A bicycle wheel of radius .7m is rolling without slipping on a horizontal surface with an angular speed of 2 rev/s when the cyclist begins to uniformly apply the brakes. The bicycle stops in 5 s. Through how many revolutions did the wheel turn during the 5 s of braking?
I tried it but got the wrong answer and I'm not sure what to do. Please help!
To determine the number of revolutions the wheel turned during the 5 seconds of braking, we can use the formula:
Number of revolutions = (change in angular displacement) / (2π)
First, let's calculate the change in angular displacement:
We know that the initial angular speed of the wheel is 2 rev/s. Since the wheel is rolling without slipping, the linear speed of any point on the circumference of the wheel is the product of its angular speed and the radius of the wheel.
Linear speed = angular speed * radius
= 2 rev/s * 0.7 m
= 1.4 m/s
During the 5 seconds of braking, the wheel gradually stops rotating. At the end of the 5 seconds, the angular speed is 0 rev/s.
To find the change in angular displacement, we can use the formula:
Change in angular displacement = (final angular speed - initial angular speed) * time
= (0 rev/s - 2 rev/s) * 5 s
= -2 rev/s * 5 s
= -10 rev
Notice that the change in angular displacement is negative because the wheel is rotating in the opposite direction during braking.
Finally, substitute the values into the formula to find the number of revolutions:
Number of revolutions = (-10 rev) / (2π)
≈ -1.59 rev
The negative sign indicates that the wheel turned in the opposite direction during braking. However, revolutions cannot be negative in this context, so it means that the wheel did not complete any full revolutions during the braking.