A 4.1-m-diameter merry-go-round is initially turning with a 4.2 period. It slows down and stops in 24 .
How many revolutions does the merry-go-round make as it stops?
To find the number of revolutions the merry-go-round makes as it stops, we can first calculate the initial angular velocity (ω) of the merry-go-round using the given information.
The formula to calculate angular velocity is:
ω = 2π / T
Where:
ω is the angular velocity
T is the period
Given that the period (T) is 4.2 s, we can substitute the value into the formula:
ω = 2π / 4.2
Next, we can calculate the final angular velocity (ω_f) of the merry-go-round when it stops. Since the merry-go-round slows down and stops, the final angular velocity would be 0 (ω_f = 0).
Now, we can use the equation of motion for rotational motion to calculate the angular displacement (θ) covered by the merry-go-round as it slows down and comes to a stop:
ω_f^2 = ω_i^2 + 2 α θ
Where:
ω_f is the final angular velocity
ω_i is the initial angular velocity
α is the angular acceleration
θ is the angular displacement
Since ω_f = 0, we can rearrange the equation and solve for θ:
θ = (ω_f^2 - ω_i^2) / (2 α)
Since α is also equal to the average angular acceleration, we can calculate it using the formula:
α = (ω_f - ω_i) / t
Where:
t is the time taken to stop
Given that the time taken to stop (t) is 24 s, we can calculate α using the formula:
α = (0 - ω_i) / t
Now, substitute the values of ω_i and t into the formula:
α = (0 - (2π / 4.2)) / 24
Once we have calculated α, we can substitute it into the equation for θ:
θ = (0 - (2π / 4.2))^2 / (2 * ((0 - (2π / 4.2)) / 24))
Finally, we can calculate the number of revolutions by dividing the angular displacement by 2π (since one revolution is equal to 2π radians):
Number of revolutions = θ / (2π)
Plug in the calculated value of θ into the equation to find the number of revolutions.