Chapter 4 Work and Energy
The concept of energy is one of the most important in the world of science. In everyday usage, the term energy has to do with the cost of fuel for transportation and heating, electricity for lights and appliances, and the foods we consume. However, these ideas do not really define energy. They tell us only that fuels are needed to do a job and that those fuels provide us with something we call energy.
Energy is present in the Universe in a variety of forms, including mechanical energy, chemical energy, electromagnetic energy, heat energy, and nuclear energy. Although energy can be transformed from one form to another, the total amount of energy in the Universe remains the same. If an isolated system loses energy in some form, then by the principle of conservation of energy, the system must gain an equal amount of energy in other form. The transformation of energy from one form into another is an essential part of the study of physics, chemistry, biology, geology, and astronomy.
In this chapter we are concerned only with mechanical energy. We introduce the concept of kinetic energy, which is defined as the energy associated with motion, and the concept of potential energy, the energy associated with position. We shall see that the ideas of work and energy can be used in place of
Newton’s law to solve certain problems.
4.1 Work and Power
1. Work W done by a constant force is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement.
ϕcos FS S F W =⋅=
Where the force makes an angle of ϕ with displacement .S
The SI unit of work is the joule (J), named for James Prescott Joule, an English scientist of the 1800s. It is derived directly from the units for mass and velocity:
1joule=1J=1(kg) (m/s 2) (m)=1 kg m 2/s 2
2. Work done by a variable force
(1). The increment of work dW done on the particle by F during the displacement d r is
r d F dW ⋅=, where Force F is function of its position.
(2). The work W done by F while the particle moves from an initial position a to a final position b is then
⎰⎰⋅==b a b a r d F dW W
(3). We use the components of
F and r d to express the force
and displacement, then we have
dz
F dy F dx F k dz j dy i dx k F j F i F r d F W z y b a x z y b a x b a ++=++⋅++=⋅=⎰⎰⎰
)()( 3. Work done by multiple forces : If there are several forces act on a particle, we can replace F in above equation with the net force ∑F , where +++=∑321F F F F , where j F are the
individual forces. Then
+++=⋅+++=⋅=⎰⎰321321)(W W W r d F F F r d F W b a
b
a 4. Power
(1). The rate at which work is done by a force is said to be the power due to the force . If an amount of work W is done in a time interval
t ∆ by a force, then the average power due to the force is t W
P ∆=.
(2). The instantaneous power P is the instantaneous rate of doing work, which can be written as
v F dt
r d F dt dW P ⋅=⋅==. (3). The SI unit of power is the joule per second . This unit is used so often that it has a special name, the watt (W), after James Watt, who greatly improved the rate at which steam engines could do work.
4.2 Kinetic Energy and Work-Kinetic Energy Theorem
Energy is a scalar quantity that is associated with a state of one
