Physics

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Physics

• Conservative and
• Non-Conservative
• Forces
• Teacher Luiz Izola

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Chapter Preview

• Conservative Forces
• Non-Conservative Forces
• Potential Energy
• Work Done by Conservative Forces
• Conservation of Mechanical Energy
• Work Done by Non-Conservative Forces
• Potential Energy Curves and Equipotentials

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Introduction

• One of the greatest physics concepts is the
  conservation of energy.
• Energy has several forms (mechanical, thermal).
  The universe has a constant amount of energy that
  flows from one form to another.
• In this chapter we will focus on the
  conservation of energy, the first conservation
  law.

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Conservative Forces

• The work done by a conservative force can be
  stored in the form of energy and released at a
  later time.
• Simplest case of conservative force ? Gravity

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Conservative Forces

• Definition1 A conservative force does zero work
  on a closed path.
• Definition2 Work done by a conservative force
  in going from points A to B is independent of the
  path taken.

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Conservative Forces

• W1 W2 0
• W1 W3 0
• Therefore W2 W3

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Non-Conservative Forces

• The work done by a non-conservative force cannot
  be recovered later as kinetic energy.
• It is converted to other forms of energy such as
  heat. See the work done by friction below.

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Non-Conservative Forces

• Considering the same closed path but analyzing
  the effect of friction on the total work, we
  would have Wtotal -4µkmgd

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Example

• A 4.57kg box is moved with constant speed from A
  to B along the two paths below. Calculate the
  work done by gravity on each one of these paths.
  Also, calculate the work done by friction (µk
  0.63) along the two paths.

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Potential Energy (U)

• Potential energy is a storage system for energy.
• Energy is never lost as long as the separation
  remains the same. For example, when we lift a
  ball we produce an amount of potential energy
  related to the height we lift it. The ball can
  rest on a shelf for a million years and when it
  falls, it will gain the same amount of kinetic
  energy.
• Work done against friction is not stored as
  potential energy. It dissipates as heat or sound.
• Only conservative forces have the
  potential-energy storage system.

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Work by Conservative Forces

• When a conservative force does an amount of work
  Wc (c conservative), the corresponding
  potential energy U is changed according to
• Wc Ui Uf -?U
• The work done by a conservative force is equal
  to the negative of the change in potential
  energy.
• For example, when an object falls, gravity does
  positive work on it and its potential energy
  decreases.

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Work by Conservative Forces

• Gravity Lets apply our definition of potential
  energy to the force of gravity near the Earths
  surface.

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Work by Conservative Forces

• Gravitational Potential Energy
• U mgy
• y height
• m mass
• g acceleration of gravity
• Example Find the gravitational potential energy
  of a 65-kg person on a 3.0meter diving board. Let
  U 0 be the water level.

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Work by Conservative Forces

• Example An 82kg mountain climber is in the
  final stage of a 4301-meter-high peak. What is
  the change in potential energy as the climber
  gains the last 100 meters of altitude? Let U 0
  be (a) at sea level (b) at the top of the peak.

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Work by Conservative Forces

• Springs Consider a spring that is stretched from
  its equilibrium position a distance x. The work
  required to cause this stretch is W 1/2Kx2.
• From our definition of potential energy, we have
• Wc 1/2Kx2 Ui Uf
• Assuming that at x0 (equlibirum position), U
  0, we can simplify the formula Ui 1/2Kx2 .
• Spring Potential Energy U 1/2Kx2

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Work by Conservative Forces

• Example Find the potential energy of a spring
  with force constant k680N/m if it is (a)
  stretched 5cm and (b) compressed 7cm.
• Example When a force of 12.0N is applied to a
  certain spring, it causes a stretch of 2.25cm.
  What is the potential energy of this spring when
  it is compressed by 3.50cm?

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Practice Session

1. Calculate the work done by the gravity as a 2.6kg
   object is moved from point A to point B, along
   paths 1,2, and 3.

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Practice Session

1. Calculate the work done by friction as a 2.6kg
   box is slid along a floor from point A to point B
   along paths 1,2, and 3. Assume the kinetic
   coefficient of friction is 0.23. Based on
   previous picture.
2. A 4.1kg block is attached to a spring with force
   constant of 550N/m. Find the work done by the
   spring on the block as the block moves from A to
   B along paths 1 and 2.

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Practice Session

• Calculate the work done by gravity as a 5.2kg
  object is moved from A to B along paths 1 and 2.
  How does the mass of the object affects the
  results?

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Practice Session

1. As an Acapulco cliff diver drops to the water
   from a height of 40.0meters, his gravitational
   potential energy decreases by 25,000J. How much
   does the diver weighs?
2. Find the gravitational potential energy of an
   80.0kg person standing atop Mt. Everest, at an
   altitude of 8848meters. Use the sea level as the
   location y 0.
3. Compressing a spring 0.50cm produces a potential
   energy equals 0.0035J. Which compression is
   required to generate a energy equals 0.080J?

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Practice Session

1. A force of 4.7N is required to stretch a certain
   spring by 1.30cm. (a) How far must the spring be
   stretched for its potential energy to be 0.020J?
   (b) How much stretch is required for the spring
   potential energy to be 0.080J?
2. A 0.33kg pendulum bob is attached to a string
   1.2meters long. What is the bobs potential
   energy change from points A to B?

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Conservation of Mechanical Energy

• Mechanical Energy is the sum of potential and
  kinetic energies of an object.
• E U K
• Mechanical energy is conserved on systems
  involving ONLY conservative forces. E is constant
  in this case.
• Proving that E is constant for conservative
  forces
• Wtotal ?K Kf Ki and Wtotal Wc
• We know that Wc Ui Uf . Then, replacing
  both,
• we get Kf Ki Ui Uf .
• Rearranging, we get Ef Ei.

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Conservation of Mechanical Energy

• In terms of physical systems, conservation of
  mechanical energy means that energy can be
  converted between potential and kinetic forms,
  but the SUM REMAINS THE SAME.
• In systems with conservative forces only, the
  mechanical energy E is conserved, that is
• E U K constant

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Example
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Example

• Ei Ef
• Ui Ki Uf Kf
• From left side of previous picture, we have
• mgh 0 0 1/2mv2
• Therefore v (2gh)1/2
• Now Prove for right side of previous picture.

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Practice

• At the end of a graduation ceremony, graduates
  fling their caps into the air. Suppose a 0.12kg
  cap is thrown straight upward with an initial
  speed of 7.85m/s and there is no friction. (a)
  Use kinematics to find the speed of the cap when
  is 1.18m above the release point. (b) Show that
  mechanical energy is the same at the release and
  at 1.18m

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Practice

• In the bottom of the ninth inning, a player hits
  a 0.15kg baseball over the outfield fence. The
  ball leaves the bat with a speed of 36m/s, and a
  fan in the bleachers catches it 7.2m above the
  point where it was hit. Find (a) Kinetic Energy
  when the ball is caught. (b) The speed when it is
  caught.

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Practice

• A 55kg skateboarder enters a ramp horizontally
  with a speed of 6.5m/s, and leaves the ramp
  vertically with a speed of 4.1m/s. Find the
  height of the ramp.

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Practice

• A 1.7kg block slides on a horizontal,
  frictionless surface until it encounters a spring
  with a force constant k955n/m. The block comes
  to rest after compressing the spring a distance
  of 4.60m. Find the initial speed of the block.

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Practice

• Suppose the spring and block are oriented
  vertically. Initially, the spring is compressed
  4.60cm and the block is at rest. When the block
  is released, it accelerates upward. Find the
  speed of the block when the spring returns to the
  equilibrium position.

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Homework

1. If a 30.0J of work is required to stretch a
   spring from 4.00cm to 5.00cm., how much work is
   necessary to stretch it from 5.00cm to 6.00cm?
2. A spring scale has a spring with a force constant
   of 250N/M and a weighing pan with a mass of
   0.075kg. During the first weighing, the spring is
   stretched a distance of 12cm from equilibrium.
   The second time is stretched 18cm. (a) How much
   greater is the elastic potential energy of the
   spring during the second than the first weighing?
   (b) If the spring is released after each
   weighing. What is the ratio of the pans maximum
   speed between second and first weighing?

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Homework

1. An 80.0N box of clothes is pulled 20.0m up a 30o
   ramp by a force of 115N that points along the
   ramp. If the coefficient of kinetic friction
   between the box and the ramp is 0.22, calculate
   the change in the boys kinetic energy.
2. A 0.60kg rubber ball has speed of 2.0m/s at point
   A and kinetic energy of 7.5J at point B.
   Determine the following (a) Balls kinetic
   energy at A. (b) Balls speed at B. (c) Work done
   by ball from A to B.

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Homework

1. Starting from rest, a 5.0kg block slides 2.5m
   down a rough 30o incline in 2.0s. Determine the
   following (a) The work done by gravity. (b)
   The mechanical energy lost due to friction
2. At a park, a swimmer uses a water slide to enter
   the main pool. If the swimmer starts at rest,
   slides without friction, and falls through a
   vertical height of 2.61m, what is her speed at
   the bottom of the slide?
3. A player passes a 0.60kg ball. The ball leaves
   the players hands with a 8.30m/s speed and slows
   to 7.10m/s at its highest point. How high is the
   ball at the highest point from release?

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Homework

1. An 18kg child plays on a slide that drops through
   a height of 2.2m. The child starts at rest. On
   the way down a non-conservative work of -373J is
   done on the child. What is the childs speed at
   the bottom of the slide?
2. A 17,000kg airplane lands with a speed of 82m/s
   on a 115m long carried deck. Find the work done
   by non-conservative forces in stopping the plane.
3. A 5.0kg rock is dropped and allowed to fall
   freely. Find the initial kinetic energy, the
   final kinetic energy, and the change in kinetic
   energy for (a) first 2 meters of the fall. (b)
   next two meters of the fall

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