Concepts: rolling motion and exercise practice
System of Particles and Rotational Motion24 min readLesson 6 of 8
6.1 Rolling without slipping
A wheel rolling without slipping has its contact point instantaneously at rest, and the speed of the centre of mass equals the angular speed times the radius. The top of the wheel moves at twice the speed of the centre, which is why the top of a rolling wheel is blurred in a photograph. Slipping breaks the relation between the linear and angular speeds, and the friction becomes kinetic rather than static.
For rolling down an incline the energy method is quick: the loss of potential energy becomes the translational plus rotational kinetic energy, and the rolling condition gives the angular speed in terms of the linear speed. The acceleration follows as g sin theta divided by one plus the moment of inertia divided by mass times radius squared. The factor depends only on the shape, which is why a solid sphere, a disc and a ring race differently down the same incline.
The static friction is essential: without it the body would slide instead of rolling. In ideal rolling the friction does no work because the contact point is momentarily at rest, so mechanical energy is conserved. When the body slips as well as rolls, the friction does negative work and the mechanical energy falls, so the two cases must be distinguished in a numerical.
6.2 Which body wins the race
Substitute the moment of inertia for each shape into the acceleration formula. A solid sphere has I over m R squared equal to two fifths, so its acceleration is five sevenths of g sin theta. A disc or solid cylinder has one half, giving two thirds of g sin theta. A ring or hollow cylinder has one, giving half of g sin theta. The solid sphere accelerates most and the ring least, independent of their masses and radii.
The same order appears in the final speed at the bottom of the incline: the body with the smallest fraction of its kinetic energy in rotation is fastest. A useful check is that a body with all its mass at the rim, the ring, wastes the largest share of the energy on rotation, so it is slowest. These rankings are a favourite examination question and need no calculation if the fractions are known.
A rolling body that is projected up an incline slows down with the same magnitude of acceleration as it would have rolling down, provided the friction is enough to maintain rolling. The energy method can find how far it rises, and the answer depends on the initial speed and the shape factor in the same way.
6.3 Exercise practice: centre of mass and torque
The first exercise locates the centre of mass of a pair of particles, and the second does the same for three particles sitting at the corners of a triangle. The third treats a rod whose linear density changes along its length, which calls for the integral form. Exercise 6.4 finds the centre of mass of a plate with a hole, treated as a negative mass. Exercise 6.5 uses the motion of the centre of mass for a projectile exploding at the top of its path.
Exercise 6.6 gives the position of a particle and asks for its angular momentum about the origin, using the cross product. Exercise 6.7 asks for the torque of a force about a point, and exercise 6.8 checks the equilibrium of a rod under several forces. Exercise 6.9 finds the force needed to keep a ladder from slipping, the standard equilibrium problem. Exercise 6.10 and 6.11 determine whether a rigid body is in equilibrium from its forces and torques.
Exercises 6.12 and 6.13 use the parallel axis theorem for a rod and a ring about off-centre axes. Exercise 6.14 finds the moment of inertia of a composite body, and exercise 6.15 compares the moments of inertia of a ring, a disc and a sphere of the same mass and radius. Exercise 6.16 gives the moment of inertia about a diameter from the central value using the perpendicular axis theorem.
6.4 Exercise practice: dynamics and angular momentum
Exercise 6.17 uses the rotational second law for a flywheel and asks for the torque needed to bring it to a given speed. Exercise 6.18 connects a falling mass to a pulley and asks for the acceleration and the tension, so both the torque equation and the second law are needed. Exercise 6.19 is a rod hinged at one end and released from horizontal, which needs the torque about the hinge and then the kinematics. Exercise 6.20 finds the kinetic energy of a rolling body and the fraction that is rotational.
Exercise 6.21 is the conservation of angular momentum for a person on a rotating platform, and exercise 6.22 applies the same principle to a dancer or a diver changing shape. Exercise 6.23 is a rolling body moving up an incline and returning, with the energy method the fastest route. Exercise 6.24 asks for the acceleration of a solid cylinder and of a hollow cylinder down the same incline, which the shape factor answers directly.
Through all these, the choice of axis and the identification of the conserved quantity do most of the work. A quick energy check and a dimension check close each numerical, and the limit cases of a very small or very large moment of inertia give a feel for whether the answer is sensible.
Exam
In a rolling question, decide first whether the body rolls without slipping or slips. The relation v equals omega R and the conservation of energy hold only for rolling without slipping, and using them for a slipping body is the commonest error.