Question

A machine part has the shape of a solid uniform sphere of mass 220 g and diameter 2.90 cm . It is spinning about a frictionless axle through its center, but at one point on its equator it is scraping against metal, resulting in a friction force of 0.0200 N at that point.

Part A

Find its angular acceleration. Let the direction the sphere is spinning be the positive sense of rotation.

Part B

How long will it take to decrease its rotational speed by 19.0 rad/s ?

Answer 1

Mass of the solid uniform sphere = M = 220 g = 0.22 kg

Diameter of the solid uniform sphere = D = 2.9 cm = 0.029 m

Radius of the solid uniform sphere = R = D/2 = 0.029/2 = 0.0145 m

Moment of inertia of the solid uniform sphere = I

I = 1.85 x 10^-5 kg.m²

Friction force on the sphere = F = 0.02 N

The sphere is being scraped at the equator therefore the distance of the friction force from the axis of rotation is equal to the radius of the sphere.

Torque on the sphere due to the friction force =

= -FR (Negative as the torque is acting in the opposite direction of the motion of the sphere)

= -(0.02)(0.0145)

= -2.9 x 10^-4 N.m

Angular acceleration of the sphere =

= -15.7 rad/s²

Time taken to reduce the rotational speed of the sphere by 19 rad/s = t

Change in angular speed of the sphere = = -19 rad/s (Negative as it has decreased)

t = 1.21 sec

A) Angular acceleration of the sphere = -15.7 rad/s²

B) Time taken to reduce the rotational speed of the sphere by 19 rad/s = 1.21 sec