Question

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what is the average angular velocity of the seconds hand of a clock?
$\bigcirc \omega = 0$
$\bigcirc \omega = - 0.10 rad/s$
$\bigcirc \omega = - 2\pi rad/s$
$\bigcirc \omega = 2\pi rad/s$
$\bigcirc \omega = 0.10 rad/s$

the initial angular velocity of a ceiling fan is 6.28 rad/s. it accelerates uniformly to 3.14 rad/s in 4.0 s. what is the angular acceleration?
$\bigcirc - 0.79 rad/s^{2}$
$\bigcirc 0.39 rad/s^{2}$
$\bigcirc - 0.39 rad/s^{2}$
$\bigcirc 0.79 rad/s^{2}$

Explanation:

Step1: Recall the formula for angular velocity

The formula for average angular velocity is \(\omega=\frac{\Delta\theta}{\Delta t}\). For a seconds - hand, it makes a full - circle (\(\Delta\theta = 2\pi\) radians) in \(t = 60\) s. Also, the direction is clock - wise, so the angular velocity is negative. \(\omega=\frac{- 2\pi}{60}\approx - 0.10\) rad/s.

Step2: Recall the formula for angular acceleration

The formula for angular acceleration is \(\alpha=\frac{\omega_f-\omega_i}{t}\). Given \(\omega_i = 6.28\) rad/s, \(\omega_f=3.14\) rad/s, and \(t = 4.0\) s. Then \(\alpha=\frac{3.14 - 6.28}{4.0}=\frac{-3.14}{4.0}=- 0.785\approx - 0.79\) rad/s².

Answer:

For the first question: \(\omega=-0.10\) rad/s.
For the second question: \(-0.79\) rad/s².