WORK, POWER AND ENERGY
MODULE 7
COMPETENCE/S
A-II/2 F1. Navigation at the operational Level
COURSE
OUTCOMES
CO 2. Cite concrete applications of concepts in
Mechanics particularly in marine navigation.
CO 3. Demonstrate analytical skills in performing problem
solving activities
KUP1: Understanding of fundamental principles of ship
construction and the theories and factors affecting trim
KNOWLEDGE
and stability and measures necessary to preserve trim
UNDERSTANDING
and stability.
PROFICIENCY
KUP 2: Knowledge of the effect on trim and stability of cargoes
and cargo operations.
1. Define work, power and energy
2. Solve simple problems in work, power and energy
3. Apply the conservation of energy concept as an
alternative approach to the Newtonian method of solving
kinematic problem.
LEARNING
OUTCOMES
7.1 Work
7.2 Power
7.3 Energy
TOPICS
LESSON PRESENTATION AND ACTIVITIES
KEY WORDS
Mechanical Work
Kinetic Energy
Energy
Elastic Potential Energy
Mechanical Energy
Datum
Gravitational Potential
Energy
Power
Work-Energy Theorem
Pre-test
Crossword Puzzle
1
6
5
2
3
4
Across
1 dot product of force and displacement
2 ability to do work
3 energy by virtue of position
4 unit of power
Down
3 rate of doing work
5 work done by the normal force when
a block is pushed along a flat
horizontal floor
6 energy by virtue of motion
7.1
WORK
Work is the effort exerted on something that will change its energy. Two ingredients
to work: force and displacement. In order for a force to qualify as having done work
on an object, there must be a displacement and the force must cause or hinder the
displacement. A force perpendicular to the displacement does not do work.
Doing push-ups is an example of work. When you do push-ups, you do work against
your weight. A ship accelerating through water is also an example of work. There is
force (the ship’s propeller pushes the water backwards which will result to a reaction
force which is equal in magnitude but opposite in direction) which causes the ship to
be displaced through water. A power tool falling off a shelf and free falls to the floor is
another example of work. There is a force (gravity) that acts on the power tool which
causes it to be displaced in a downward direction
Pushing on a wall of a room is not an example of work because the wall is not
displaced. In carrying a bag on your shoulder while walking across the room, work is
not done on the bag because even if there is a force (the bag was pulled up towards
the shoulder) and there is a displacement (the bag is moved horizontally across the
room), yet the force does not cause the displacement. There must be a component of
force in the direction of displacement.
Work transfers energy from one place to another or one form to another. Work can
change the potential energy of an object, energy in a thermal system, or the electrical
energy in an electrical device.
Work Equation
Consider an object acted upon by a force, F. Work done by F on the object is defined
as the dot product of the object’s displacement, d and the force, F. The dot product
of two vectors is a scalar quantity. Work is a scalar quantity.
Mathematically, work can be expressed by the following equations:
W = F. d
= /F//d/cos θ
where:
W – work done by F, N.m or J
F - force, N
d – displacement, m
θ – smaller angle between F and d
Work can either be positive, negative or zero.
1. W = +
If 0o < θ < 90o or if θ = 0o, W is positive because cos θ is positive. The component
of F parallel to d has the same direction as d.
2. W = 0
Work is zero if there is no displacement or the force is perpendicular to the
displacement (θ = 90o).
3. W = If 90o < θ < 180o or if θ = 180o, work is negative because cos θ is negative. The
force acts in the direction opposite the object’s motion, usually to slow it down.
Common units of work are: N.m or Joules, J; Dyn.cm or erg; and lb.ft2/s2
Examples:
1. A cargo box with a mass of 250kg is pulled a distance 3m, to the right at
constant speed by constant force F of 50N. What is the work done by each
force?
2. A man holds a 50 lb box at waist level for 10 minutes. Has he done any work
during this time?
Solution:
W = /F//d/cos θ
d= 0
W = /F/(0)cos θ
=0
Work is done on the books when they are being lifted, but no work is done
on them when they are being held or carried horizontally.
7.1
ENERGY
Energy is the ability to do work. There are may forms of energy. Below are some
examples:
1. Mechanical – energy of the object because of its position, motion, or
deformation. When work is done to raise a heavy shipping container, the
container acquires the ability to do work on the object it hits when it falls. When
work is done to wind a spring mechanism, the spring acquires the ability to do
work on various gears. Something has been acquired in each case that enables
the object to do work. This something that enables an object to do work is
energy. The two forms of mechanical energy are potential energy and kinetic
energy.
2. Potential Energy - Energy may be stored by an object by virtue of its position.
The energy that is stored and held in readiness is called potential energy (PE)
because in the stored state, it has the potential for doing work. A stretched or
compressed spring has a potential for doing work to another object. The
chemical energy in fuels is potential energy. It is energy by virtue of position at
the sub microscopic level. This energy is available when the positions of
electric charges within and between molecules are altered. This is when a
chemical change occurs.
1. Gravitational Potential Energy, PEg
Energy stored in an object as the result of its vertical position or height.
The energy is stored as the result of gravitational attraction of the Earth
for the object. There is a direct relation between gravitational potential
energy and the mass of an object. Objects which are more massive have
greater gravitational potential energy. Between the height where the
object was brought and gravitational potential energy, there is also a direct
relation. The higher the object is elevated, the greater its gravitational
potential energy. These relationships can be expressed by the equation:
PEg=mgh
where:
m – mass of the object
g - acceleration due to gravity
h - distance from a reference point or datum, a zero height
position arbitrarily assigned
.
2. Elastic Potential Energy, PEe
Energy of position of a spring or an elastic object
Ue = kX2/2
Where:
k - spring constant or Hooke’s Law constant
x - displacement of the spring from its equilibrium or
unstretched position
3. Kinetic Energy, KE
Energy that an object possesses due to its motion. If an object is
moving, then it is capable of doing work. It has energy in motion. If a
ship is sailed, work is done on it to give it speed. The moving ship can
then collide with something and push it, doing work on what it hits. The
amount of kinetic energy that an object has depends upon two
variables: the mass of the object and the speed of the object. The
kinetic energy of an object is directly proportional to the square of its
speed. That means for a double increase in speed, the kinetic energy
will increase by a factor of four. There is also a direct relation between
the mass of an object and its kinetic energy. Objects which are more
massive will have greater kinetic energy. The equation used to
represent kinetic energy is:
KE = mV2/2
where:
m - mass of the object
v – speed of the object
3. Electrical Energy
It is the kinetic energy of the moving electrons or charged particles
4.
Light
Light is the common name for electromagnetic radiation with wavelengths of
400-750 nm. Electromagnetic radiation is a form of energy that is propagated
through free space or through a material medium in the form of
electromagnetic waves, such as radio waves, visible light, and gamma rays.
Common units of energy are the same as that of work. Other units of energy are:



calorie, cal
kilocalorie, kcal
British Thermal Unit, BTU
Work and Gravitational Potential Energy
Work done against the gravitational force goes into a form of stored energy –
gravitational potential energy. If an object is lifted straight up, then the force needed
to lift it up is equal to its weight. The work done on the object then changes its
potential energy. When the force does positive work, it increases the gravitational
potential energy of the object. Work equals change in potential energy
𝑾=Δ𝑷𝑬
=𝑷𝑬𝒇−𝑷𝑬𝒊
=𝒎𝒈𝒉𝟐−𝒎𝒈𝒉𝟏
Work and Change in Kinetic Energy (Work-energy Theorem)
The Work-Energy Theorem states that the work done by the net force on an object
is equal to its change in kinetic energy. In mathematical terms, the Work-energy
Theorem can be expressed as:
𝑾 = Δ𝑲𝑬
= 𝑲𝑬𝒇−𝑲𝑬𝒊
= 𝟏/𝟐(𝒎𝒗𝒇𝟐)−𝟏/𝟐(𝒎𝒗𝒊𝟐)
where:
W – total work or work done by the net force
𝑲𝑬𝒇 − final kinetic energy
𝑲𝑬𝒊 – initial kinetic energy
This theorem provides a direct link between the net force acting on an object and its
energy. Furthermore, it provides an alternative method to studying the dynamics of a
system that would be too difficult to treat with Newton’s 2nd law.
A consequence of the Work-Energy Theorem is the law of Conservation of Mechanical
Energy. This law states that the total initial energy (the sum of the initial kinetic and
potential energy) of a system equals the total final energy (the sum of the final kinetic
and potential energy) of the system as long as all forces involved are conservative.
7.3
POWER
Power is the rate of doing work or transfer of energy. It is equal to the amount of
work done per time it takes to do it. Power is calculated using the equation:
P = W/t
where:
P – power
W – work
t - time
Common unit of power is J/s or Watt, W. The horsepower is occasionally used to
describe the power delivered by a machine. One horsepower is equivalent to
approximately 746 Watts.
Examples:
1. A 20N horizontal force is used to pull a box along a surface. How much work
does it do in pulling the box through a distance of 0.5 m?
Solution:
𝑊=𝐹𝑑cos𝜃
= (20N)(0.5m)(cos 0)
= 10 J
2. What is the work done against gravity in lifting a 10-kg object through a distance
of 0.45 m?
Solution:
F=W
= mg
= (10 kg)(9.8 m/s2)
= 98 N
𝑊=𝐹𝑑cos𝜃
= (98 N)(0.45m)(cos 0)
= 44 J
3. A 20,000 kg shipping container is lifted by a crane to a height of 10 m from the
ground. Using the ground as the datum, what is its potential energy?
Solution:
PEg = mgh
= (20,000 kg) (9.8 m/s2) (10 m)
= 1,960,000 J
4. A ship with a mass of 40,000 tons is sailing at a speed of 10 knots. Find:
a. ship’s kinetic energy at 10 knots
b. work required to stop the ship, and
c. the force needed to stop the ship over a 1.8 nautical mile distance.
Solution:
a. KE = ½(mV2)
40,000 T (1,000kg/1T) = 40,000,000kg
10 knots (1 nmi/h/1 knot) (1852m/1 nmi) (1 h/3,600 s) = 5.14 m/s
KE = ½ (40,000,000 kg) (5.14 m/s)2
= 528,392,000 J
b. W = ΔKE
= KEf - KEi
= 𝟏/𝟐(𝒎𝒗𝒇𝟐)−𝟏/𝟐(𝒎𝒗𝒊𝟐)
= ½ (40,000,000kg)(0m/s)2 – ½ (40,000,000kg)(5.14m/s)2
= -528,392,000 J
c. W = Fdcosθ
F = W/dcosθ
= -528,392,000 J/(1.8 nmi)(1852m/1 nmi) (cos 0)
= 158,504.92 N
5. What power must the engine produce for a 20,000,000 kg ship to accelerate
from rest to a speed of 5 knots in a time of 120 seconds?
Solution:
P = W/t
W = ΔKE
= KEf - KEi
= 𝟏/𝟐(𝒎𝒗𝒇𝟐)−𝟏/𝟐(𝒎𝒗𝒊𝟐)
= ½ (20,000,000kg)((5 knots)(1852m/s/knot))2 – ½(20,000,000kg)(0m/s)2
= 66,049,000 J
P = 66,049,000J/120s
= 550,408.33 W