Unit 6: Energy and Momentum of Rotating Systems

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

• Rotational kinetic energy

  The energy an object has because it spins, $K = \frac{1}{2}I\omega^2$. Doubling the angular speed makes it four times as large.

  Example. A flywheel with $I = 2.0$ kg·m² spinning at 10 rad/s stores $\frac{1}{2}(2.0)(10)^2 = 100$ J.

• Work done by a torque

  A constant torque that turns an object through an angle does work $W = \tau\Delta\theta$, with the angle in radians.

  Example. A 5.0 N·m torque that turns a crank through 4.0 rad does 20 J of work.

• Work-energy for rotation

  The net work done by torques on a rigid object equals its change in rotational kinetic energy.

  Example. If friction torque does -30 J of work on a spinning wheel with 50 J of rotational energy, the wheel is left with 20 J.

• Angular momentum of a rigid object

  For an object spinning about a fixed axis, $L = I\omega$, in kg·m²/s. Its direction is set by the sense of the rotation.

  Example. A wheel with $I = 0.50$ kg·m² spinning at 8.0 rad/s has $L = 4.0$ kg·m²/s.

• Angular momentum of a point mass

  A small object moving with momentum mv has angular momentum $L = mvr_\perp$ about a point, where $r_\perp$ is the perpendicular distance from the point to its line of motion.

  Example. A 0.20 kg ball moving at 5.0 m/s along a line passing 2.0 m from a post has $L = 2.0$ kg·m²/s about the post.

• Angular impulse

  A torque acting over a time changes angular momentum: $\tau\Delta t = \Delta L$. On a torque-time graph, the area is the angular impulse.

  Example. A 0.30 N·m torque acting for 10 s raises a turntable’s angular momentum by 3.0 kg·m²/s.

• Conservation of angular momentum

  If the net external torque on a system is zero, its total angular momentum stays constant, so $I_i\omega_i = I_f\omega_f$.

  Example. A cat twisting in midair can change its shape, but its total angular momentum stays the same because no outside torque acts on it.

• Changing rotational inertia

  With angular momentum conserved, pulling mass toward the axis lowers I and raises $\omega$; rotational kinetic energy rises, with the extra energy supplied by the work of pulling in.

  Example. A diver who pulls into a tight tuck spins faster, and the work her muscles do pulling inward supplies the added rotational kinetic energy.

• Rolling without slipping

  Rolling in which the contact point does not slide, so the center moves at $v = \omega R$ and the contact point is momentarily at rest.

  Example. A tire of radius 0.40 m rolling at 10 m/s spins at 25 rad/s, and the bottom of the tire is at rest on the road.

• Kinetic energy of a rolling object

  A rolling object has both translational and rotational kinetic energy: $K = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$.

  Example. A rolling 1.0 kg hoop at 2.0 m/s has 2.0 J of translational and 2.0 J of rotational energy, 4.0 J in all.

• Shape factor in rolling

  Writing $I = \beta MR^2$, an object rolling from rest down a drop h reaches $v = \sqrt{2gh/(1 + \beta)}$; a smaller $\beta$ means a faster roll, whatever the mass or radius.

  Example. A solid ball ($\beta = 2/5$) beats a hollow ball ($\beta = 2/3$) down any ramp, even if the hollow ball is larger or heavier.

• Friction in rolling

  Static friction at the contact point makes a rolling object spin, but it does no work when there is no slipping, so mechanical energy is conserved.

  Example. A ball rolling down a rough ramp without slipping arrives with the same total kinetic energy as the gravitational energy it lost.

• Orbital speed

  In a circular orbit, gravity supplies the centripetal force, so $GMm/r^2 = mv^2/r$ and $v = \sqrt{GM/r}$, which does not depend on the satellite’s mass.

  Example. A small probe and a large space station in the same circular orbit move at the same speed.

• Orbital period

  For a circular orbit, $T = 2\pi r/v$, which gives $T^2 = 4\pi^2r^3/(GM)$: larger orbits take much longer.

  Example. A satellite in an orbit four times as wide as another’s around the same planet takes eight times as long to go around.

• Angular momentum in an elliptical orbit

  Gravity points toward the central body, so it exerts no torque about it, and the orbiting object’s angular momentum about that body stays constant.

  Example. A comet moves fastest when it swings closest to the Sun and slowest when it is farthest away.

• Energy of an orbit

  With gravitational potential energy $U = -GMm/r$ (zero far away), the sum of kinetic and potential energy stays constant along an orbit under gravity alone.

  Example. As a satellite climbs from its closest point to its farthest point, its kinetic energy falls by exactly as much as its potential energy rises.

• Escape speed

  The launch speed that gives a total energy of zero, $v = \sqrt{2GM/R}$, just enough to coast away from the body without returning.

  Example. Escape speed from a planet’s surface is $\sqrt{2}$ times the speed of an orbit skimming that same surface.