Topic 5: Torque and rotational dynamics

Up to now you have tracked objects sliding and falling along straight lines, but the world also spins: wheels, gears, pulleys, and balanced beams. This unit recasts familiar ideas like velocity, mass, and force into their rotational cousins, then shows that a single law, torque = I alpha, governs everything that turns.

Rotational kinematics and angular quantities

When an object rotates, every point on it sweeps through the same angle in the same time, so we describe the motion with angular quantities rather than tracking each point separately. Angular position theta is measured in radians, where one full turn equals 2 pi rad. Angular velocity omega (rad/s) is how fast theta changes, and angular acceleration alpha (rad/s^2) is how fast omega changes. These three play exactly the roles that x, v, and a play in straight-line motion. Because of that parallel, the constant-acceleration kinematics equations carry over directly: just swap x for theta, v for omega, and a for alpha. For example, omega_final = omega_initial + alpha t, and theta = omega_initial t + (1/2) alpha t^2. A spinning fan slowing to a stop, a wheel speeding up from rest, and a record turning at steady speed are all handled by these rotational kinematics equations.

Linking linear and rotational motion

A point on a rotating object also moves through space, so each angular quantity has a linear partner connected by the radius r, the distance from the axis of rotation. The arc length traveled is s = r theta, the linear (tangential) speed is v = r omega, and the tangential acceleration is a = r alpha. The key insight is the factor of r: points farther from the axis cover more distance and move faster even though every point shares the same omega and alpha. This is why the rim of a merry-go-round whips by while the center barely moves. The relationship v = r omega is also the bridge to rolling motion: when a wheel rolls without slipping, the speed of its center equals r omega, tying the spin of the wheel to how fast it travels down the road. Remember these equations require omega and alpha in radian-based units.

Torque and the lever arm

A force makes something rotate only if it is applied off the axis and not pointed straight at or away from it. The rotational effect of a force is called torque, written tau, with units of newton-meters (N m). Torque depends on three things: how big the force is, how far from the axis it is applied, and the angle at which it pushes. The general expression is tau = r F sin(theta), where theta is the angle between the force and the line from the axis to the point of application. The quantity r sin(theta) is the lever arm (or moment arm), the perpendicular distance from the axis to the line of the force. Torque is largest when the force is perpendicular to that line (theta = 90 degrees) and zero when the force points straight through the axis. This is everyday experience: pushing a door near the hinge, or at a glancing angle, barely turns it, while pushing perpendicular at the handle swings it easily. Torque has a sign or direction, conventionally counterclockwise positive and clockwise negative.

Moment of inertia

In straight-line motion, mass measures how strongly an object resists changes in velocity. In rotation, the analogous quantity is the moment of inertia I, which measures how strongly an object resists changes in its spin. Crucially, I depends not just on how much mass there is but on how that mass is distributed relative to the axis. Mass placed far from the axis contributes far more, because for a single point mass I = m r^2: the distance is squared. Extended objects are sums of such contributions, which is why standard shapes have standard formulas, such as I = (1/2) M R^2 for a solid disk and I = M R^2 for a thin hoop about its center. The same object can have different moments of inertia about different axes. This is why a figure skater spins faster when pulling their arms in: bringing mass closer to the axis lowers I, and the spin responds. The units of moment of inertia are kg m^2.

Newton's second law for rotation: torque = I alpha

Linear motion obeys F_net = m a; rotation obeys the parallel law tau_net = I alpha. The net torque on an object equals its moment of inertia times its angular acceleration. The structure is identical: torque plays the role of force, moment of inertia plays the role of mass, and angular acceleration plays the role of linear acceleration. To use it, add up every torque acting on the object (with proper signs for clockwise versus counterclockwise), set the total equal to I alpha, and solve. A larger net torque produces a faster change in spin, while a larger moment of inertia produces a slower change for the same torque. This single equation predicts how quickly a wheel spins up, how fast a beam rotates after a push, and how a pulley responds to unequal tensions on its two sides.

Worked example: a torque problem

A uniform solid disk pulley of mass M = 2.0 kg and radius R = 0.10 m can turn freely about its center. A rope wrapped around the rim is pulled with a steady force F = 6.0 N tangent to the rim. Find the angular acceleration. Step 1, find the torque: the force is tangent, so the lever arm is the full radius and tau = F R = (6.0 N)(0.10 m) = 0.60 N m. Step 2, find the moment of inertia of a solid disk: I = (1/2) M R^2 = (1/2)(2.0 kg)(0.10 m)^2 = 0.010 kg m^2. Step 3, apply tau = I alpha and solve for alpha: alpha = tau / I = (0.60 N m) / (0.010 kg m^2) = 60 rad/s^2. The disk speeds up its spin at 60 rad/s^2. Notice the recipe mirrors a linear problem: find the net push, find the inertia, then divide.

Rotational (static) equilibrium and balanced beams

An object is in complete equilibrium when it has no tendency to accelerate either in translation or in rotation. That means two conditions must both hold: the net force is zero (F_net = 0) and the net torque is zero (tau_net = 0). The second condition is what keeps a beam, bridge, or shelf from tipping or rotating. A powerful trick is that you may compute torques about any axis you choose, since a balanced object is balanced about every point. Choosing the axis at an unknown force makes that force's torque zero, removing it from the equation and simplifying the algebra. For a classic balanced seesaw, set the clockwise torque from one weight equal to the counterclockwise torque from the other: m1 g d1 = m2 g d2. A lighter person sitting farther out can balance a heavier person sitting closer in, because torque depends on distance, not just weight. For a uniform beam, treat its entire weight as acting at its center of mass when computing its torque.