Angular speed and revolutions question.

[shanest]

I've attached the problem as an image. I solved 20 and 21 very easily. For 22, my obvious instinct was to set dtheta/dt to 0, but that's an unsolvable equation. I'm obviously missing something key in solving 22. What is it?

 

[chanvincent]

obviously, your obvious instinct is not pointing you to the right direction...

Try to set up a definite integral with appropiate limit... (what is time t equal to when the wheel coming to rest?)

 

[m2003]

Thank you for sharing your solution to problems 20 and 21. Regarding problem 22, you are correct that setting dtheta/dt to 0 will not lead to a solvable equation. This is because the problem is asking for the time it takes for the wheel to stop rotating, which means the angular speed would be 0 at that time. However, this does not mean that the angular speed is constantly 0 throughout the rotation.

To solve problem 22, we can use the formula for angular speed, w= theta/t, where w represents the angular speed, theta is the angle of rotation, and t is the time taken. We know that the wheel makes 4 revolutions, or 8*pi radians, in 2 seconds. Therefore, we can set up the equation:

w = 8*pi radians / 2 seconds

Solving for w, we get w = 4*pi radians/second. We also know that the wheel stops rotating after 4 revolutions, or 8*pi radians. So, using the same formula, we can set up the equation:

4*pi radians/second = 8*pi radians / t seconds

Solving for t, we get t = 2 seconds. Therefore, it takes 2 seconds for the wheel to stop rotating.

In summary, to solve problem 22, we use the formula for angular speed and set it equal to the given values. This allows us to find the angular speed, and then use it to find the time it takes for the wheel to stop rotating. I hope this helps clarify the key step in solving this problem.