The angular acceleration of a wheel is , with is in radians per second-squared and in seconds. At time , the wheel has an angular velocity of and an angular position of .
Write expressions for (a) the angular velocity (rad/s) and (b) the angular position (rad) as functions of time (s).
The angular acceleration of a wheel is,
The angular velocity of a wheel at t = 0 is,
The angular position of a wheel at t = 0 is,
Find the angular velocity by taking time integral of angular acceleration and angular position by taking the time integral of angular velocity.
Angular velocity is a time integral of angular acceleration. Hence,
Angular velocity of the wheel is .
Angular position is a time integral of angular velocity. Hence:
Therefore, angular position of the wheel is
Angular speed and angular position of an object is calculated from its angular acceleration.
In Fig., a wheel of radius is mounted on a frictionless horizontal axle. A massless cord is wrapped around the wheel and attached to a box that slides on a frictionless surface inclined at angle with the horizontal. The box accelerates down the surface at . What is the rotational inertia of the wheel about the axle?
A yo-yo-shaped device mounted on a horizontal frictionless axis is used to lift a box as shown in Fig. . The outer radius R of the device is , and the radius r of the hub is . When a constant horizontal force of magnitude is applied to a rope wrapped around the outside of the device, the box, which is suspended from a rope wrapped around the hub, has an upward acceleration of magnitude .What is the rotational inertia of the device about its axis of rotation?
In Fig., two blocks, of mass and , are connected by a massless cord that is wrapped around a uniform disk of mass and radius . The disk can rotate without friction about a fixed horizontal axis through its centre; the cord cannot slip on the disk. The system is released from rest. Find (a) the magnitude of the acceleration of the blocks, (b) the tension in the cord at the left, and (c) the tension in the cord at the right.
In Fig. 10 - 37 , two particles, each with mass m = 0.85KG , are fastened to each other, and to a rotation axis at O , by two thin rods, each with length d = 5.6cm and mass M = 1.2kg . The combination rotates around the rotation axis with the angular speed v = 0.30rad/s . Measured about O , what are the combination’s (a) rotational inertia and (b) kinetic energy?
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