Chapter 7
Work and Energy
7.0 Introduction
Suppose you want to find the speed of a bullet that has been fired from a gun. You apply Newton’s laws and other problem solving techniques involving equations of motion. You will agree that forces involved in propelling the bullet are complex, also too the mathematics involved. However, there are other simple methods for dealing with such complex mechanics problems.
Conservation of Energy
The new problem solving technique we will introduce in this chapter involves use of work and energy. The importance of the energy idea stems from the principle of conservation of energy, which states that:
Energy is a quantity that can be converted from one form to another but cannot be created or destroyed.
At the petrol station you fuel your car with gasoline, which is a chemical energy. The car’s engine converts partially the chemical energy stored in the fuel to vehicle’s motion, and partially to thermal (heat) energy. In a microwave oven, electromagnetic energy obtained from Power Company is converted to thermal energy of the food being cooked. In all these and all other process that involve work-energy relation, the total energy i.e., the sum of all energy present in all different forms remains constant.
In this chapter, we will concern ourselves with problems involving mechanics. We will learn one important form of energy called kinetic energy, or energy due to motion, and how it relates to the concept of work. We will also consider power, which is time rate of doing work. In Chapter 8 we will expand the ideas of work and kinetic energy and also introduce the general formulation of conservation of energy. Conservation laws play very important role in our interpretation of physical properties.
7.1 Work
When you pull a table, lift a chair or push
a car you do work by exerting a force on a body while that body
moves from one place to another, i.e., it undergoes a displacement.
(Effort times distance, or force times height for vertical forces).
Let us consider an object that undergoes a displacement 
along a straight line, when acted upon by a constant
force 
, which is in the same direction as the displacement, as
in Fig. 7.1.
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Fig. 7.1 Work done by a constant force.
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The W done by the force on the
object is defined as the product of the magnitude of force and the magnitude of displacement through which the force act, i.e.,
(7.1)
where the constant force is in direction of straight-line displacement. The SI unit of work is the joule (symbol: J). In SI units, force is the newton and that of distance is meter, so one joule is equivalent to one newton-meter (N×m):
1 joule = (1 newton)(1 meter) or 1 J = 1 Nm
The units of work for British Engineering is: ft-lbs. The units of work in the cgs system is an, erg, which is equal to: one dyne.cm.
When dealing with work, as with force, it is necessary to specify whether you are talking about work done by a specific object or done on a specific object. It is also important to specify whether the work done is due to one particular force (and which one), or the total (net) work done by the net force on the object.
Case
11: - When the force and the displacement have different directions, we take the component of in the direction of displacement , and define the work as the product of this component and
the magnitude of displacement, see Fig. 7.2.
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From Fig. 7.2(b) for the FBD, the
component of force in the direction of is, 
, so from definition of work, Eq. (7.1):
(constant force)
(7.2)
Work is a scalar quantity. From the definition of the scalar dot product (A.B = ABcosq), in general work done can be written as:
(constant force)
(7.3)
Which is always in the direction of the force. From Eq. (7.2) we can deduce the following consequences:
- If q = 0o, cos0o = 1, Þ W = +Fs .
- If q = 90o, cos90o = 0, Þ W = 0 (no work is done)
- If s = 0, no work done. (NOTE: If the object doesn’t move, then there no work done, i.e., no work is done when you simply lift an object with no displacement.)
- Work can be negative, e.g., if q = 180o, cos180o = -1 implies W = -Fs.
From the above consequences, we can see that work can be either positive or negative. Positive work increases the energy of the system and it can be said that work was done on the object. Negative work, on the other hand, decreases the energy of the system, and it can be said that work was done by the object.
