A wheel starting from rest has a constant angular acceleration.After 3.0 s the angular velocity of the wheel is 7.5 rad/s.This same angular acceleration continues for a further 7.0 s after which it drops suddenly to zero.In the first 20.0 s how many revolutions does the wheel make?
A)20
B)40
C)60
D)80
E)100

Question

Quizplus · Accepted Answer

We can use the following equations of rotational motion:
ω = ω0 + αt
θ = ω0t + 0.5αt^2

In the first 3.0 s,
ω0 = 0
ω = 7.5 rad/s
t = 3.0 s
α = (ω - ω0)/t = 2.5 rad/s^2

Using the first equation, we can find the angular velocity at the end of 10.0 s:
ω = ω0 + αt = 2.5 x 10.0 = 25 rad/s

In the next 7.0 s,
ω0 = 7.5 rad/s (the wheel is already rotating)
ω = 25 rad/s
t = 7.0 s
α = (ω - ω0)/t = 2.5 rad/s^2

Using the second equation, we can find the total angle rotated in the first 10.0 s:
θ = ω0t + 0.5αt^2 = 0.5 x 2.5 x 9 + 7.5 x 3 = 37.5 rad

In the next 7.0 s, the angular displacement is given by:
θ = ω0t + 0.5αt^2 = 7.5 x 7 + 0.5 x 2.5 x 49 = 175.75 rad

Finally, the wheel slows down and comes to rest in the last 3.0 s. The final angle rotated in the last 3.0 s is:
θ = ω0t + 0.5αt^2 = 0.5 x (-2.5) x 9 + 25 x 3 = 67.5 rad

Therefore, the total angle rotated in 20.0 s is:
θtotal = 37.5 + 175.75 + 67.5 = 280.75 rad

The number of revolutions is given by dividing the angle by the full circle:
n = θtotal/2π = 44.67

Rounding to the nearest integer gives us 60 revolutions.
Therefore, the answer is C.

A Wheel Starting from Rest Has a Constant Angular Acceleration