WORK AND KINETIC ENERGY

III - Dalton

Introduction :In this chapter we will introduce the concepts of

work and kinetic energy. These tools will significantly

simplify the manner in which certain problems can be

solved.

Figure 7.1. A force F acting on a body. The resulting displacement is indicated by the vector d.

Work: constant forceSuppose a constant force F acts on a

body while the object moves over a distance d. Both the force F and the displacement d are vectors who are not necessarily pointing in the same direction (see Figure 7.1). The work done by the force F on the object as it undergoes a displacement d is defined as

The work done by the force F is zero if:* d = 0: displacement equal to zero* [phi] = 90deg.: force perpendicular to

displacement

The work done by the force F can be positive or negative,depending

on [phi]. For example, suppose we have an object moving withconstant velocity. At time t = 0 s, a force F is applied. If F is the onlyforce acting on the body, the object will either increase or decrease

itsspeed depending on whether or not the velocity v and the force F arepointing in the same direction (see Figure 7.2). If (F * v) > 0, the

speedof the object will increase and the work done by the force on theobject is positive. If (F * v) < 0, the speed of the object will decreaseand the work done by the force on the object is negative. If (F * v) =

0we are dealing with centripetal motion and the speed of the objectremains constant. Note that for the friction force (F * v) < 0 (always)and the speed of the object is always reduced !

Per definition, work is a scalar. The unit of work is the Joule (J).From the definition of the work it is clear that:

1 J = 1 N m = 1 kg m2/s2

Work: Variable Force

In the previous discussion we have assumed that the force acting on the object is constant (not dependent on position and/or time). However, in many cases this is not a correct assumption. By reducing the size of the displacement (for example by reducing the time interval) we can obtain an interval over which the force is almost constant. The work done over this small interval (dW) can be calculated

The total work done by the force F is the sum of all dW

Kinetic EnergyThe observation that an object is

moving with a certain velocity indicates that at some time in the past work must have been done on it. Suppose our object has mass m and is moving with velocity v. Its current velocity is the result of a force F. For a given force F we can obtain the acceleration of our object:

Assuming that the object was at rest at time t = 0 we can obtain the velocity at any later time:

Therefore the time at which the mass reaches a velocity v can be calculated:

If at that time the force is turned off, the mass will keep moving with a constant velocity equal to v. In order to calculate the work done by the force F on the mass, we need to know the total

distance over which this force acted. This distance d can be found easily from the

equations of motion:

The work done by the force F on the mass is given by

The work is independent of the strength of the force F and depends only on the mass of the object and its velocity. Since this work is related to the motion

of the object, it is called its kinetic energy K:

If the kinetic energy of a particle changes from some initial value Ki to some final value Kf the amount of work done on the particle is given by

W = Kf - Ki

This indicates that the change in the kinetic energy of a particle is equal to the total work done on that particle by all the forces that act on it.

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