Torque and Rotational Dynamics
Torque and Rotational Dynamics
Torque measures a force's ability to rotate a body about a chosen axis: τ = r F sin θ, with the sign set by the right-hand rule or a stated convention. The rotational analogue of Newton's second law is τ_net = I α, where the moment of inertia I = Σ m r² depends on both the mass distribution and the chosen axis; the parallel-axis theorem, I = I_cm + M d², shifts I between axes. Static equilibrium requires both ΣF = 0 and Στ = 0. For extended bodies, weight acts at the center of mass, which also determines stability.
Traps: computing torque with the full distance r but forgetting the sin θ factor; forgetting that I changes when the axis changes; leaving gravity out of the torque sum in a ladder or beam problem; and treating a body as a point mass when its rotation matters.
Worked example: a uniform rod of length L and mass m is pivoted at one end and released from horizontal. About the pivot, τ = mg(L/2) and I = mL²/3, so α = τ/I = 3g/(2L). The result is independent of m, and exceeds g/L because the mass sits closer to the pivot than the tip.
Rule of thumb: place the pivot at the point where an unknown force acts, so that force contributes zero torque and drops out of the equation.