A uniform disk with radius 0.320m and mass 28.0kg rotates in a horizontal plane on a frictionless vertical axle that passes through the center of the disk. The angle through which the disk has turned varies with time according to è(t)= (1.20rad/s)t + (8.70rad/s^2)t^2

Question

A uniform disk with radius 0.320m and mass 28.0kg rotates in a horizontal plane on a frictionless vertical axle that passes through the center of the disk. The angle through which the disk has turned varies with time according to è(t)= (1.20rad/s)t + (8.70rad/s^2)t^2

my solution:
w(t) = 1.2 + 17.4t
a = (17.4)*(0.32) = 5.568 rad/s2
is my answer CORRECT ??

Answer 1

Update ==>

theta (t) = (1.20rad/s)t + (8.70rad/s^2)t^2

Answer 2

To determine if your answer is correct, we need to verify that you correctly derived the equation for angular velocity, and then calculate the acceleration based on that equation.

First, let's verify the equation for angular velocity, ω(t), that you derived. The equation you provided, ω(t) = 1.2 + 17.4t, appears to be the correct expression. It shows that the angular velocity of the disk is increasing linearly over time, with an initial value of 1.2 rad/s and a slope of 17.4 rad/s^2. Therefore, your equation for angular velocity is correct.

Next, let's calculate the acceleration, a(t), at time t = 0. We can use the equation for angular acceleration, α(t), which is the derivative of the angular velocity with respect to time:

α(t) = dω(t)/dt = 17.4 rad/s^2

Since the disk is rotating about a frictionless axle, its net torque (τ) will be equal to the moment of inertia (I) multiplied by the angular acceleration (α):

τ = Iα

The moment of inertia of a uniform disk rotating about its central axis is given by the equation:

I = (1/2)MR^2,

where M is the mass and R is the radius. Plugging in the values for M = 28.0 kg and R = 0.320 m, we can calculate the moment of inertia:

I = (1/2)(28.0 kg)(0.320 m)^2 = 1.792 kg·m^2.

Now, we can calculate the torque:

τ = Iα = (1.792 kg·m^2)(17.4 rad/s^2) = 31.1808 N·m.

Since there are no other torques acting on the disk, the torque is equal to the moment of inertia multiplied by the angular acceleration:

τ = Iα = 31.1808 N·m.

Therefore, your calculation for the acceleration is incorrect. The correct value of acceleration is 31.1808 N·m. Please recheck your work and make sure to use the correct equations and units in your calculations.