Rotational work and power for rigid object

1. rotational work

When a force causes a torque by rotating an object, the agent that exerted this force does a work. If the agent rotates the object by an angle dθ, during a time interval dt; the work done by the force F is:
dW = F . ds = F ds cos θ = F cos θ ds =
F_t ds = F_t r dθ = τ dθ

Where F_t is the tangential component of the force F, and τ = r F_t



This torque is the net torque. That is the sum of all the torques in the object τ = Σ τ_i.

The net work done by this torque rotating the object OP from an angle θ_i to an angle θ_f is:

W_net = ∫ τ dθ
From θ_i to θ_f

If the torque τ is constant, we will have:
W_net = τ (θ_f - θ_f)

The general case is:
W _net= ∫ τ dθ = ∫ I α dθ
From θ_i to θ_f
Where I is the moment of inertia of the object.

Since α = dθ²/dt² = dω/dt and
ω = dθ/dt
Then:
α dθ = (dω/dt) dθ = dω (dθ/dt) = dω ω

Therefore:
W_net = I (dθ/dt) = I ∫ dω ω =
From ω_i to ω_f
= (1/2) I (ω_f² - ω_i²)

2. Power delivered by a toque

From the relationship dW = τ dθ, we obtain the expression of the power( the rate at which work is done by the torque):
P = dW/dt = τ dθ/dt = τ ω

The work done by a torque and its power delivered may be negative. This is the case when the torque is slowing the rotation : the angle θ and the torque due to the force have opposite signs.