Question

Dario, a prep cook at an Italian restaurant, spins a salad spinner and observes that it rotates 20.0 times in 5.00 seconds and then stops spinning it. The salad spinner rotates 6.00 more times before it comes to rest. Assume that the spinner slows down with constant angular acceleration. What is the angular acceleration of the salad spinner as it slows down in d/s^2?

Answer 1

Concepts and reason

This problem is based on the concept of angular velocity and angular acceleration.

First, calculate the angular velocity by using the relation between angular displacement, angular velocity, and time. Finally, calculate the angular acceleration by using the appropriate kinematic expression.

Fundamentals

Angular velocity of a body is given by the expression,

$\omega = \frac{\theta }{t}$

Here, $\omega$ is angular velocity, $\theta$ is angle swept by body and, $t$ is time interval

The angular displacement of an object is given as,

$\theta = n2\pi$

Here, n is the number of rotations.

Angular acceleration of a body is given by the expression,

$\alpha = \frac{{\partial \omega }}{{\partial t}}$

Here, $\omega$ is angular velocity, $\alpha$ is angular acceleration and $t$ is time interval

The kinematic expression for angular acceleration is given as,

${\omega _{\rm{f}}}^2 = {\omega _{\rm{i}}}^2 + 2\alpha (\theta )$

Angular velocity of a body is given by the expression,

$\omega = \frac{\theta }{t}$

Here, $\omega$ is angular velocity, $\theta$ is angle swept by body and $t$ is time interval.

The angular displacement of the object is,

$\theta = n2\pi$

Here, n is the number of rotations.

Substitute 20 for n in equation $\theta = n2\pi$ .

$\begin{array}{c}\\\theta = \left( {20} \right)2\pi {\rm{ rad}}\\\\ = 40\pi {\rm{ rad}}\\\end{array}$

Substitute $40\pi {\rm{ rad}}$ for $\theta$ and ${\rm{5 s}}$ for $t$ in the above equation.

$\begin{array}{c}\\\omega = \frac{{40\pi {\rm{ rad}}}}{{5{\rm{ s}}}}\\\\ = 8\pi \;{\rm{rad/s}}\\\end{array}$

The kinematic expression for angular acceleration is given by

${\omega _{\rm{f}}}^2 = {\omega _{\rm{i}}}^2 + 2\alpha (\theta )$

Substitute 6 for n in equation $\theta = n2\pi$ .

$\begin{array}{c}\\\theta = \left( 6 \right)2\pi {\rm{ rad}}\\\\ = 12\pi {\rm{ rad}}\\\end{array}$

Substitute 0 rad/s for ${\omega _{\rm{f}}}$ , $8\pi \;{\rm{rad/s}}$ for ${\omega _{\rm{i}}}$ and $12\pi {\rm{ rad}}$ for $\theta$ in equation ${\omega _{\rm{f}}}^2 = {\omega _{\rm{i}}}^2 + 2\alpha (\theta )$ .

$\begin{array}{c}\\0{\rm{ rad/s}} = {\left( {8\pi {\rm{ rad/s}}} \right)^2} + 2\alpha (12\pi {\rm{ rad}})\\\\\alpha = - 8.38\;{\rm{rad/}}{{\rm{s}}^{\rm{2}}}\\\end{array}$

Ans:

The angular acceleration of salad spinner is $8.38\;{\rm{rad/}}{{\rm{s}}^{\rm{2}}}$ .