterparts in which x = θ, v = ω, a = α. A system undergoing uniform circular motion has a constant angular velocity, but points at a distance r from the rotation axis have a linear centripetal acceleration. A system undergoing nonuniform circular motion has an angular acceleration and therefore has both a linear centripetal and linear tangential acceleration at a point a distance r from the axis of rotation.
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22.4: Newton’s Second Law for Rotation
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Newton’s second law for rotation says that the sum of the torques on a rotating system about a fixed axis equals the product of the moment of inertia and the angular acceleration. In the vector form of Newton’s second law for rotation, the torque vector is in the same direction as the angular acceleration. If the angular acceleration of a rotating system is positive, the torque on the system is also positive, and if the angular acceleration is negative, the torque is negative.
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22.5: Examples
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