A cyclist starts from rest and pedals so that the wheels make 7.7 revolutions in the first 5.1 s. What is the angular acceleration of the wheels (assumed constant)?
(1/2)*alpha*t^2 = 2 pi * 7.7 radians
alpha = 4 pi * 7.7/ t^2
alpha is the angular acceleration in radians/s^2
To find the angular acceleration of the wheels, we need to use the formula:
Angular acceleration (α) = Δω / Δt
Where:
Δω is the change in angular velocity
Δt is the change in time
In this case, we are given the number of revolutions made by the wheels and the time it took, so we need to convert these values into angular velocity.
The number of revolutions can be converted into radians by multiplying it by 2π (since 1 revolution = 2π radians).
Given:
Number of revolutions = 7.7
Time = 5.1 s
Converting the number of revolutions to radians:
Angular displacement (θ) = Number of revolutions * 2π
θ = 7.7 * 2π
To find the angular velocity, we divide the angular displacement by the time taken:
Angular velocity (ω) = θ / t
ω = (7.7 * 2π) / 5.1
Now, we can calculate the angular acceleration using the formula mentioned earlier:
Angular acceleration (α) = Δω / Δt
In this case, Δω is the change in angular velocity, which is the final angular velocity minus the initial angular velocity. Since the cyclist starts from rest, the initial angular velocity is 0.
So, α = (ω - 0) / Δt
α = ω / Δt
Plugging in the values:
α = [(7.7 * 2π) / 5.1] / 5.1
Simplifying this expression will give us the answer, which represents the angular acceleration of the wheels.