Rolling

FIGURE 2.16

FIGURE 2.17

A common example of rotation is that of a ball or wheel rolling on a surface. Figure 2.16 shows a wheel of radius R rolling without slipping. In one revolution it covers a distance equal to its circumference and takes a time equal to one period T. Thus, the speed of the center is υ _c = (2π R)/ T=ωR, where ω is the angular velocity of the wheel. From Eq.11.5 this is equal to the tangential speed υ_t of a point on the rim relative to the center:

υ_с = υ_t = ω R (2.28)

Rolling is a combination of translation of the center and rotation about the center. The velocity of any point on the rim is the vector sum v = v _c + v _t. At the top of the wheel these two velocities are in the same direction, so υ = 2ω R, as shown in Fig. 2.17 a. At the bottom, they are in opposite directions, so υ = 0. Since the wheel does not slip, the point of contact with the surface is instanta­neously at rest and the wheel momentarily rotates about this point: The particles appear to describe circular paths with angular velocity ω about P as the center, as indicated in Fig. 2.17 b.

CHAPTER 3

Particle Dynamics

3.1 NEWTON'S FIRST LAW

Based on the work of Galileo and the French philosopher Rene Descartes, Isaac Newton published his first law of motion in 1687. According to Newton's first law:

Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.

This law involves a property of bodies called inertia:

The inertia of a body is its tendency to resist any change in its state of motion.

In other words, objects at rest tend to stay at rest, and if moving, they tend to keep moving at constant velocity. They display the same resistance to slowing down as to speeding up. Later we will connect the concept of inertia with the concept of mass, which is a measure of the inertia of a body.

The first law implies that a change in velocity, and therefore an acceleration, is produced by "forces." It does so without defining what force is. At this stage we have to use our intuitive understanding of force as either a push or a pull. The first law does not imply anything about the functional relationship between force and acceleration. Nonetheless, it does tell us when no net force acts on an object. It does not distinguish between cases for which there are no external forces at all, and those for which the forces balance to produce a zero resultant. For example, a block is being pulled along a rough floor. It experiences two horizon­tal forces—the tension in the rope and the frictional force due to the floor. If the magnitudes of these two forces are equal, the block will move at constant ve­locity.

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