What is the Moment of Inertia for a Uniform Disk with a Falling Object?

[tsnikpoh11]

Homework Statement

A uniform disk of radius 0.12 m is mounted on a frictionless, horizontal axis. A light cord wrapped around the disk supports a 1 kg object, as shown in the figure from rest the object falls with a downward acceleration of 2.3 m/s^2.The acceleration of gravity is 9.8 m/s^2 When released What is the moment of inertia of the disk? in units of kg m^2

Homework Equations

Moment of Inertia of a disk by itself= 1/2MR^2
I= Torque/Angular Acceleration

The Attempt at a Solution

1/2*(M)*(.12^2) = I

If there is downward acceleration of 2.3 m/s^2 is that added to the 9.8 m/s^2 from gravity? 12.1 m/s^2

Im having a hard time knowing where to start with how this 1kg object affects the moment of inertia of the disk, and how I find M in the disk.

 

[Doc Al]

You'll need to separately analyze the forces on the hanging mass and on the disk.

Start with the hanging mass. Apply Newton's 2nd law to find the tension in the cord.

 

[Moetasim]

The moment of inertia of a disk is a measure of its resistance to rotational motion. It is defined as the sum of the masses of all the particles in the disk multiplied by the square of their perpendicular distance from the axis of rotation. In this case, the disk is mounted on a frictionless, horizontal axis, so we can use the formula for the moment of inertia of a disk by itself, which is 1/2MR^2, where M is the mass of the disk and R is the radius.

To solve this problem, we first need to find the mass of the disk. We know that the disk has a radius of 0.12 m, but we do not have any information about its mass. However, we can use the information about the object that is attached to the disk to find the mass of the disk.

The object attached to the disk has a mass of 1 kg and is accelerating downward at 2.3 m/s^2 when released. This means that there must be a net downward force acting on the object. This force is caused by the gravitational force and the tension in the light cord. We can use Newton's second law (F=ma) to determine the net downward force on the object:

F = m*a
F = 1 kg * 2.3 m/s^2
F = 2.3 N

Since the gravitational force is acting downward, we can assume that the tension in the cord is also acting downward. This means that the tension in the cord is equal to the net downward force on the object, which is 2.3 N.

Now, we can use this information to find the mass of the disk. Since the tension in the cord is equal to the net downward force on the object, we can use the formula for the tension in a cord (T = mg) to find the mass of the disk:

T = mg
2.3 N = m * 9.8 m/s^2
m = 2.3 N / 9.8 m/s^2
m = 0.234 kg

Now that we have the mass of the disk, we can plug it into the formula for the moment of inertia of a disk by itself, along with the given radius of 0.12 m, to find the moment of inertia of the disk:

I = 1/2 * 0.234 kg * (0.12 m