A machine part has the shape of a solid uniform sphere of mass 250 g and diameter 4.50 cm. It is...

Question:

A machine part has the shape of a solid uniform sphere of mass {eq}250 \ g {/eq} and diameter {eq}4.50 \ cm {/eq}. 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 {eq}0.0200 \ N {/eq} at the point.

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

(b) How long will it take to decrease its rotational speed by {eq}21.0 \ rad / s {/eq}?

Torque

Torque is referred to as the moment of force. When torque is present, there is rotational acceleration. Torque is equal to the product of the moment of inertia and the angular acceleration of the rigid body.

Answer and Explanation:

First, we solve for moment of inertia of the sphere. We write:

{eq}I = \frac{2}{5}mR^2 \\ I = \frac{2}{5}(0.250)(0.0225)^2 \\ I = 5.06\times10^{-5} {/eq}

(A) Using the equation for torque, we write:

{eq}I\alpha = -f\times R {/eq}

where:

f is the force due to friction

R is the radius

Inserting the values of the available parameters, we write:

{eq}(5.06\times10^{-5})\alpha = -(0.02)\times (0.0225) \\ \alpha = -8.89 \ rad/s^2 {/eq}

(B) The time is given as:

{eq}t = \frac{21.0}{8.89} \\ t = 2.36 \ s {/eq}


Learn more about this topic:

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Torque in Physics: Equation, Examples & Problems

from Physics: Middle School

Chapter 3 / Lesson 13
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