Unit 5: Torque and Rotational Dynamics

17 terms. Say what each one means, then open the card to check.

• Angular displacement

  The angle an object turns through, measured in radians; an arc of length s on a circle of radius r spans $\Delta\theta = s/r$. One revolution is $2\pi$ rad.

  Example. A clock’s minute hand turns through $\pi/2$ rad, a quarter turn, between 12:00 and 12:15.

• Angular velocity

  How fast the angle changes, $\omega = \Delta\theta/\Delta t$, in rad/s. Every point of a rigid spinning object has the same angular velocity.

  Example. A record turning at 33 revolutions per minute has $\omega = 33(2\pi)/60 \approx 3.5$ rad/s.

• Angular acceleration

  How fast the angular velocity changes, $\alpha = \Delta\omega/\Delta t$, in rad/s².

  Example. A drill bit that goes from rest to 300 rad/s in 0.50 s has an average angular acceleration of 600 rad/s².

• Rotational kinematics equations

  With constant angular acceleration, the linear motion equations carry over: $\omega = \omega_0 + \alpha t$, $\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ and $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$.

  Example. A reel starting from rest with $\alpha = 2.0$ rad/s² turns through $\frac{1}{2}(2.0)(3.0)^2 = 9.0$ rad in 3.0 s.

• Tangential speed

  The speed of a point on a spinning object, $v = r\omega$; points farther from the axis move faster, though all share the same $\omega$.

  Example. On a fan spinning at 10 rad/s, a point 0.30 m from the hub moves at 3.0 m/s and a point 0.10 m out moves at 1.0 m/s.

• Tangential acceleration

  The part of a point’s acceleration along its circular path, $a_t = r\alpha$; it changes the point’s speed, while the centripetal part, $\omega^2 r$, turns its direction.

  Example. A point 0.50 m from the axis of a wheel with $\alpha = 4.0$ rad/s² has a tangential acceleration of 2.0 m/s².

• Torque

  The turning effect of a force about an axis, $\tau = rF\sin\theta$, where r is the distance from the axis to where the force acts and $\theta$ is the angle between r and F. Units: N·m.

  Example. A 20 N push at right angles to a gate, 1.5 m from its hinge, gives a torque of 30 N·m about the hinge.

• Lever arm

  The perpendicular distance from the axis to the line along which a force acts. Torque equals the force times its lever arm.

  Example. A force whose line of action passes 0.20 m from a bolt has a lever arm of 0.20 m, however long the wrench is.

• Sign of a torque

  Torques are given a direction: pick one sense of turning, such as counterclockwise, as positive, and the opposite sense as negative.

  Example. On a seesaw, a child on the left gives a counterclockwise torque and a child on the right gives a clockwise torque.

• Rotational inertia

  An object’s resistance to changes in its rotation about an axis. It depends on the mass and on how far that mass is from the axis.

  Example. A tightrope walker’s long pole has a large rotational inertia, so it turns slowly when she starts to tip.

• Rotational inertia of point masses

  For small masses at distances r from the axis, $I = \sum mr^2$, in kg·m². Doubling a mass’s distance from the axis makes its share four times as large.

  Example. A 0.10 kg bead 0.50 m from an axis adds $(0.10)(0.50)^2 = 0.025$ kg·m² to the rotational inertia.

• Rotational equilibrium

  The state of an object whose net torque is zero, so its angular velocity does not change; it may be at rest or spinning at a steady rate.

  Example. A ceiling fan turning at a steady speed is in rotational equilibrium: the motor’s torque balances the air’s drag torque.

• Static equilibrium

  An object at rest that stays at rest needs both a net force of zero and a net torque of zero about every axis.

  Example. A bookshelf bracket holds a shelf still only if the upward forces match the weight and the torques about the wall cancel.

• Choosing a pivot point

  In an equilibrium problem you may take torques about any point; picking the point where an unknown force acts removes that force from the torque equation.

  Example. For a plank on two supports, taking torques about the left support leaves the right support’s force as the only unknown.

• Center of gravity of a uniform object

  For torque purposes, the whole weight of a uniform beam, rod or plank acts at its middle.

  Example. A uniform 3.0 m board’s weight acts 1.5 m from either end, wherever the board is supported.

• Newton’s second law for rotation

  The net torque on an object equals its rotational inertia times its angular acceleration: $\tau_{\text{net}} = I\alpha$.

  Example. A net torque of 3.0 N·m on a wheel with $I = 0.50$ kg·m² gives it an angular acceleration of 6.0 rad/s².

• Pulley with mass

  A pulley with rotational inertia needs a net torque to speed up, so the string tensions on its two sides differ, and the hanging masses accelerate less than with a light pulley.

  Example. When a heavy block falls and turns a massive pulley, the string tension is less than the block’s weight, and the gap supplies the pulley’s torque.