Chapter 5
Energy
Lecture Notes
5.1 Work
In order for work to be accomplished, an object must undergo a displacement; the force associated with the work must have a component parallel to the direction of the displacement.
Work is a scalar quantity and can be either positive or negative (positive when the component
F cos

is in the same direction as the displacement). The SI unit of work is the newton-meter
(Nm) or joule (J).
5.2 Kinetic Energy and the Work-Energy Theorem
Any object that has mass m and speed v has kinetic energy. Kinetic energy is a scalar quantity and has the same units as work; kinetic energy of an object will change only if net work is done on the object by external forces. The relationship between work and change in kinetic energy is stated in the work-energy theorem.
A force is conservative if the work it does on an object moving between two points is independent of the path the object takes between the points. The work done on an object by a conservative force depends only on the initial and final positions of the object. The gravitational force is an example of a conservative force.
A force is non-conservative if the work it does on an object moving between two points depends on the path taken. Kinetic friction is an example of a non-conservative force.
5.3 Gravitational Potential Energy
The work done on an object by the force of gravity is equal to the object’s initial potential energy minus its final potential energy ( W g
  
PE g
). The gravitational potential energy associated with an object depends only on the object’s weight and its vertical height above the surface of the Earth. If the height above the surface increases, the potential energy will also increase, but the work done by the gravitational force will be negative. In this case the direction of the displacement is opposite the direction of the gravitational force. In working problems involving gravitational potential energy, it is necessary to choose an arbitrary reference level, or location, at which the potential energy is taken to be zero.

Chapter 5
Energy
5.4 Spring Potential energy
Elastic potential energy is the energy associated with a spring that is compressed from or stretched beyond its equilibrium position. The spring force is a conservative force and has a magnitude that is proportional to the displacement of the spring from the equilibrium position.
5.5 Systems and Energy Conservation
The change in the kinetic energy of a physical system equals the sum of the work done by conservative forces and the work done by the non-conservative forces. The sum of the kinetic energy plus the potential energy is called the total mechanical energy. Since the work done by conservative forces equals the negative of the change in potential energy ( W c
  
PE ), the work done by non-conservative forces equals the change in total mechanical energy of the system. The mechanical energy of a system remains constant if only conservative forces do work on the system.
5.6 Power
Power delivered to an object is defined as the rate at which energy is transferred to the object or the rate at which work is being done on the object. The average power delivered to an object during a time interval can be expressed as the product of the average speed during the time interval and the component of the force in the direction of the velocity.
5.7 Work Done by a Varying Force
If the value of a non-constant force is known as a function of position, the work done by the varying force during a displacement can be determined by calculating the area under the force vs. displacement curve.

Chapter 5
Energy
Equations & Concepts o The work done by a constant force

F , (constant in both magnitude and direction) is defined to be the product of the component of the force in the direction magnitude of the displacement.
W

( F cos

)
 x o Work can be positive, negative or zero, depending on the value of θ, the angle between the direction of the force and the direction of the displacement. o Work is a scalar quantity and the SI unit of work is the newton-meter or joule.
1 N m = 1 J joule (J) = newton .
meter = kg .
m 2 /s 2 o Kinetic energy, defined by the equation below, is a scalar quantity and is the energy associated with the motion of a mass m. Kinetic energy has the same units as work.
KE

1
2 mv
2 o The work-energy theorem states that the net work done on a particle equals the change in kinetic energy of the particle. The work-energy theorem is valid for a particle or for a system that can be modeled as a particle.
W net

1
2 mv 2 
1
2 mv
0
2

Chapter 5
Energy o The net work done on an object is the sum of the work done by the non-conservative forces and the conservative forces.
W nc

W c
 
KE o Gravitational potential energy of a mass-Earth system near the surface of the earth is defined as below. The quantity y in the equation is the vertical position of the mass relative to the Earth, or other arbitrarily chosen reference level where PE g
= 0.
PE
 mgy o The work done b y the force of gravity can be expressed in terms of initial and final values of the y-coordinates. The units of energy (kinetic and potential) are the same as the units of work. The difference in potential energy between two points is independent of the location of the origin.
W g




( PE f
( mgy f


PE i
) mgy i
) o Hook’s law expresses the force exerted by a spring that is stretched or compressed from the equilibrium position. The negative sign signifies that the force is always directed opposite the displacement from equilibrium.
F s k

  kx spring constant in N/m
Each spring has a characteri stic value of k o Elastic potential energy is associated with a mass-spring system that has been stretched or compressed from the equilibrium position. The elastic potential energy of a deformed spring is always positive.
PE s

1
2 kx
2 x = displacement from equilibrium

Chapter 5
Energy o The work done by all non-conservative forces equals the change in the total mechanical energy of a system, change in kinetic energy plus changes in the gravitational and elastic potential energies.
W nc

( KE f

KE i
)

( PE gf

PEgi )

( PE sf

PE si ) or W nc
 
KE
 
PE g
 
PE s o The law of conservation of mechanical energy holds when only conservative forces act on a system; the total mechanical energy of the system remains constant.
( KE

PE g

PE s
) i

( KE

PE g

PE s
) f o The average power supplied by a force is the ratio of the work done by the force to the time interval over which the force exists. The average power can also be expressed in terms of the force and the average speed of the object on which the force acts. In the equation below F is the component of the force along the direction of the velocity.
P
P

W
 t

F v o The SI unit of power is the watt; in the US customary system, the unit of power is the horsepower.
1 W = 1 J/s = I kg .
m 2 /s 2 1 hp = 550 ft
 lb s
= 746 W o Work done by a variable force is equal to the area under the force vs. displacement curve. The figure below illustrates the case of a force acting along the x-axis.

Chapter 5
Energy
Problem Solving Hints
1.
Choosing a zero level for potential energy – In working problems involving gravitational potential energy, it is always necessary to choose a location at which to set the gravitational potential energy to zero. This choice is completely arbitrary because the important quantity is the difference in potential energy, and that difference is independent of the location of zero. It is often convenient, but not essential, to choose the surface of the Earth as the reference position for zero potential energy. In most cases, the statement of the problem suggests a convenient level to use.
2.
Conservation of energy – Take the following steps in applying the principle of conservation of energy:
3.
Define your system, which may consist of more than one object.
4.
Select a reference level for the zero point of gravitational potential energy. This level must not be changed during the solution of a specific problem.
5.
Determine whether or not non-conservative forces are present. o If mechanical energy is conserved, ie., if only conservative forces are present, you should use the Law of conservation of energy equation. Usually, except when a quantity is equal to zero, it is better to solve for the unknown algebraically using symbols before substituting numerical values. o If non-conservative forces such as friction are present, and therefore mechanical energy is not conserved, first write expressions for the total initial and total final mechanical energies. In this case, the change in the two total energies is equal to the work done by the non-conservative force(s).

Chapter 5
Energy
Final Problem Solving Comment
 Define the work done by a constant force and work done by a force that varies with position. Recognize that work done by a force can be positive, negative or zero.
 Understand that the work done by a conservative force in moving a body between any two points is independent of the path taken. Non-conservative forces are those for which the work done on a particle moving between two points depends on the path.
Account for non-conservative forces acting on a system using the work-energy theorem.
In this case, the work done by all non-conservative forces equals the change in total mechanical energy of the system.
 Relate the work done by the net force on an object to the change in kinetic energy. The relation W ne
 
KE

KE f

KE i
is called the work-energy theorem, and it is valid whether or not the resultant force is constant. That is, if we know the net work done on a particle as it undergoes a displacement, we also know the change in its kinetic energy.
This is the most important concept in this chapter, so you must understand it thoroughly.

Recognize that the gravitational potential energy of the mass-Earth system, PE g
 mgy can be positive, negative or zero, depending on the location of the reference level used to measure y. Be aware of the fact that although PE depends on the origin of the coordinate system, the change in potential energy, (PE) f
– (PE)
I
, is independent of the coordinate system used to define PE.
 Calculate average power when work is accomplished over a period of time interval and instantaneous power when an applied force acts on an object moving with speed v.