Question

Starting from rest, a disk rotates about its central axis with constant angular acceleration. In 5.00 s, it rotates 10.1 rad. During that time, what are the magnitudes of (a) the angular acceleration and (b) the average angular velocity? (c) What is the instantaneous angular velocity of the disk at the end of the 5.00 s? (d) With the angular acceleration unchanged, through what additional angle (rad) will the disk turn during the next 5.00 s?

Answer 1

Answer:

a

The angular acceleration is  $\alpha = 0.808 rad/s^2$

b

The average angular velocity is $w = 2.02 \ rad/s$

c

The instantaneous angular velocity is  $w = 4.04 \ rad/s$

d

The additional angle (rad) will the disk turn during the next 5.00 s is

      $\theta_n = 30.30 \ rad$

Explanation:

From the question we are told that

    The duration of rotation is  $t = 5.00s$

     The angular displacement is  $\theta= 10.1 rad$

From newtons Law the angular displacement is mathematically represented as

           $\theta = w_o t +\frac{1}{2} \alpha t^2$

Where $\alpha$ is the angular acceleration  

            $w_o$ is the initial angular velocity and its value is  0

     Making $\alpha$ the subject of formula

               $\alpha = \frac{\theta}{0.5 * t^2}$

Substituting value  

               $\alpha = \frac{10.1}{0.5 * 5^2 }$

               $\alpha = 0.808 rad/s^2$

The average angular velocity is mathematically represented as

             $w = \frac{\theta }{t}$

   Substituting value  

              $w = \frac{10.1}{5}$  

              $w = 2.02 \ rad/s$

From the equations of motion  the instantaneous  angular velocity is mathematically represented as  

          $w = w_o + \alpha t$

Substituting value  

         $w = 0 + 0.808 * 5$

         $w = 4.04 \ rad/s$

Modifying the equation for angular displacement is mathematically represented as

        $\theta_{n} = w_o t + 0.5 \alpha t^2$

From the question $\alpha$ is not changed

                                t = 5s

  the initial velocity would be the instantaneous velocity

     So  $w_o = 4.04 \ rad/s$

Substituting value

      $\theta_n = 4.04 * 5 + (0.5 * 0.808 * 5^2)$

     $\theta_n = 30.30 \ rad$