Question

In testing an automobile tire for proper alignment, a technician marks a spot on the tire 0.190 m from the center. He then mounts the tire in a vertical plane and notes that the radius vector to the spot is at an angle of 33.0° with the horizontal. Starting from rest, the tire is spun rapidly with a constant angular acceleration of 1.50 rad/s2. (Assume the spot's position is initially positive, and assume the angular acceleration is in the positive direction.) (a) What is the angular speed of the wheel after 1.20 s? rad/s

Answer 1

Complete Question

In testing an automobile tire for proper alignment, a technician marks a spot on the tire 0.190 m from the center. He then mounts the tire in a vertical plane and notes that the radius vector to the spot is at an angle of 33.0° with the horizontal. Starting from rest, the tire is spun rapidly with a constant angular acceleration of 1.50 rad/s2. (Assume the spot's position is initially positive, and assume the angular acceleration is in the positive direction.)

(a) What is the angular speed of the wheel after 1.20 s? rad/s

b) What is the tangential speed of the spot after 1.20 s?

(c) What is the magnitude of the total acceleration of the spot after 1.20 s?

(d) What is the angular position of the spot after 1.20 s?

Answer:

a

   $w = 1.8 rad/sec$

b

 $v =0.342 \ m/s$

c

 $a_r = 0.6156\ m/s^2$

d

$\theta = 1.656 \ rad$

Explanation:

From the question we are told that

      The distance of the spot from the center is $r = 0.190 \ m$

      The angle of the radius vector with the horizontal is  $\theta_o = 33^o = \frac{33\pi}{180} rad$

      The acceleration is  $\alpha = 1.50 \ rad/s^2$

The  angular speed of the wheel after 1.20 s  is  mathematically represented as

             $w =w_o + \alpha t$

Where $w_o$ is the initial angular speed of the wheel which is zero

  Substituting values

          $w = 0 + 1.5(1.20)$

           $w = 1.8 rad/sec$

the tangential speed of the spot after 1.20 is mathematically represented as

       $v = r w$

substituting values

       $v = 0.190 * 1.8$

       $v =0.342 \ m/s$

The magnitude of the total acceleration of the spot after 1.30 s is mathematically represented as

   $a = \sqrt{a_r ^2 + a_t ^2}$

The radial acceleration is  

        $a_r = \frac{v^2}{r}$

substituting values

        $a_r = \frac{(0.342)^2}{0.190}$

        $a_r = 0.6156\ m/s^2$

The tangential  acceleration is  

         $a_t = r \alpha$

substituting values

         $a_t = 0.19 * 1.5$

         $a_t = 0.285 \ m/s^2$

=>    $a = \sqrt{0.6156^2 + 0.285 ^2}$

      $a = 0.6784 \ m/s^2$

the angular position of the spot after 1.20 s is mathematically represented as

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

substituting values  

       $\theta = \frac{33 \pi}{180} + 0 + 0.5 * 1.5 * 1.2^2$

         $\theta = 1.656 \ rad$