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EF 101: Module 5, Lecture 2
Engage Freshmen Engineering Program
R. M. Bennett & D. R. Raman
The University of Tennessee
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Competencies you should have after this lecture: Know the relationship between force, displacement,
and work (i.e., know the definition of mechanical work). Know the equation for kinetic energy and
understand its relation to work done on an object; be familiar with the work-energy theorem. Be able to
state the principle of conservation of energy, and explain the meaning of efficiency in the context of
energy transformation processes. Be able to work simple work-energy balance problems.
The work done by a constant force F (one with constant magnitude and direction), acting on an object
that moves a displacement s, depends upon the magnitude of the force, the magnitude of the
displacement of the object, and the angle between F and s. This can be expressed as W = |F||s|cos(θ),
where θ is the angle between vectors F and s. Alternatively, we can write W = Fx s x + Fy s y , in other words,
we can multiply the x and y components of the force and displacement, then sum them, to get the total
work done. {You don’t need to know this for the test, but doing such a term-by-term multiplication of two
vectors is known as taking the “dot product” of the two vectors. This is written symbolically as W = F • s.}
Note that although work is computed by multiplying two vectors, the work done is not a vector; it is a
scalar. The SI units of work are the Newton-meter (N·m), or Joule (J), where 1 J = 1 N·m. The USC units
of work are the ft·lb, with the “lb” being interpreted as lbf . Another useful unit of work (or energy) is that of
the calorie, with 1 cal = 4.182 J. To confuse matters a bit, the food industry expresses the energy content
of foods in units of kilo-calories, but doesn’t write kcal on packages! A 250 cal soda actually has an
energy content of 250 kcal, or 1.05 MJ.
With the term “energy” defined as the ability to do work, it becomes possible to write equations linking
various forms of mechanical energy. For example, the kinetic energy of an object is a measure of the
energy inherent in the objects motion (stationary objects have zero kinetic energy). The kinetic energy of
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an object is given by the expression KE = ½mv . We can interpret kinetic energy as either the energy
required to accelerate the object to velocity v, starting from rest, or as the energy available if we stop an
object that is moving at velocity v. The Work-Energy Theorem states that the work done on an object by a
the net force acting on an object is equal to the change in kinetic energy of the object, and can be used to
solve certain problems in dynamics.
The Law of Conservation of Energy, discussed in Lecture 5.1, states that Energy can be neither created
nor destroyed, but can only be converted from one form to another. Many engineering and science
problems revolve around understanding, designing, maintaining, or developing new energy conversion
processes. A critical concept in all energy conversion processes is that of efficiency. The efficiency (η) of
an energy conversion process is a measure of the ratio of energy delivered by the process, to energy
required by the process. Efficiency values cannot exceed 1.0 (or be less than 0) without violating the
principle of Conservation of Energy. Efficiencies may be reported in fractional or percentage format.
Practice Problems:
1. What is the conversion factor from J to ft·lb?
2. A constant net force F = (24î – 7.5j) lbf acts on an object while the displacement of the object is
s = (-6.4î + 2.1j)ft. What work is done on the object by this force (ft·lb)?
3. Three students pushed a stalled car 17 m to get it off the road. Student A pushed with a constant
force of 440 N directly along the path of motion. Student B pushed with a constant force of 605 N
at an angle of 24° off the path of motion. Student C pushed with a constant force of 355 N at an
angle of 11° off the path of motion. What is the total work done by the students on the car (kJ)?
4. How much work is done by gravity on a 2.5-kg ball falling from a height of 19.62 m (J)? Using the
constant acceleration equations developed in Module 4, what is the velocity of the ball at impact
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(m s )? What is the kinetic energy of the ball at impact (J), and how does it compare to the kinetic
energy predicted by the work-energy theorem?
5. A minivan driving along a road at a constant velocity of 45 mph, overcomes a net force of 195 lbf .
Assuming that the engine is 21.7% efficient, and that the energy content of gasoline is
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92 x 10 ft·lb gal , what is the fuel economy of the car (mpg)?
Hint: Figure out how much energy is needed to run for a fixed time, say 1 h or 1 min or 1 s.
EF 101: Module 5, Lecture 2
Engage Freshmen Engineering Program
Answers to Practice Problems:
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1. 1.3558 J (ft·lb)
2. –169.4 ft·lb.
3. 22.8 kJ
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4. 481 J; 19.62 m s ; 481 J – the same as the KE predicted by WET.
5. 19.4 mpg
R. M. Bennett & D. R. Raman
The University of Tennessee