A torque of 77.7Nm causes a wheel to start from rest , completes 5.55revolutions and attains a final angular velocity of 88.8rad/sec . What is the moment of inertia of the wheel?
To find the moment of inertia of the wheel, we can use the formula that relates torque, moment of inertia, and angular acceleration.
The formula is: τ = I * α
Where:
τ is the torque,
I is the moment of inertia,
and α is the angular acceleration.
Given:
τ = 77.7 Nm
α = Δω/Δt = (ωf - ωi) / t = (88.8 rad/s - 0 rad/s) / t
ϴ = 5.55 revolutions = 5.55 * 2π radians (since 1 revolution = 2π radians)
ti = 0 s (initial time)
tf = t (final time)
We need to calculate the angular acceleration α first. From the given information, we can rearrange the formula:
α = (ωf - ωi) / t = (88.8 rad/s - 0 rad/s) / t = 88.8 rad/s / t
Now, let's convert the revolutions to radians:
ϴ = 5.55 revolutions * 2π radians/revolution = 5.55 * 2π radians
Since the wheel starts from rest, the initial angular velocity (ωi) is 0 rad/s. And the final angular velocity (ωf) is given as 88.8 rad/s.
Using the formula for the angular displacement ϴ = ωi * t + (1/2) * α * t^2, we can solve for t:
ϴ = ωi * t + (1/2) * α * t^2
5.55 * 2π radians = 0 rad/s * t + (1/2) * (88.8 rad/s / t) * t^2
5.55 * 2π = 0 + (1/2) * 88.8 * t
5.55 * 2π = 44.4 * t
t = (5.55 * 2π) / 44.4
Now that we have the value of t, we can substitute it back into the formula for α to solve for α:
α = 88.8 rad/s / t = 88.8 rad/s / ((5.55 * 2π) / 44.4)
Finally, we can substitute the values of torque (τ) and angular acceleration (α) into the formula τ = I * α, and solve for I:
I = τ / α = 77.7 Nm / (88.8 rad/s / ((5.55 * 2π) / 44.4))
Calculating this expression will give us the moment of inertia of the wheel.