A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3.00 cm . It

step 1 So here, since we're dealing with rotational motion, we can use torque equal to moment of inerti

A machine part has the shape of a solid uniform sphere of...

A machine part has the shape of a solid uniform sphere of mass $225 \mathrm{~g}$ and diameter $3.00 \mathrm{~cm} .$ 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 $0.0200 \mathrm{~N}$ at that point. (a) Find its angular acceleration. (b) How long will it take to decrease its rotational speed by $22.5 \mathrm{rad} / \mathrm{s} ?$

A machine part has the shape of a solid uniform sphere of mass $225 \mathrm{~g}$ and diameter $3.00 \mathrm{~cm} .$ 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 $0.0200 \mathrm{~N}$ at that point.
(a) Find its angular acceleration.
(b) How long will it take to decrease its rotational speed by $22.5 \mathrm{rad} / \mathrm{s} ?$

Key Concepts

Moment of Inertia

The moment of inertia measures an object's resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a uniform solid sphere, there is a specific formula that quantifies this distribution, which is essential for calculating how the sphere responds to applied torques.

Torque

Torque is the measure of the turning force acting on an object. It is calculated as the product of a force and its lever arm, or the distance from the force's point of application to the axis of rotation. Torque is central to rotational dynamics because it directly causes angular acceleration according to the relation ? = I?.

Angular Acceleration

Angular acceleration is the rate at which an object's angular velocity changes over time. It is directly related to the net applied torque and inversely related to the moment of inertia of the object, making it a key concept in understanding how and how quickly rotation speeds up or slows down.

Angular Kinematics

Angular kinematics involves the study of the relationships between angular displacement, angular velocity, angular acceleration, and time. It provides the framework for analyzing how rotational speeds evolve under constant or variable angular accelerations, which is analogous to linear kinematics but for rotational motion, and is used for calculating time intervals for changes in rotational speed.

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