Two spheres are each rotating at an angular speed of 24 rad / s about axes that pass through the

step 1 Hello, my name is David. In this video, we'll cover the angular acceleration of a solid sphere a

Two spheres are each rotating at an angular speed of 24 rad...

Two spheres are each rotating at an angular speed of $24 \mathrm{rad} / \mathrm{s}$ about axes that pass through their centers. Each has a radius of $0.20 \mathrm{m}$ and a mass of 1.5 kg. However, as the figure shows, one is solid and the other is a thin-walled spherical shell. Suddenly, a net external torque due to friction (magnitude $=0.12 \mathrm{N} \cdot \mathrm{m}$ ) begins to act on each sphere and slows the motion down. Concepts: (i) Which sphere has the greater moment of inertia and why? (ii) Which sphere has the angular acceleration (a deceleration) with the smaller magnitude? (iii) Which sphere takes a longer time to come to a halt? Calculations: How long does it take each sphere to come to a halt?

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a solid sphere and a thin-walled spherical shell of the same mass and radius, the shell has a larger moment of inertia because more of its mass is distributed farther from the rotational axis.

Torque and Angular Acceleration

Torque is the rotational equivalent of force and is related to angular acceleration through the relation ? = I?, where ? is the net torque, I is the moment of inertia, and ? is the angular acceleration. When the same torque is applied, an object with a larger moment of inertia will experience a smaller angular acceleration, and vice versa.

Rotational Kinematics

Rotational kinematics involves the relationships among angular displacement, initial angular velocity, angular acceleration, and time. When a constant net torque (and thus a constant angular deceleration) is applied, these relationships can be used to determine the time required for a rotating object to come to a stop by considering the decrease of angular velocity over time.

Transcript

00:01 Hello, my name is david.

00:02 In this video, we'll cover the angular acceleration of a solid sphere and a holosphere.

00:07 So for this problem, we have a thin spherical shell and a solid sphere both rotating with an initial angular velocity of 24 rats per second.

00:21 And both of them have a radius of 0 .20 meters with a mass of 1 .5 kilograms.

00:31 And suddenly an external torque due to friction starts to slow them down.

00:37 And this torque is equal to negative 0 .12 newton's meter.

00:44 So for the first part, we want to know which sphere has the moment of inertia with the greater magnitude.

00:53 So the moment of inertia for a solid sphere is equal to two fits times the mass times the radius square and the moment of inertia for a holosphere or a thin walled spherical shell is equal to two thirds times the mass times the radius square.

01:17 So the thing wall spherical shell has the greater moment of inertia.

01:28 Now for the second part, we want to know which of these spheres has the angular acceleration with the smaller magnitude.

01:38 So we know that the net torque is equal to the moment of inertia times the angular acceleration.

01:48 But since we only have one torque acting on the spheres, which is the external torque due to friction.

01:54 We get that the external torque is equal to the moment of inertia times the angular acceleration.

02:02 So we saw for the angular acceleration, we get that the angular acceleration is equal to the external torque divided by the moment of inertia.

02:13 So for a solid sphere, we get that the angular acceleration is equal to the external torque divided by two feet, mr square.

02:31 So with plug in our values, we get negative 0 .12 newton meters divided by two fits times 1 .5 kilograms times 0 .2 meters squared.

02:50 So we clean it up, we get negative 0 .12 newton's meter divided by 0 .024 kilograms meter square...

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