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Torque and Rotational Inertia: What Starts and Stops a Spin

A force alone does not tell you whether something will spin — where and how you push matters. Meet torque, meet the moment of inertia (rotation's version of mass), and put them together into Newton's second law for rotation.

Torque: the turning effect of a force

Push a door right at the hinge and it barely moves; push at the outer edge and it swings easily. Same force, wildly different result — because rotation cares about where the force acts. The turning effect of a force is the torque \tau, and it depends on the force, the distance from the axis, and the angle.

A wrench turned by a force: torque is the force times the perpendicular lever arm, so a longer arm or a squarer push turns harder.

\tau = r\,F\sin\theta = F\,r_{\perp}

Torque equals force times the perpendicular lever arm r⊥ = r sinθ; it is largest when the force is perpendicular to the arm.

The quantity r_{\perp} = r\sin\theta is the lever arm: the perpendicular distance from the axis to the line of the force. Pushing straight toward or away from the hinge (\theta = 0) gives zero torque — which is why you instinctively push a door at right angles. Torque is measured in newton-metres (N·m).

Balance: when torques cancel

On a seesaw, a small child far out can balance a heavy adult near the middle. Balance is not about equal weights — it is about equal and opposite torques. When the net torque about the pivot is zero, there is no angular acceleration: the object is in rotational equilibrium.

Slide the weights and adjust their masses: the lever balances only when the clockwise and counter-clockwise torques are equal. Try a small mass far out against a large mass near the pivot.

\sum \tau = 0 \quad\Longleftrightarrow\quad m_{1}g\,d_{1} = m_{2}g\,d_{2}

Rotational equilibrium: the total torque about the pivot is zero, so each side's weight times its lever arm must match.

True static equilibrium also needs zero net force (translational equilibrium). The two conditions together — no net force and no net torque — are the foundation of statics, from balancing beams to standing bridges.

Moment of inertia: rotation's version of mass

In straight-line motion, mass measures how hard it is to change an object's velocity. In rotation, the analogous 'rotational mass' is the moment of inertia I. But there is a twist: I depends not just on how much mass there is, but on how far that mass sits from the axis.

The same mass placed far from the axis has a much larger moment of inertia than when packed near the axis — distance counts twice (r²).

I = \sum_{i} m_{i} r_{i}^{2} \qquad\Longrightarrow\qquad I_{\text{hoop}} = MR^{2},\ \ I_{\text{disk}} = \tfrac{1}{2}MR^{2},\ \ I_{\text{sphere}} = \tfrac{2}{5}MR^{2}

Moment of inertia sums each mass times its squared distance from the axis; the standard shapes differ only by a numerical factor.

Because the distance is squared, moving mass outward is doubly punishing: a hoop (all mass at radius R) has twice the moment of inertia of a solid disk of the same mass and radius. This single fact — that where the mass sits matters — will decide the rolling race in Guide 5. If you know I about the center of mass, the parallel-axis theorem gives I about any parallel axis: I = I_{\text{cm}} + Md^{2}.

Newton's second law, rotational form

Now assemble the pieces. Force causes linear acceleration through F = ma; torque causes angular acceleration through the exact rotational parallel, Newton's second law for rotation.

\tau_{\text{net}} = I\,\alpha \qquad\longleftrightarrow\qquad F_{\text{net}} = m\,a

Net torque equals moment of inertia times angular acceleration — the rotational F = ma.