A wheel with rotational inertia I initially rotates with a angular speed ωi . A torque is applied, speeding the wheel up to three times the angular speed in time Δt . A) Find the magnitude of the applied torque. B) Find the work done on the wheel during this time.
To solve this problem, we'll need to use the concept of rotational kinematics and energy.
Let's start with part A, finding the magnitude of the applied torque.
The relationship between torque (τ), moment of inertia (I), and angular acceleration (α) is given by the equation:
τ = I * α
In this case, the wheel rotates with an initial angular speed ωi, and it speeds up to three times this angular speed in time Δt. We need to find the torque that causes this acceleration.
The angular acceleration (α) can be calculated using the following equation:
α = (ωf - ωi) / Δt
where ωf is the final angular speed.
Since the wheel speeds up to three times the initial angular speed, ωf = 3ωi.
Plugging this value into the equation for α, we have:
α = (3ωi - ωi) / Δt = 2ωi / Δt
Now, we can substitute this value of α into the torque equation:
τ = I * α = I * (2ωi / Δt)
This gives us the expression for torque in terms of the given quantities.
Moving on to part B, finding the work done on the wheel during this time:
The work done on an object is given by the product of the torque and the angular displacement (θ). However, since the angle is not given, we need to find it.
The angular displacement (θ) can be calculated using the equation:
θ = ωi * Δt + (1/2) * α * (Δt)^2
Substituting the value of α:
θ = ωi * Δt + (1/2) * (2ωi / Δt) * (Δt)^2
= ωi * Δt + ωi * Δt
= 2ωi * Δt
Now, we have the value for θ in terms of the given quantities.
The work done (W) can be calculated using the following equation:
W = τ * θ
Substituting for τ and θ:
W = (I * (2ωi / Δt)) * (2ωi * Δt)
= 4I * ωi^2
This gives us the expression for the work done on the wheel.
To summarize:
A) The magnitude of the applied torque is given by: τ = I * (2ωi / Δt)
B) The work done on the wheel during this time is given by: W = 4I * ωi^2