In the device a 50g mass is hung from a pulley and released. A disk is placed on the platform. The platform has a radius of 5cm.

The disk and platform go from rest to an angular velocity of 8.5rad/s in 2.5 seconds. What is the moment of inertia of the disk? What is its mass? Ignore the mass of the platform.

To solve this problem, we can use the equation for the angular acceleration of an object rotating about an axis:

α = ωf - ωi / Δt

where α is the angular acceleration, ωf is the final angular velocity, ωi is the initial angular velocity, and Δt is the time interval.

Given:
- Δt = 2.5 seconds
- ωi = 0 rad/s (since the disk and platform start from rest)
- ωf = 8.5 rad/s

We can substitute these values into the equation and solve for α:

α = (8.5 rad/s - 0 rad/s) / 2.5 s
α = 8.5 rad/s / 2.5 s
α = 3.4 rad/s²

Next, we need to determine the net torque acting on the disk. Since there are no external torques acting on the system (ignoring air resistance and friction), the net torque is simply equal to the torque due to the hanging mass. The torque on a disk can be calculated using the equation:

τ = I * α,

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

Given:
- τ = m * g * r, where m is the mass, g is the acceleration due to gravity, and r is the radius of the disk (5 cm = 0.05 m)
- α = 3.4 rad/s²

We can substitute the values into the equation and solve for I:

m * g * r = I * α
I = (m * g * r) / α

Since m = 50 g = 0.05 kg and g = 9.8 m/s², we can calculate I:

I = (0.05 kg * 9.8 m/s² * 0.05 m) / 3.4 rad/s²
I ≈ 0.036 kg·m²

Therefore, the moment of inertia of the disk is approximately 0.036 kg·m².

To find the mass of the disk, we can rearrange the equation to solve for m:

m = I * α / (g * r)

Substituting the known values:

m = (0.036 kg·m² * 3.4 rad/s²) / (9.8 m/s² * 0.05 m)
m ≈ 0.024 kg

Therefore, the mass of the disk is approximately 0.024 kg.

To find the moment of inertia of the disk, we can use the rotational kinetic energy formula:

KE = (1/2) * I * ω^2

Where KE is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.

Given that the disk and platform go from rest to an angular velocity of 8.5 rad/s in 2.5 seconds, we can calculate the angular acceleration (α) using the formula:

α = ωf - ωi / t

Where ωf is the final angular velocity, ωi is the initial angular velocity (which is 0 since it starts from rest), and t is the time.

α = (8.5 rad/s - 0) / 2.5 s
α = 3.4 rad/s^2

Now, we can use the formula for angular acceleration (α) in terms of moment of inertia (I) and torque (τ):

τ = I * α

In this case, the only force causing torque on the disk is the gravitational force acting on the mass. The torque (τ) due to gravity can be calculated as the product of the mass (m), gravitational acceleration (g), and the radius of the platform (r):

τ = m * g * r

Where m is the mass of the disk and platform, g is the gravitational acceleration (approximately 9.8 m/s^2), and r is the radius of the platform.

Since we know the torque (τ) and the angular acceleration (α), we can equate the two equations and solve for the moment of inertia (I):

τ = I * α
m * g * r = I * α
I = m * g * r / α

To find the mass of the disk, we can rearrange the equation for moment of inertia (I) and solve for the mass (m):

I = m * r^2
m = I / r^2

Now that we have formulas for the moment of inertia (I) and mass (m), we can substitute the given values to calculate them.

Given:
ωf = 8.5 rad/s
t = 2.5 s
r = 0.05 m (5 cm)
g = 9.8 m/s^2

First, calculate the angular acceleration (α):
α = (8.5 rad/s - 0) / 2.5 s
α = 3.4 rad/s^2

Calculate the moment of inertia (I):
I = m * g * r / α
I = (50 g / 1000 g/kg) * 9.8 m/s^2 * 0.05 m / 3.4 rad/s^2

Finally, calculate the mass (m):
m = I / r^2

Now substitute the calculated values into the respective formulas and solve for the moment of inertia (I) and mass (m).