torque and moment of inertia explained
Just as force produced acceleration in straight-line motion, torque produces angular acceleration in rotation. A door opens easily when you push far from its hinges, at the handle, because torque depends on the distance to the axis as well as on the size of the force. And some objects turn readily under the same torque while others are sluggish; what measures the difference is the moment of inertia I, the mass of rotation, which describes how far the mass is spread from the axis. This lesson sets up torque and moment of inertia in turn and joins them into Newton's law for rotation. It also shows how the moment of inertia changes with shape and with the position of the axis.
What makes something turn is not only the size of the force but where it is applied. Torque M = F × d is the force times the lever arm d, the distance to the axis, so the same force turns more strongly the farther from the axis it is applied. Increase with the slider the distance r to the point where the force acts, and the dashed lever arm lengthens while the torque in the readout grows in proportion to it. That happens even though the orange force arrow keeps the same size. A long wrench loosens a tight bolt, and a light child at the end of a seesaw balances a heavy adult, for the same reason.
In rotation, the role of mass is played by the moment of inertia I = Σ m r². The same mass contributes more the farther it sits from the axis, with the square of its radius r, raising the moment of inertia I, how hard the body is to set turning. Increase r with the slider to send the two gold masses away from the axis, and although the mass stays the same, the blue bar grows rapidly with the square of the distance. Double the distance and the moment of inertia becomes four times larger. That is why a figure skater spins faster with arms pulled in and slower with arms spread.
Combine the two and Newton's second law for rotation, ΣM = Iα, appears. The net torque produces angular acceleration, and the constant of proportionality is the moment of inertia I, so it has the same form as ΣF = ma for straight-line motion, with only the letters changed. Increase the torque M with the slider and the angular acceleration α = MI grows in proportion, the gold bar lengthening with it. The moment of inertia is held fixed, so doubling the torque doubles the angular acceleration. Just as mass resists acceleration in a straight line, the moment of inertia resists angular acceleration in rotation.
Even with the same mass and radius, the factor β in I = βMR² depends on shape, decreasing in the order hoop (1), disk (½), sphere (⅖). The hoop, with its mass gathered at the rim, has the largest value, and the sphere, filled evenly to the center, the smallest. Roll these objects down a slope and the smaller β is, the less they are held back by rotation and the faster they go down; the rolling acceleration depends on β (acceleration ∝ 1(1+β)). Switch shapes with the buttons and the green bar shows the rolling-acceleration factor for that shape, longest for the sphere and shortest for the hoop. So a ball and a hoop released together on the same slope always end with the ball arriving first.
Even for the same object, moving the axis of rotation changes the moment of inertia I. The parallel-axis theorem I = Icm + M d² gives the new value: move the axis a distance d away from the axis through the center of mass, and M d² is added on. In the figure, the green dashed line is the axis through the center of mass and the gold line the shifted axis of rotation. Increase d with the slider and the blue bar, I, grows with the square of the distance. Because the axis through the center of mass always gives the smallest value, a door swung about its hinges feels heavier than one turned about its middle would.