Physics 1020 Lecture 22 Chapter 7. Work and Kinetic Energy 2 lectures in total

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Physics 1020 Lecture 22Chapter 7. Work and
Kinetic Energy(2 lectures in total)

• Work Done by a Constant Force (Sect 7-1)
• Kinetic Energy and the Work-Kinetic Energy
  Theorem (Sect. 7-2)
• Work Done by a Varying Force (Sect. 7-3)
• Power (Sect. 7-4)

2
Energy Approach to Problems

• The concept of energy is one of the most
  important topics in science
• The energy approach to describing motion is
  particularly useful when the force is not
  constant
• Note In this case, our standard constant
  acceleration kinematic equations do not apply

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Work

• Work is done on a system where an applied force
  results in some net displacement
• A force that results in no displacement does no
  work
• A displacement that results with no applied force
  has had no work done (orbital motion, for example)

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Work

• W F d cos q definition of work

d - the displacement of the point of application
of the force ? - is the angle between the force
and the displacement vectors
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Work

• W F d cos q definition of work
• Units Nm Joule J 1(kgm/s2)m kgm2/s2

force
displacement
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Work

• W F d cos q
• A force does no work on the object if the force
  does not move through a displacement
• The work done by a force on a moving object is
  zero when the force applied is perpendicular to
  the displacement of its point of application
  (think about an object in orbit) !cos 90 0

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The Scalar Product of Two Vectors

• Our definition of work W F d cos q is messy
  because of the cosine factor
• What we are actually doing is performing an
  operation with two vectors called the scalar
  product

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Scalar Product of Two Vectors

• The scalar product of two vectors is written as
• It is also called the dot product
• q is the angle between A and B

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Scalar Product, cont

• The scalar product is commutative
• The scalar product obeys the distributive law of
  multiplication

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Dot Products of Unit Vectors

• Dot product of orthogonal vectors is 0!
• Using component form with and

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Dot product / Work

• Why did we introduce the dot product?
• Because
• Work is a scalar quantity!!!! (Not a vector)

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More About Work

• The sign of the work depends on the direction of
  the displacement relative to the force
• Work is positive when the projection of F onto
  d is in the same direction as the
  displacement
• Work is negative when the projection is in the
  opposite direction

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More About Work

• When more than one force acts on an object

or
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Work - Example

• The normal force, n, and the gravitational force,
  mg, do no work on the object
• cos q cos 90 0
• The force does do work on the object
• There is a component along the direction of
  motion

W F d cos q
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Example Pr. 12 p. 194

• Water skiers often ride to one side of the center
  line of a boat, as shown in Figure. In this case,
  the ski boat is traveling at 15 m/s and the
  tension in the rope is 75 N. If the boat does
  3500 J of work on the skier in 50.0 m, what is
  the angle between the tow rope and the center
  line of the boat?

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Example Pr. 13 p. 194

• To keep her dog from running away while she
  talks to a friend, Susan pulls gently on the
  dogs leash with a constant force given by
• How much work does she do on the dog if its
  displacement is (a)
• or
• (c)

Home!
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Work Done by a Varying Force

• We know that for a constant force the work is
  just
• But if the force is not constant in space we have
  a problem
• We will use an approach similar to that used to
  determine net displacement given a time varying
  velocity!

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Work Done by a Varying Force

• Consider a force in the x-direction with a
  magnitude that changes depending on the exact
  position.
• We can carve that (the displacement xf-xi)
  displacement up into a bunch of small (very
  small) displacements.

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Work Done by a Varying Force

• For each of these small displacements, the force
  will be nearly constant, in that case, W1
  FxDx (the area of the rectangle)
• This will be true for all the intervals so we can
  form

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Work Done by a Varying Force

• In fact, this sum gives us the area under the
  Force-position curve!

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Work Done by a Varying Force, cont

• Again, we make the intervals vanishingly small
• Therefore,
• The work done is equal to the area under the
  curve

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Work Done by a Spring

• Calculate the work done on the block by the
  spring as the block moves from xi - xmax to xf
  0
• W area 1/2 kx2
• (Recall Hooks law F - k x)
• Calculate the work done on the block by the
  spring as the block moves from xi 0 to xf
  xmax
• W area 1/2 k x2
• The total work done as the block moves from
  xmax to xmax is zero

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Work Done by a Spring

• The work done for an arbitrary displacement is
  given by
• For any motion that results in no net
  displacement (xi xf) there is no work done!
• More intriguing, you get the same result if the
  displacement is symmetric (xi -xf).

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Work Kinetic Energy

• A force F 10 N pushes a box across a
  frictionless floor for a distance ?x 5 m. The
  speed of the box is v1 before the push and v2
  after the push.

v1
v2
F
m
i
?x
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Work Kinetic Energy...

• Since the force F is constant, acceleration a
  will be constant. We have shown that for
  constant a
• v22 - v12 2a(x2-x1) 2a?x.
• multiply by 1/2m 1/2mv22 - 1/2mv12 ma?x
• But F ma 1/2mv22 - 1/2mv12 F?x
• F?x W
  1/2mv22 - 1/2mv12 W

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Kinetic Energy and the Work-Kinetic Energy
Theorem

• So we have
• That is, the work done by the net force on a
  particle of mass m is equal to the difference
  between the initial and final values of a
  quantity

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Kinetic Energy and the Work-Kinetic Energy
Theorem

• The quantity is so important that we
  give it a special name.
• The kinetic energy K of an object of mass m
  moving with a speed v is
• Kinetic energy is the energy of a particle due to
  its motion.
• Kinetic energy is a scalar and has the same units
  as work (Joules)

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Kinetic Energy and the Work-Kinetic Energy
Theorem

• A change in kinetic energy is one possible result
  of doing work to transfer energy into a system.
• If the net work done on a particle is positive,
  the speed (and hence the kinetic energy) of the
  particle increases.
• If the net work done on a particle is negative,
  the speed (and hence the kinetic energy) of the
  object decreases.

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Kinetic Energy and the Work-Kinetic Energy
Theorem

• The equation relating work and kinetic energy is
  often written in the form
• Wnet Kf Ki DK
• This result is known as the Work-Kinetic Energy
  Theorem
• When work is done on a system and the only
  change in the system is in its speed, the work
  done by the net force equals the change in
  kinetic energy of the system.
• Important The Work-Kinetic Energy Theorem
  applies even if the force varies.

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Work-Kinetic Energy Theorem Example

• A 6.00 kg block initially at rest is being pulled
  to the right along a horizontal frictionless
  surface by a constant, horizontal force F of
  magnitude 12.0 N as shown. Find the speed of the
  block after it has moved 3.00 m.

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Work-Kinetic Energy Theorem Example

• Three external forces interact with the block.
• The normal and gravitational forces do no work
  since they are perpendicular to the direction of
  the displacement.
• There is no friction and so resultant external
  force is F.

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Work-Kinetic Energy Theorem Example

• The work done by the force F is
• WF F Dx (12.0 N)(3.00) m
• 36.0 J
• Apply Work-Kinetic Energy Theorem

Exercise Find the acceleration of the block and
determine its final speed using the constant
acceleration kinematic equations. Answer a
2.0 m/s2, vf 3.46 m/s
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HOMEWORK! Example 7-5 page 183
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Example Pr.18-19 p.194

• A 0.21-kg pine cone falls 14 m to the ground,
  where it lands with a speed of 13 m/s. (a) With
  what speed would the pine cone have landed if
  there had been no air resistance? (b) Did air
  resistance do positive work, negative work, or
  zero work on the pine cone? Explain.
• In the previous problem, (a) how much work was
  done on the pine cone by air resistance? (b) What
  was the average force of air resistance exerted
  on the pine cone?

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Power

• It is also interesting to know the rate at which
  energy is transferred.
• The time rate of energy transfer is called power
• If an external force is applied to an object and
  if the work done by this force is W in the time
  interval ?t, the average power during this
  interval is defined as

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Units of Power

• The SI unit of power is called the Watt
• 1 Watt 1 joule / second 1 kg . m2 / s2
• The common unit of power is the horsepower
• 1 hp 550 ft .lb/s 746 W
• In fact the rate at which a horse could do work!
• The unit of energy that appears on your household
  electricity bill is defined in terms of the Watt
• 1 kWh (1000 W)(3600 s) 3.6 x106 J
• See the electricity meter on your house or your
  electricity bill!

These sorts of constructions are an abuse of the
SI system
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Example Pr. 46 p. 195

• A certain car can accelerate from rest to the
  speed v in T seconds. If the power output of the
  car remains constant, (a) how long does it take
  for the car to accelerate from v to 2v? (b) How
  fast is the car moving at 2T seconds after
  starting? (home)