Question

Mary is an avid game show fan and one of the contestants on a popular game show. She spins the wheel, and after 5.5 revolutions, the wheel comes to rest on a space that has a $1500 value prize. If the initial angular speed of the wheel is 3.15 rad/s, find the angle through which the wheel has turned when the angular speed reaches 1.80 rad/s

Answer 1

The angle is 23.2 radians, equivalent to 3.69 revolutions.

Step-by-step explanation:

First, we need to find the angular acceleration of the wheel. This can be done using one of the kinematic formulas:

`\omega^(2)=\omega_0^(2)+2\alpha\theta\\\\\implies \alpha=(\omega^(2)-\omega_0^(2))/(2\theta)`

Since the final angular velocity is zero after 5.5 revolutions (equivalent to 11π radians) we have that:

`\alpha=(-(3.15rad/s)^(2))/(2(11\pi rad))\\\\\alpha=-0.144rad/s^(2)`

Now, using the same equation, we can solve for the requested angle:

`\theta=(\omega^(2)-\omega_0^(2))/(2\alpha)\\\\\theta=((1.80rad/s)^(2)-(3.15rad/s)^(2))/(2(-0.144rad/s^(2)))\\\\\theta=23.2rad`

Finally, it means that the angle through which the wheel has turned when the angular speed reaches 1.80 rad/s is 23.2 radians, equivalent to 3.69 revolutions.