Answer
(a) $\alpha_z(t) = (-1.60~rad/s^3)~t$
(b) $\alpha_z = -4.80~rad/s^2$
$\alpha_{ave} = -2.40~rad/s^2$
The two quantities are different because the angular acceleration changes as a function of time. The angular acceleration is not constant.
Work Step by Step
(a) $\omega_z(t) = \gamma -\beta ~t^2$
$\alpha_z(t) = \frac{d\omega}{dt} = -2\beta ~t$
$\alpha_z(t) = (-2)(0.800~rad/s^3)~t$
$\alpha_z(t) = (-1.60~rad/s^3)~t$
(b) When t = 3.00 s:
$\alpha_z = (-1.60~rad/s^3)(3.00~s)$
$\alpha_z = -4.80~rad/s^2$
When t = 0:
$\omega_z(t) = \gamma -\beta ~t^2$
$\omega_z(t) = (5.00~rad/s) -(0.800~rad/s^3)(0)^2$
$\omega_z(t) = 5.00~rad/s$
When t = 3.00 s:
$\omega_z(t) = \gamma -\beta ~t^2$
$\omega_z(t) = (5.00~rad/s) -(0.800~rad/s^3)(3.00~s)^2$
$\omega_z(t) = -2.20~rad/s$
We can find the average angular acceleration.
$\alpha_{ave} = \frac{\omega_2-\omega_1}{t}$
$\alpha_{ave} = \frac{(-2.20~rad/s)-(5.00~rad/s)}{3.00~s}$
$\alpha_{ave} = -2.40~rad/s^2$
The instantaneous angular acceleration at t = 3.00 seconds is double the magnitude of the average angular acceleration between 0 and 3.00 seconds.
The two quantities are different because the angular acceleration changes as a function of time. The angular acceleration is not constant.
