In Figure (a), an irregularly shaped plastic plate with uniform thickness and de

Question

In Figure (a), an irregularly shaped plastic plate with uniform thickness and density (mass per unit volume) is to be rotated around an axle that is perpendicular to the plate face and through point O. The rotational inertia of the plate about that axle is measured with the following method. A circular disk of mass 0.47 kg and radius 2.1 cm is glued to the plate, with its center aligned with point O (see Figure (b)). A string is wrapped around the edge of the disk the way a string is wrapped around a top. Then the string is pulled for 5.2 s. As a result, the disk and plate are rotated by a constant force of 0.40 N that is applied by the string tangentially to the edge of the disk. The resulting angular speed is 110 rad/s. What is the rotational inertia of the plate about the axle?

Explanation / Answer

Given:- F = 0.40N, R = 2.1cm = 0.021m, M = 0.47kg,

Angular speed = 110rad/sec

t = 5.2s.

For the torque you have

Torque = F x R = 0.4 x 0.02 = 0.008.

For the angular acceleration,

Angular acceleration = 110/5.2

= 21.15rad/s^2.

For the moment of inertia you have Ip + Id, where Ip is the unknown and

Id = (1/2)M R^2

Id = (1/2)(0.47)(0.021)^2

Id = 1036 × 10^-3 kg-m^2

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