A uniform disk with mass 40.0 kg and radius 0.200 m is...

A uniform disk with mass 40.0 kg and radius 0.200 m is pivoted at its center about a horizontal, frictionless axle that is stationary. The disk is initially at rest, and then a constant force $F =$ 30.0 N is applied tangent to the rim of the disk. (a) What is the magnitude $v$ of the tangential velocity of a point on the rim of the disk after the disk has turned through 0.200 revolution? (b) What is the magnitude $a$ of the resultant acceleration of a point on the rim of the disk after the disk has turned through 0.200 revolution?

Key Concepts

Torque

Torque is the measure of the rotational force acting on an object. In problems involving rotational dynamics, torque is calculated as the product of the force and the lever arm (distance from the pivot to the point where the force is applied). It is a crucial factor in determining the angular acceleration of the object.

Moment of Inertia

The moment of inertia quantifies how an object’s mass is distributed relative to a rotational axis and determines how much resistance it offers against angular acceleration. For example, in a uniform disk rotated about its center, the moment of inertia is calculated based on its mass and radius, affecting the rate at which the disk responds to applied torques.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity with time. It is determined by the net torque acting on a rotating object divided by its moment of inertia. This concept is key to analyzing how rotational motion evolves, especially when a constant force is applied, as it influences the speed at which the object rotates.

Angular Kinematics

Angular kinematics describes the motion of objects undergoing rotation, similar to linear kinematics but using angular displacement, angular velocity, and angular acceleration. It includes equations that relate these quantities over time, allowing one to determine how far an object rotates and what its angular velocity will be after a given period or angular displacement.

Tangential and Centripetal Acceleration

In rotational motion, tangential acceleration is associated with the change in the magnitude of the tangential velocity of a point on a rotating object, while centripetal acceleration is related to the change in the direction of the velocity, keeping the point in circular motion. The resultant acceleration is the vector sum of these two components, and it is essential for determining the overall acceleration of a point on a rotating body.

Transcript

00:01 So in this case we have this pulley rotated about its center and it weighs 40 kilograms.

00:06 It has a radius of 0 .2 meters and we have a force that's applied that's always acting tangential.

00:13 So whether it's actually a force that's moving on here or maybe a cable that is unwrapping around here, but it's always a tangential force.

00:22 And we're asked to figure out, what are we asked to figure out? the magnitude of velocity of the tangential velocity on the point of the rim after the rim after the disk has spun through 0 .2 revolutions and that is 1 .5, 1 .427 radiance.

00:42 So again, we need to remember to convert our angle from revolutions to radians.

00:48 Now, we know from kinematics that the velocity, angular velocity squared, equals the initial angular velocity square plus two times the angular acceleration minus the change in angle.

01:04 And since i'm just going to measure the start as initial angle at zero, then this is just the change in angle.

01:12 For a uniform disk, j, and i'm going to use j here instead of i, that's from more of an engineering point of view because in mechanical engineering, i is an area moment and j is a mass moment.

01:25 And those things get really, really confused later on in higher level classes.

01:33 And so an area moment has units of length to the fourth.

01:41 And so when you start talking about beams in engineering, you're going to see i a lot.

01:47 And then you're going to see j a lot in dynamics class because to distinguish the two, engineers we use j for the mass moment.

01:58 So i'm going to use j.

02:01 And that's 0 .800 kilogram meter squared.

02:05 So the angular acceleration, so it just, you know, torque equals j times the angle of acceleration because it's the angle of acceleration is the torque divided by j...

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