The combination of an applied force and a frictional force produces a constant total torque of 38.1 Nm on a wheel rotating about a fixed axis. The applied force acts for 7.00 s, during which time, the angular speed of the wheel increases from 2.40 rad/s to 12.5 rad/s. The applied force is then removed. The wheel comes to rest in 78.4 s. What is the total number of revolutions of the wheel? Answer in units of revolutions.
To find the total number of revolutions of the wheel, we need to first calculate the angular acceleration of the wheel when the applied force is acting on it.
We know that torque (τ) is given by the equation:
τ = I * α
where I is the moment of inertia of the wheel and α is the angular acceleration.
The applied force (F) and the frictional force (f) produce a constant total torque (τ) of 38.1 Nm. So we have:
τ = F * r + f * r
where r is the radius of the wheel.
Since the applied force acts for 7.00 s, we can find the change in angular speed (Δω) using the formula:
Δω = α * Δt
We can rearrange the torque equation to solve for α:
α = τ / I
To find the moment of inertia of the wheel (I), we need to know its mass (m) and the radius (r):
I = 0.5 * m * r^2
Next, we need to find the initial angular acceleration (α_initial) when the applied force is acting on the wheel. We can use the initial angular speed (ω_initial) and the final angular speed (ω_final) along with the time taken (Δt) to calculate α_initial:
α_initial = (ω_final - ω_initial) / Δt
Once we have α_initial, we can calculate the angular displacement (θ_initial) during the time the applied force acts on the wheel using the formula:
θ_initial = ω_initial * Δt + 0.5 * α_initial * (Δt)^2
Next, we need to find the final angular acceleration (α_final) when the wheel comes to rest. We can use the final angular speed (ω_final = 0) and the time taken (Δt_final = 78.4 s) to calculate α_final:
α_final = -ω_final / Δt_final
Now, we can calculate the total angular displacement (θ_final) during the time the wheel comes to rest using the formula:
θ_final = ω_initial * Δt_final + 0.5 * α_final * (Δt_final)^2
Finally, we can calculate the total angle covered by the wheel in revolutions (θ_total_revolutions):
θ_total_revolutions = (θ_initial + θ_final) / (2π)
Just substitute the given values into the equations and perform the calculations to find the total number of revolutions of the wheel.