A disk turns through an angle of β(t)=Ct2 - Bt3 where C=3.20 rad/s2 and B= 0.500 rad/s3. Calculate the angular acceleration α(t) and velocity w(t) as a function of time.

ω(t) = 6.4 t - 1.5 t²

α(t) = 6.4  - 3 t

Explanation:

The angular displacement of the disk is given as the function of time:

β(t)=Ct² - B t³

where,

C = 3.2 rad/s²

B = 0.5 rad/s³

Therefore,

β(t) = 3.2 t² - 0.5 t³

Now, for angular velocity ω(t), we must take derivative of angular displacement with respect to t:

ω(t) = dβ/dt = (d/dt)(3.2 t² - 0.5 t³)

ω(t) = 6.4 t - 1.5 t²

Now, for angular acceleration α(t), we must take derivative of angular velocity with respect to t:

α(t) = dω/dt = (d/dt)(6.4 t - 1.5 t²)

α(t) = 6.4  - 3 t