A uniform disk of radius 0.45 m is mounted on a frictionless, horizontal axis. A light cord wrapped around the disk supports a 1 kg object, as shown. When released from rest the object falls with a downward acceleration of 4.5 m/s^2. What is the moment of inertia of the disk? The acceleration of gravity is 9.8 m/s^2. Answer in units of kg m^2.

Torque = I alpha

Tension in line = T

weight = m g = 1 * 9.8 = 9.8 N

so 9.8 - T = 1(4.5)

T = 5.3 N or kg m/s^2

torque = T * R = 5.3 R Nm or kg m^2/s^2
so
5.3 R kg m^2/s^2 = I alpha
but a = alpha R so alpha = a/R
so
5.3 R^2 kg m^3/s^2 = I kg m^2 * 4.5 m/s^2
I = 5.3 * .45^2 /4.5 = .212 kg m^2

We can use Newton's second law to solve this problem. The net force in the downward direction is equal to the gravitational force acting on the object.

The net force is given by the equation:

Net force = Mass × Acceleration

In this case, the mass of the object is 1 kg, and the downward acceleration is 4.5 m/s^2.

So the net force is:

Net force = 1 kg × 4.5 m/s^2
= 4.5 N

The gravitational force acting on the object is given by the equation:

Gravitational force = Mass × Gravity

In this case, the mass of the object is 1 kg, and the acceleration due to gravity is 9.8 m/s^2.

So the gravitational force is:

Gravitational force = 1 kg × 9.8 m/s^2
= 9.8 N

Since the net force is equal to the gravitational force, we have:

4.5 N = 9.8 N

To find the moment of inertia, we need to use the equation for the torque:

Torque = Moment of inertia × Angular acceleration

In this case, the torque is equal to the net force multiplied by the radius of the disk, and the angular acceleration is the downward acceleration of the object divided by the radius of the disk.

So the equation can be written as:

4.5 N × 0.45 m = Moment of inertia × (4.5 m/s^2 / 0.45 m)

Simplifying the equation, we can solve for the moment of inertia:

Moment of inertia = (4.5 N × 0.45 m) / (4.5 m/s^2 / 0.45 m)

Moment of inertia = (4.5 N × 0.45 m) / (10 m/s^2)

Moment of inertia = 0.2025 kg m^2

Therefore, the moment of inertia of the disk is 0.2025 kg m^2.

To find the moment of inertia of the disk, we can use the equation:

τ = Iα

Where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

In this problem, the torque is created by the gravitational force acting on the object hanging from the disk. The torque can be calculated using the equation:

τ = r * F

Where r is the radius of the disk and F is the force acting on the object due to gravity. The force acting on the object due to gravity can be calculated using the equation:

F = m * g

Where m is the mass of the object and g is the acceleration due to gravity.

From the problem, we are given the following information:
- The radius of the disk (r) is 0.45 m
- The mass of the object (m) is 1 kg
- The acceleration due to gravity (g) is 9.8 m/s^2
- The angular acceleration (α) is 4.5 m/s^2

First, we can calculate the force acting on the object due to gravity:

F = m * g
F = 1 kg * 9.8 m/s^2
F = 9.8 N

Next, we can calculate the torque:

τ = r * F
τ = 0.45 m * 9.8 N
τ = 4.41 N*m

Finally, we can use the equation τ = I * α to solve for the moment of inertia (I):

4.41 N*m = I * 4.5 m/s^2

Simplifying for I:

I = (4.41 N*m) / (4.5 m/s^2)
I = 0.98 kg*m^2

Therefore, the moment of inertia of the disk is 0.98 kg m^2.

torque=momentdisk*angular scceleration

moment disk=torque/(acceleration/radius)
torque=force*distance=1*9.8*radius

mmomentdisk=9.8(radius^2)/4.5m/s^2