doPhysics · Rotational Mechanics
Rolling Motion
Bodies that move and turn at once. The velocity of any point as translation plus rotation, and the instantaneous axis. The rolling condition, with its skids and spins. The velocities and accelerations of a rolling wheel, the cycloid, and the kinetic energy of a rolling body.
Start reading0%done
Why this chapter
Why this chapter exists
The wheel may be the most useful shape people ever made, and rolling is how it works. A rolling wheel moves along the road and turns at the same time, and the two motions are locked together. This chapter first adds a translation and a rotation, point by point. It then finds the one point about which a body turns at each instant. Rolling then follows as a single condition, and with it the whole picture. The top of a wheel moves at twice the axle's speed, a valve cap draws arches, and the spin carries energy.
You will learn
By the end of this chapter you will
- Find the velocity of any point of a body as translation plus rotation, and explain why the body has a single \(\omega\).
- Locate the instantaneous axis of rotation from two velocities, and find every speed as \(\omega\) times the distance from it.
- State the rolling condition \(v = \omega R\), \(a = \alpha R\), tell a forward skid from a wheel spin, and apply the condition on a moving surface.
- Give the velocity of every point of a rolling wheel: \(2v\) at the top, \(\sqrt2\,v\) at the sides, \(2v\sin(\theta/2)\) on the rim.
- Find the accelerations of a rolling wheel, including the acceleration of the contact point, which is at rest.
- Describe the cycloid traced by a point on the rim, and compare its path of \(8R\) with its displacement of \(2\pi R\) in one turn.
- Split kinetic energy about the centre of mass, and find it for a rolling ring, disc or sphere by two routes.
Before you begin
Before you begin
From Rigid Bodies and Moment of Inertia you should know general motion as translation plus rotation. You should also know the moment of inertia, the radius of gyration and the parallel-axis theorem. From Kinematics of Circular Motion you should know angular velocity, \(v = \omega r\), and the centripetal and tangential accelerations. You should be able to add velocities as vectors, and to integrate \(\sin x\).
Lessons
1
Translation Plus Rotation
The velocity of any point of a moving, turning body is the velocity of a chosen reference point plus a rotation part \(\omega r\). That part is at right angles to the line from the reference point. The body has one \(\omega\), whatever point is chosen.
Velocity is reference velocity plus rotationOne angular velocity for every reference pointVelocities of a wheel that moves and spins
2
The Instantaneous Axis of Rotation
At each instant a body moving in a plane turns about one point that is at rest, where the perpendiculars to any two velocities meet. Every speed is then \(\omega\) times the distance from that point. The axis moves, and it holds for velocities only.
A point at rest at every instantPerpendiculars to two velocities meet at the axisThe axis holds for velocities only
3
The Rolling Condition
A body rolls without slipping when the point touching the surface does not slide over it. On a fixed road that forces \(v = \omega R\) and \(a = \alpha R\) for the centre. A forward skid and a wheel spin break both, and on a moving surface the condition is relative.
Rolling without slipping means the contact is at restA cm equals alpha r while rolling lastsForward skid and wheel spinRolling on a moving surface
4
Velocities and Accelerations in Rolling
A rolling wheel is at rest at the contact, moves at \(v\) at the centre, \(\sqrt{2}\,v\) at the sides and \(2v\) at the top. Every speed is \(\omega\) times the distance from the contact point, which gives \(2v\sin(\theta/2)\) on the rim. The contact point is at rest, yet it accelerates at \(\omega^2R\) towards the centre.
Velocity field of a rolling wheelSpeed is omega times distance from the contactAccelerations in steady rollingAccelerations while rolling gains speed
5
The Path of a Point on a Rolling Wheel
A point on the rim of a rolling wheel traces a cycloid. Each arch is \(2R\) high and \(2\pi R\) long, and meets the road in a sharp cusp. In one turn the point covers a path of \(8R\) while its displacement is \(2\pi R\). A point inside the rim traces a smooth wave and never stops.
A rim point traces a cycloidPath length 8r against displacement 2 pi rPoints inside the rim trace smooth waves
6
Kinetic Energy of a Rolling Body
A rigid body's kinetic energy is the energy of its centre of mass plus the energy of its spin about that point. For a rolling body this is \(\tfrac12 Mv^2(1 + k^2/R^2)\), a different split for every shape. The same total is \(\tfrac12 I\omega^2\) with \(I\) about the contact point.
Kinetic energy splits at the centre of massOnly the centre of mass splits so cleanlyRolling energy shape by shapeThe same energy about the contact point
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- angular velocity
- The rate at which a rigid body turns, \(\omega = d\theta/dt\), in rad/s. A rigid body has one angular velocity, whichever point is used to describe its motion.
- chord
- A straight line joining two points of a circle. A rim point of a rolling wheel moves at \(\omega\) times the length of its chord from the contact point.
- contact point
- The point of a rolling body that touches the surface at a given instant. In rolling without slipping it is at rest relative to the surface, but it accelerates.
- curtate cycloid
- The path of a point inside the rim of a rolling wheel. It is a smooth wave with no cusps, because the point never touches the road.
- cusp
- A sharp point of a curve. A cycloid has a cusp wherever the rim point touches the road and stops for an instant.
- cycloid
- The path of a point on the rim of a wheel rolling on a straight line. It is a chain of arches, each \(2R\) high, \(2\pi R\) long along the road and \(8R\) long along the curve.
- forward skid
- Sliding in which the centre of a wheel moves faster than its rim turns, \(v_{cm} > \omega R\), so the contact point slides forward. Locked brakes cause it.
- instantaneous axis of rotation
- The line, at right angles to the plane of motion, through the point that is at rest at a given instant. At that instant every point of the body moves at \(\omega\) times its distance from it.
- parallel-axis theorem
- The rule \(I = I_{cm} + Md^2\), for an axis parallel to one through the centre of mass and a distance \(d\) from it. About the contact point of a rolling body, \(d = R\).
- radius of gyration
- The radius \(k\) of a ring that would hold the whole mass of a body with the same moment of inertia, \(I = Mk^2\). In rolling, \(k^2/R^2\) fixes how the energy divides.
- reference point
- The point of a moving body whose velocity is taken as the translation part. Every other point adds a rotation part about it; any point of the body may be chosen.
- rolling condition
- The relation \(v_{cm} = \omega R\) for a round body rolling without slipping on a fixed surface, with \(a_{cm} = \alpha R\) while rolling lasts.
- rolling sense
- The sense of turning of a wheel that rolls forward: clockwise for a wheel moving to the right, as seen with the road below it.
- rolling without slipping
- Motion in which the point of a body touching a surface does not slide over it. On a fixed surface it gives \(v_{cm} = \omega R\).
- rotational kinetic energy
- The kinetic energy \(\tfrac12 I_{cm}\omega^2\) of a body turning about its centre of mass, the energy of the spin.
- translational kinetic energy
- The kinetic energy \(\tfrac12 Mv_{cm}^2\) of the whole mass of a body moving with its centre of mass, the energy of the ride.
- wheel spin
- Sliding in which the rim of a wheel turns faster than its centre moves, \(\omega R > v_{cm}\). The contact point slides backward, as with a tractor in mud.