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Dynamics of Rolling

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Why this chapter
Why this chapter exists

A wheel on a road, a ball on a slope and a cue ball on a snooker table all move and spin at once. The chapter on Rolling Motion described that motion. This chapter asks what causes it. The answer is mostly friction, but not the friction you know. It can push a car forward, it can vanish on a level road, and in pure rolling it wastes no energy. With three equations and one number for the shape of a body, you will predict races and move pulled spools. You will also follow a skidding ball until it rolls. The chapter ends with angular momentum about a well chosen point and the sweet spot of a cricket bat. It closes the dictionary between straight-line motion and rotation.

You will learn
By the end of this chapter you will
  1. Explain why the friction on a rolling body is static, does no work, and can point forward, backward or nowhere.
  2. Find the acceleration and the friction of a rolling body with three equations, and the direction of friction from the sign.
  3. Use \(a = g\sin\theta/(1 + k^2/R^2)\), the least \(\mu_s\) for rolling and the energy route down and up a slope, and predict a rolling race.
  4. Solve pulled rollers, spools pulled at any height, wheels driven by a torque, a cylinder on a moving plank and a yo-yo.
  5. Follow a skidding body until it rolls: the time, the distance, the final speed and the energy the skid costs.
  6. Find the angular momentum of a body that moves and spins about any point, and choose the point that silences an unknown force.
  7. Solve a particle striking a free rod, elastic or sticking, and find the centre of percussion of a hinged body.
Before you begin
Before you begin

From the chapter on Rigid Bodies and Moment of Inertia you should know the moments of inertia of standard bodies. You should also know the radius of gyration and the parallel-axis theorem. From the chapter on Angular Momentum you should know \(L = I\omega\), the angular impulse of a blow and impacts on a pivoted body. You should know static and kinetic friction, and energy conservation. You should know \(\tau_{cm} = I_{cm}\alpha\) from the chapter on Dynamics of Rotation. You should know the rolling condition \(v = \omega R\) from the chapter on Rolling Motion. Each is recalled where it is used.

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Lessons

1
Friction in Rolling
The friction at the contact of a body that rolls without slipping is static. It can point either way, it takes only the size the motion needs, and it does no work. Steady rolling on level ground needs none at all. One procedure finds its direction every time.
Static friction at a rolling contactSteady rolling needs no frictionAssume friction and read the sign
14 min0%
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2
Rolling Down an Incline
A body rolling down a slope accelerates at \(g\sin\theta/(1 + k^2/R^2)\), whatever its mass and radius. Then the least friction for rolling, the energy route, the race of the shapes, and a climb up rough and smooth slopes.
Acceleration down a slopeLeast friction for rolling on a slopeThe race of the shapesThe energy route on a slope
15 min0%
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3
Pulled Rollers, Driven Wheels and Yo-yos
The three equations applied to pulled rollers, spools, driven wheels, a cylinder on a moving plank and a yo-yo. The agent that drives the body decides the direction of the friction.
Pull height decides the frictionA driven wheel is pushed by the roadA cylinder on a moving plankThe yo yo falls slower than g
15 min0%
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4
Rolling with Slipping
When \(v\) is not equal to \(\omega R\) the contact slides, and kinetic friction changes both until they agree. Then the struck ball, the cost of the skid, a spinning disc set down, and a shortcut by angular momentum.
Kinetic friction brings v and omega r togetherThe struck ball rolls at five seventhsAngular momentum about a point on the ground
15 min0%
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5
Angular Momentum of a Body That Moves and Spins
About a fixed point, a body that moves and spins has orbital plus spin angular momentum, added as vectors. For a rolling body it is \(I_P\omega\) about the contact. Choose the point that silences an unknown force.
Orbital plus spin angular momentumThe two terms add as vectorsRolling body l equals i p omegaChoose the point that silences the unknown force
14 min0%
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6
Free Collisions and the Centre of Percussion
A particle strikes a free rod on smooth ice, and three conservation laws give three unknowns. The particle that sticks instead. The centre of percussion of a hinged rod, 2L/3 from the hinge, where a blow leaves the hinge feeling nothing. The translation-rotation dictionary, complete.
A particle strikes a free rodWhen the particle sticksCentre of percussionThe complete translation rotation dictionary
15 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

centre of percussion
The point of a hinged body where a blow makes the hinge give no impulse, \(x_p = I_{hinge}/(Md)\). For a uniform rod it is \(2L/3\) from the hinge.
free body
A body held by no pivot, hinge or support in the direction of an impact, such as a rod lying on smooth ice. Linear and angular momentum are both conserved when it is struck.
kinetic friction
The friction between two surfaces that slide over each other, of size \(\mu_k N\), opposing the sliding of the contact. It turns mechanical energy into heat.
orbital angular momentum
The part of a body's angular momentum about a point that comes from the motion of its centre of mass, \(\vec r_{cm}\times M\vec v_{cm}\), of size \(Mv_{cm}d\). It depends on the point chosen.
radius of gyration
The distance \(k\) from an axis at which the whole mass of a body could be placed to give the same moment of inertia, \(I = Mk^2\).
rolling resistance
The small torque that slows a real wheel on a level road. It arises because the tyre and the road deform, which moves the normal force slightly ahead of the centre. It is not friction.
rolling without slipping
Motion in which the point of a body touching a surface has no velocity relative to that surface, so that \(v_{cm} = \omega R\) on a fixed surface.
shape number
The ratio \(k^2/R^2\) of a rolling body, fixed by its shape alone. It is \(1\) for a ring, \(\tfrac12\) for a disc, \(\tfrac23\) for a hollow sphere and \(\tfrac25\) for a solid sphere.
spin angular momentum
The part of a body's angular momentum that comes from its turning about its own centre of mass, \(I_{cm}\vec\omega\). It is the same about every point.
static friction
The friction between two surfaces that do not slide over each other. It takes whatever size and direction are needed to prevent sliding, up to \(\mu_s N\). On a rolling body it acts at a point at rest and does no work.
sweet spot
The point of a bat or racket where a struck ball gives no jolt to the hands: the centre of percussion about the hands.