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Chapter 6
Work and Energy
Energy
• What is energy?
• Energy – is a fundamental, basic notion in physics
• Energy is a scalar, describing state of an object or a
system
• Description of a system in ‘energy language’ is
equivalent to a description in ‘force language’
• Energy approach is more general and more effective
than the force approach
• Equations of motion of an object (system) can be
derived from the energy equations
Forms of energy
• Mechanical
• Chemical
• Electromagnetic
• Nuclear
• Energy can be transformed from one form to
another
Work
• Work is a scalar
• Work is done on the object by a force
• Work (W) done by a constant force on an object:
product of the component of the force along the
direction of displacement and the magnitude of the
displacement
W  (F cos  )  x
• SI unit: kg*m2/s2 = J (Joule)
James Prescott Joule
(1818 - 1889)
Work
• If there are multiple forces acting on an object, the
total work done is the algebraic sum of the amount of
work done by each force
• Work is positive (negative) if the force and the
displacement are in the same (opposite) direction
• The work done by a force is zero when the force is
perpendicular to the displacement: cos 90° = 0
Work
• If there are multiple forces acting on an object, the
total work done is the algebraic sum of the amount of
work done by each force
• Work is positive (negative) if the force and the
displacement are in the same (opposite) direction)
• The work done by a force is zero when the force is
perpendicular to the displacement: cos 90° = 0
Kinetic energy
• Kinetic energy is a scalar
• Kinetic energy describes object’s ‘state of motion’
• Kinetic energy:
mv 2
K
2
• SI unit: kg*m2/s2 = J (Joule)
• Work is related to kinetic energy
Work-kinetic energy theorem
• When work is done by a net force on an object and
the only change in the object is its speed, the work
done is equal to the change in the object’s kinetic
energy
• Work is equal to the change in kinetic energy, i.e.
work is required to produce a change in kinetic
energy
Wnet  KE f  KE i   KE
• Speed will increase (decrease) if work is positive
(negative)
• Kinetic energy can also be thought of as the amount
of work the moving object could do in coming to rest
Work-kinetic energy theorem
Chapter 6
Problem 26
Under the influence of its drive force, a snowmobile is moving at a constant
velocity along a horizontal patch of snow. When the drive force is shut off, the
snowmobile coasts to a halt. The snowmobile and its rider have a mass of 136
kg. Under the influence of a drive force of 205 N, it is moving at a constant
velocity whose magnitude is 5.50 m/s. The drive force is then shut off. Find (a)
the distance in which the snowmobile coasts to a halt and (b) the time required
to do so.
Work done by the gravitational force
• Gravity force is ~ constant near the surface of the
Earth
Wg  mgd cos 
• If the displacement is vertically up
Wg  mgd cos 180   mgd
• In this case the gravity force does a negative work
(against the direction of motion)
Lifting an object
• We apply a force F to lift an object
• Force F does a positive work Wa
• The net work done
Wnet  K  K f  K i  Wa  Wg
• If in the initial and final states the object is at rest,
then the net work done is zero, and the work done by
the force F is
Wa  Wg  mgd
Conservative forces
• The net work done by a conservative force on a
particle moving around any closed path is zero
Wab,1  Wba, 2  0
Wab,1  Wba, 2
Wab, 2  Wba, 2
Wab,1  Wab, 2
• The net work done by a conservative force on a
particle moving between two points does not depend
on the path taken by the particle, only on the initial
and final positions of the object
Conservative forces: examples
• Gravity force
• Spring force
• Electromagnetic force
Potential energy
• For conservative forces we introduce a definition of
potential energy U
U  W
• The change in potential energy of an object is being
defined as being equal to the negative of the work
done by conservative forces on the object
• Potential energy is associated with the arrangement
of the system subject to conservative forces
• Potential energy is a property of the system, not the
object
Potential energy
• A conservative force is associated with a potential
energy and for every conservative force a potential
energy function can be found
• There is a freedom in defining a potential energy:
adding or subtracting a constant does not change the
force
Gravitational potential energy
• Gravitational potential energy is the energy
associated with the relative position of an object in
space near the Earth’s surface
• Objects interact with the earth through the
gravitational force
• Actually the potential energy is for the earth-object
system
U g ( y )  mgy
Spring force
• Spring in the relaxed state
• Spring force (restoring force) acts to restore the
relaxed state from a deformed state
Hooke’s law
• For relatively small deformations


Fs  kd
Robert Hooke
(1635 – 1703)
• Spring force is proportional to the deformation and
opposite in direction
• k – spring constant
• Spring force is a variable force
• Hooke’s law can be applied not to springs only, but
to all elastic materials and objects
Elastic potential energy
• Hooke’s law in 1D
Fs  kx
• For a spring obeying the Hooke’s law
kx
U s ( x) 
2
2
• Elastic potential energy related to the work required
to compress a spring from its equilibrium position to
some final, arbitrary, position x
Conservation of mechanical energy
• Mechanical energy of an object is
Emec  K  U
• When a conservative force does work on the object
K  W U  W K  U
K f  U f  Ki  U i
K f  K i  (U f  U i )
Emec, f  Emec,i
• In an isolated system, where only conservative
forces cause energy changes, the kinetic and
potential energies can change, but the mechanical
energy cannot change
Conservation of
mechanical
energy:
pendulum
Chapter 6
Problem 43
A skateboarder moving at 5.4 m/s along a horizontal section of a track that is
slanted upward by 48° above the horizontal at its end, which is 0.40 m above
the ground. When she leaves the track, she follows the characteristic path of
projectile motion. Ignoring friction and air resistance, find the maximum height
H to which she rises above the end of the track.
Nonconservative forces
• A force is nonconservative if the work it does on an
object depends on the path taken by the object
between its final and starting points.
• Examples of nonconservative forces: kinetic
friction, air drag, propulsive forces
• For friction force, the work required is less on the
blue path than on the red path
Nonconservative forces
• The friction force is transformed from the kinetic
energy of the object into a type of energy associated
with temperature
• The objects are warmer than they were before the
movement
Internal energy
• The energy associated with an object’s temperature
is called its internal energy, Eint
• In this example, the friction does work and
increases the internal energy of the surface
Work done by an external force
• Work is transferred to or from the system by means
of an external force acting on that system
W  K  U  Eint
• The total energy of a system can change only by
amounts of energy that are transferred to or from the
system
• Energy is conserved: we can neither create nor
destroy energy
• If nonconservative forces are present, then the full
work-energy theorem must be used instead of the
equation for conservation of mechanical energy
Power
• Average power
Pavg
W

t
• Instantaneous power – the rate of doing work
W
P  lim
t 0 t
• SI unit: J/s = kg*m2/s3 = W (Watt)
Pavg
Fr cos 
W


 Fvavg cos 
t
t
James Watt
(1736-1819)
Chapter 6
Problem 70
A 1900-kg car experiences a combined force of air resistance and friction that
has the same magnitude whether the car goes up or down a hill at 27 m/s.
Going up a hill, the car’s engine produces 47 hp more power to sustain the
constant velocity than it does going down the same hill. At what angle is the
hill inclined above the horizontal?
Work done by varying forces
• 1D case: work done by a variable force acting on an
object that undergoes a displacement is equal to the
area under the graph of Fx(x)
Work done by a spring force
• Hooke’s law in 1D
Fs  kx
• The work is also equal to the area under the curve
• In this case, the “curve” is a triangle
2
kx
W 
 U s (x)
2
2
kx
U s ( x) 
2
Questions?