Answer : The angular acceleration is [tex]\alpha=-\beta t=-0.810\text{ t}[/tex]
Solution : Given,
[tex]\gamma=5.20rad/s[/tex]
[tex]\beta=0.810rad/s^3[/tex]
The given angular velocity equation is,
[tex]\omega_z(t)=\gamma-\beta t^2[/tex]
At t = 0, [tex]\omega_z(0)=\gamma[/tex]
At t = t, [tex]\omega_z(t)=\gamma-\beta t^2[/tex]
Angular acceleration : It is defined as the rate of change of angular velocity with respect to time.
Formula used for angular acceleration :
[tex]\alpha=\frac{\omega_z(t)-\omega_z(0)}{t}[/tex]
where,
[tex]\alpha[/tex] = angular acceleration
[tex]\omega_z(t)[/tex] = angular velocity at time 't'
[tex]\omega_z(0)[/tex] = angular velocity at time '0'
t = time
Now put all the given values in this formula, we get the angular acceleration.
[tex]\alpha=\frac{(\gamma)-(\gamma-\beta t^2)}{t}[/tex]
[tex]\alpha=-\beta t=-0.810\text{ t}[/tex]
Therefore, the angular acceleration is [tex]\alpha=-\beta t=-0.810\text{ t}[/tex]
