Angular Momentum
The moment of momentum of a particle and of a rigid body, \(L = r \times p\) and \(L = I\omega\). Torque as the rate of change of angular momentum, and angular impulse. The conservation law with its skaters, clutches and orbits, and impacts on bodies held by a pivot.
Why this chapter
Force changes momentum, and when no outside force acts, momentum is conserved. Turning has a twin of each idea. Torque changes angular momentum, and when no outside torque acts, angular momentum is conserved. This chapter builds the new quantity for a particle and for a rigid body. It then uses the twin laws on three everyday puzzles. Why does a skater spin faster with her arms drawn in? Why does a fan take a minute to stop? Why does a ball that sticks to a hinged rod set it swinging? Many problems that force and momentum cannot touch, because a pivot pushes with an unknown force, fall open to angular momentum about that pivot.
You will learn
- Find the angular momentum of a particle about a point by \(rp\sin\theta\), by the moment arm, and by components.
- Explain why a particle moving in a straight line has a constant, non-zero angular momentum about any point off its line.
- Use \(L = I\omega\) for a rigid body on a fixed axis, and the torque law \(\vec\tau_{\text{ext}} = d\vec L/dt\) for a particle, a system and a body.
- Use the angular impulse, \(\int\tau\,dt = \Delta L\), for flywheels, brakes and short blows.
- Apply the conservation of angular momentum to bodies that change shape, and account for the kinetic energy that changes.
- Solve impacts on pivoted bodies with angular momentum about the pivot, and find the share of kinetic energy kept, \(md^2/I_{\text{total}}\).
Before you begin
From the chapter on Momentum and Its Conservation you should know momentum, \(\vec p = m\vec v\), and why internal forces cannot change the total momentum of a system. From Rigid Bodies and Moment of Inertia you should know \(I = \sum mr^2\), the table of standard bodies and the parallel-axis theorem. You should know the kinetic energy of rotation, \(\tfrac12 I\omega^2\), and impulse, \(\int F\,dt = \Delta p\). You should be able to take a cross product and integrate a polynomial.
Lessons
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%Glossary
The key words of this chapter, with the meaning this chapter gives them.
- angular impulse
- The time integral of a torque, \(J = \int\tau\,dt\), the area under a torque-time graph. It equals the change in angular momentum it produces. Its unit is N m s, the same as kg m²/s.
- angular momentum
- The moment of momentum about a point, \(\vec L = \vec r\times\vec p\), of magnitude \(mvr\sin\theta\). For a rigid body turning about a fixed axis it is \(L = I\omega\) about that axis. Its unit is kg m²/s.
- clutch
- A device that brings two spinning shafts or discs into contact so that friction makes them turn together. It conserves angular momentum and loses kinetic energy as heat.
- conservation of angular momentum
- The law that a system's total angular momentum about a fixed axis stays constant when the net external torque about that axis is zero. It holds even if the system changes shape.
- flywheel
- A heavy wheel on an axle, with a large moment of inertia. It stores angular momentum and energy, and keeps a machine turning steadily.
- impulsive force
- A very large force that acts for a very short time, such as a blow or an impact. Its size is hard to know, but its impulse, \(\int F\,dt\), is easy to use.
- internal forces
- Forces that the parts of a system exert on one another. They come in equal and opposite pairs along one line, so their torques cancel and they cannot change the total angular momentum.
- moment arm
- The perpendicular distance from a point to the line along which a momentum or a force acts, \(d = r\sin\theta\). Angular momentum is momentum times moment arm, and torque is force times moment arm.
- pivot
- A fixed point or axle about which a body is held and can turn, such as a hinge or a pin. In an impact it pushes with an unknown force, but that force has no torque about the pivot.
- right-hand rule
- The rule for the direction of a cross product. Curl the fingers of the right hand from the first vector towards the second; the thumb points along the product.
- torque
- The moment of a force about a point, \(\vec\tau = \vec r\times\vec F\), of magnitude \(rF\sin\theta\), in N m. The net external torque equals the rate of change of angular momentum.