A uniform disk of radius 0.5 m and mass 3.9 kg is rotating about its center of mass. Its moment of inertia about its center of mass is I=(1/2)mR^2. Its angular velocity as a function of time is given by (t)=7kt^6/t1^7 where k = 3.8 rad and t1= 5 s. Find the magnitude of the net torque about the center of mass acting on the disk at t = 8 seconds.

How do I use moment of inertia and angular velocity to find the net torque?

To find the net torque acting on the disk, you can use the equation:

τ = I * α,

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

In this case, the moment of inertia of the disk is given as I = (1/2) * m * R^2, where m is the mass of the disk and R is its radius.

The angular velocity is given as ω(t) = 7kt^6 / t1^7, where k = 3.8 rad and t1 = 5 s.

To find the angular acceleration, we can differentiate the angular velocity with respect to time:

α(t) = dω(t) / dt.

Let's calculate the derivative:

α(t) = d/dt (7kt^6 / t1^7)
= 7k * (6t^5) / t1^7
= (42kt^5) / t1^7.

Now, we can substitute the moment of inertia and angular acceleration into the torque formula:

τ(t) = I * α(t)
= (1/2) * m * R^2 * (42kt^5 / t1^7).

To find the magnitude of the net torque at t = 8 seconds, we substitute t = 8 into the equation above:

τ(8) = (1/2) * m * R^2 * (42k * (8^5) / t1^7).

Now, you can calculate the magnitude of the net torque by substituting the given values for m, R, k, and t1 into the equation above.

To find the net torque acting on the disk, you can use the relationship between torque, moment of inertia, and angular acceleration.

The formula is:
Torque = Moment of Inertia * Angular Acceleration

To calculate the angular acceleration, you need to differentiate the given angular velocity with respect to time (t).

Given:
Angular velocity, ω(t) = 7kt^6 / t1^7
Where k = 3.8 rad and t1 = 5s

Take the derivative of the angular velocity equation with respect to time to get the angular acceleration equation:
α(t) = dω(t) / dt

Now, substitute the given values for k and t1 into the angular velocity equation:
ω(t) = 7(3.8)t^6 / (5^7)

Differentiate ω(t) with respect to t:
α(t) = d/dt [7(3.8)t^6 / (5^7)]
= 7(3.8)(6t^5) / (5^7)

Now, you have the angular acceleration equation α(t).

Next, you can find the moment of inertia (I) using the formula:
I = (1/2)mR^2
Given: m = 3.9 kg, R = 0.5 m
I = (1/2)(3.9 kg)(0.5 m)^2

Calculate the value of I.

Finally, substitute the values of I and α(t) into the formula for torque:
Torque = I * α(t)
Plug in the values of I and α(t) and calculate the torque.

This will give you the magnitude of the net torque acting on the disk at t = 8 seconds.