A uniform disk with a mass of 27.3 kg and a radius of 0.309 m is free to rotate about a frictionless axle. Forces of 90.0 N and 125 N are applied to the disk, as the drawing indicates.
(b) What is the angular acceleration of the disk?
describe the drawing. I am not certain what is the axis of rotation, is it a diameter, or is it perpendicular to the disk, through the center? Where are the forces?
To find the angular acceleration of the disk, we can apply Newton's second law for rotational motion, which states that the net torque on an object is equal to the product of its moment of inertia and its angular acceleration.
1. Calculate the net torque: The net torque can be found by summing up the torques produced by the two forces applied to the disk.
The torque produced by a force applied at some distance from the axis of rotation is given by the equation:
Torque = Force * Distance * sin(θ)
Where:
- Force is the magnitude of the force applied to the disk.
- Distance is the perpendicular distance between the axis of rotation and the line of action of the force.
- θ is the angle between the force vector and the vector perpendicular to the line connecting the point of force application and the axis of rotation.
In this case, we have two forces, 90.0 N and 125 N, each applied at a different distance from the axis of rotation. Let's call the distance for the 90.0 N force "r1" and the distance for the 125 N force "r2". From the drawing, it looks like r1 = 0.309 m and r2 = 0.618 m.
The torque produced by the 90 N force is:
Torque1 = 90.0 N * 0.309 m * sin(90°) = 90.0 N * 0.309 m = 27.81 Nm
The torque produced by the 125 N force is:
Torque2 = 125 N * 0.618 m * sin(180°) = 0 Nm (since sin(180°) = 0)
The net torque is the sum of the individual torques:
Net Torque = Torque1 + Torque2 = 27.81 Nm
2. Calculate the moment of inertia: The moment of inertia of a uniform disk about its central axis is given by the equation:
Moment of Inertia = (1/2) * Mass * Radius^2
Plugging in the values, we get:
Moment of Inertia = (1/2) * 27.3 kg * (0.309 m)^2 = 1.35 kg*m^2
3. Find the angular acceleration: Now, we can use Newton's second law for rotational motion:
Net Torque = Moment of Inertia * Angular Acceleration
Rearranging the equation, we get:
Angular Acceleration = Net Torque / Moment of Inertia
Plugging in the values, we have:
Angular Acceleration = 27.81 Nm / 1.35 kg*m^2 = 20.6 rad/s^2
So, the angular acceleration of the disk is 20.6 rad/s^2.