Question

A fan blade rotates with angular velocity given by ωz(t)= γ − β t2, where γ = 5.20 rad/s and β = 0.810 rad/s3 . calculate the angular acceleration as a function of time t in terms of β and γ.

Answer 1

Answer : The angular acceleration is $\alpha=-\beta t=-0.810\text{ t}$

Solution : Given,

$\gamma=5.20rad/s$

$\beta=0.810rad/s^3$

The given angular velocity equation is,

$\omega_z(t)=\gamma-\beta t^2$

At t = 0,    $\omega_z(0)=\gamma$

At t = t,      $\omega_z(t)=\gamma-\beta t^2$

Angular acceleration : It is defined as the rate of change of angular velocity with respect to time.

Formula used for angular acceleration :

$\alpha=\frac{\omega_z(t)-\omega_z(0)}{t}$

where,

$\alpha$ = angular acceleration

$\omega_z(t)$ = angular velocity at time 't'

$\omega_z(0)$ = angular velocity at time '0'

t = time

Now put all the given values in this formula, we get the angular acceleration.

$\alpha=\frac{(\gamma)-(\gamma-\beta t^2)}{t}$

$\alpha=-\beta t=-0.810\text{ t}$

Therefore, the angular acceleration is $\alpha=-\beta t=-0.810\text{ t}$