Chapter 6 Work & Energy
I.
Work Done by a Constant Force (6-1 )
A. The work done by a constant force is defined as the distance moved multiplied by the
component of the force in the direction of displacement:
1. Equation 6-1: W = F d cos Ɵ
2. In the SI system, the units of work are joules: 1 J = 1 N · m
B. Solving work problems:
1. Draw a free-body diagram.
2. Choose a coordinate system.
3. Apply Newton’s laws to determine any unknown forces.
4. Find the work done by a specific force.
5. To find the net work, either find the net force and then find the work it does, or find
the work done by each force and add.
C. Work done by forces that oppose the direction of motion, such as friction, will be
negative.
1. Centripetal forces do no work, as they are always perpendicular to the direction of
motion.
2. Ex 6-1: Work Done on a Crate (pg. 140)
3. Ex 6-2: Work on a Backpack (pg. 141)
4. Conceptual Ex 6-3: Does the Earth Do Work on the Moon? (pg. 142)
II.
Work Done by a Varying Force (6-2)
A. For a force that varies, the work can be approximated by dividing the distance up into
small pieces, finding the work done during each, and adding them up.
1. As the pieces become very narrow, the work done is the area under the force vs.
distance curve.
III.
Kinetic Energy & the Work-Energy Principle (6-3 )
A. Energy was traditionally defined as the ability to do work.
B. We now know that not all forces are able to do work; however, we are dealing in these
chapters with mechanical energy, which does follow this definition.
C. If we write the acceleration in terms of the velocity and the distance, we find that the
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work done here is: Wnet = 2 mv22 − 2 mv12
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D. We define the kinetic energy: KE = 2 mv 2
E. This means that the work done is equal to the change in the kinetic energy:
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1. Wnet = ΔKE = 2 mv22 − 2 mv12
2. If the net work is positive, the KE increases.
3. If the net work is negative, the KE decreases.
F. Because work and kinetic energy can be equated, they must have the same units: kinetic
energy is measured in joules.
1. Ex 6-4: Work on a Car, to Increase its KE (pg. 144)
2. Conceptual Ex 6-5: Work to Stop a Car (pg. 145)
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Chapter 6 Work & Energy
IV.
Potential Energy (6-4)
A. An object can have potential energy by virtue of its surroundings.
B. Familiar examples of potential energy:
1. A wound-up spring
2. A stretched elastic band
3. An object at some height above the ground
C. In raising a mass m to a height h, the work done by the external force is:
1. W = Fextd cos 0⁰ = mgh=mg(y2-y1)
D. We therefore define the gravitational potential energy: PEG = mgy
E. This potential energy can become kinetic energy if the object is dropped.
F. Potential energy is a property of a system as a whole, not just of the object (because it
depends on external forces).
1. For PEG, it does not matter where you measure y from as long as you are consistent
about where you choose y = 0. Only changes in potential energy can be measured.
2. Ex 6-6: PE Changes for a Roller Coaster (pg. 147)
G. Potential Energy Defined in General
1. In general, the change in PE associated with a particular force is equal to the
negative of the work done by that force when the object is moved from one point to
a second point!
2. Alternatively, we can define the change in PE as the work required of an external
force to move the object without acceleration between the two points!
H. Potential energy can also be stored in a spring when it is compressed; the figure below
shows potential energy yielding kinetic energy.
1. The force required to compress or stretch a spring is: Fs = -kx; where k is called the
spring constant, and needs to be measured for each spring and is measured in N/m.
2. The force increases as the spring is stretched or compressed further. We find that
the potential energy of the compressed or stretched spring, measured from its
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equilibrium position, can be written: PEel = 2 kx2
V.
Conservative and Nonconservative Forces (6-5)
A. If friction is present, the work done depends not only on the starting and ending points,
but also on the path taken! Therefore, Friction is called a nonconservative force.
B. Potential energy can only be defined for conservative forces.
1. Table 6-1: Conservative & Nonconservative Forces
2. Therefore, we distinguish between the work done by conservative forces and the
work done by nonconservative forces.
3. We find that the work done by nonconservative forces is equal to the total change in
kinetic and potential energies: WNC = ΔKE + ΔPE
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Chapter 6 Work & Energy
VI.
Mechanical Energy and Its Conservation (6-6)
A. If there are no nonconservative forces, the sum of the changes in the kinetic energy and
in the potential energy is zero—the kinetic and potential energy changes are equal but
opposite in sign.
B. This allows us to define the total mechanical energy: E = KE + PE
C. And its conservation: E2 = E1 = constant
VII.
Problem Solving Using Conservation of Mechanical Energy (6-7)
A. In the image on the left, the total mechanical energy is: E = KE + PE = ½ mv2 + mgy
1. Ex 6-7: Falling Rock (pg. 151)
2. Ex 6-8: Roller-coaster Car speed Using Energy Conservation (pg. 152)
3. Conceptual Ex 6-9: Speeds on Two Water Slides (pg. 153)
4. Ex 6-10: Toy Dart Gun (pg. 154)
5. Ex 6-11: Two Kinds of PE (pg. 154)
B. If there is no friction, the speed of a roller coaster will depend only on its height
compared to its starting height.
C. For an elastic force, conservation of energy tells us: ½ mv12 + ½ kx12 = ½ mv22 + ½ kx22
VIII.
Other Forms of Energy & Energy Transformations; the Law of Conservation of Energy (6-8)
A. Some other forms of energy: Electric energy, nuclear energy, thermal energy, chemical
energy.
B. Work is done when energy is transferred from one object to another.
C. Accounting for all forms of energy, we find that the total energy neither increases nor
decreases. Energy as a whole is conserved.
IX.
Energy Conservation with Dissipative Processes; Solving Problems (6-9 )
A. If there is a nonconservative force such as friction, the kinetic and potential energies
become heat; the actual temperature rise of the materials involved can be calculated.
B. Problem solving:
1. Draw a picture.
2. Determine the system for which energy will be conserved.
3. Figure out what you are looking for, and decide on the initial and final positions.
4. Choose a logical reference frame.
5. Apply conservation of energy.
6. Solve.
C. Ex 6-12: Friction on the Roller-coaster Car (pg. 158)
X.
Power (6-10)
A. Power is the rate at which work is done.
𝑤𝑜𝑟𝑘
𝑒𝑛𝑒𝑟𝑔𝑦 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑑
B. P = average power = 𝑡𝑖𝑚𝑒 =
𝑡𝑖𝑚𝑒
1. In the SI system, the units of power are watts: 1W = 1 J/s
C. Power is also needed for acceleration and for moving against the force of gravity.
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Chapter 6 Work & Energy
D. The average power can be written in terms of the force and the average velocity: P =
𝑊
𝐹𝑑
=
= Fv
𝑡
𝑡
1. Ex 6-13: Stair Climbing Power (pg. 159)
2. Ex 6-14: Power Needs of a Car (pg. 160)
XI.
Chapter 6 Vocabulary
A. Work: is done on an object when the energy of the object changes. The work done
equals the product of the net force (F), the displacement (d), and the cosine of the angle
between the force vector and the displacement vector.
B. Joule (J): is the unit associated with work and energy; 1J = 1 N·m = 1kg·m/s 2
C. Kinetic Energy (KE): is energy due to an object’s motion. Translational KE of an object
depends on the object’s mass and the square of its speed.
D. Work-Energy Principle: states that the net work done on an object is equal to its change
in kinetic energy.
E. Gravitational PE: is energy stored by an object due to its position. Near the surface of
the Earth, the gravitational PE relative to some reference point, such as the ground, is
given by the product of the object’s weight and the height above the reference level.
F. Elastic Potential Energy: is energy that results from an object being stretched or
compressed (as for a spring) or twisted (as for a wire). The change in a spring’s elastic
PE as it is stretched (or compressed) is related to the force (F) required to stretch (or
compress) the spring and the distance that the spring is stretched (or compressed).
G. Conservative Force: does the same amount of work independent of the path taken
between two points. An example of a conservative force is gravity. The work done
equals the change in PE and depends only on the initial and final positions above the
ground and not on the path taken.
H. Nonconservative Force: results in different amounts of work being expended in moving
an object. The net work required depends on the path taken in moving the object
between points.
I. Mechanical Energy: refers to an object’s KE and PE
J. Law of Conservation of Energy: states that energy is neither created nor destroyed.
Energy can be transformed from one kind to another, but the total amount remains
constant.
K. Power: is the rate at which work is done and is measured in watts, where 1 watt = 1 J/s.
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