Chapter 8 Potential Energy and Conservation of Energy

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Chapter 8 Potential Energy and Conservative
Forces

• Potential Energy and Conservation of Energy.
• Conservative and non-conservative forces
• Gravitational and Elastic Potential Energy
• Conservation of (Mechanical) Energy
• External and Internal Forces
• CONSERVATION OF ENERGY

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• Potential Energy

• Potential Energy U is a form of stored energy
  that can be associated with the configuration (or
  arrangement) of a system of objects that exert
  certain types of forces (conservative) on one
  another.
• When work gets done on an object, its potential
  and/or kinetic energy increases.
• There are different types of potential energy
• Gravitational energy
• Elastic potential energy (energy in an stretched
  spring)
• Others (magnetic, electric, chemical, )

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• We know that the work done result in a change in
  kinetic energy (WDK). Now we can ask the
  question where did the kinetic energy go (if it
  is decreased) or where did it come from (if it
  increased)! Note that the force only function as
  the agent which rearranges the configuration of
  the system (by displacing one or more of the
  object in the system). Assuming that our system
  is isolated (no external force acting on it) the
  answer, as you have already guessed, is to (or
  from) the potential energy of the system..
• When one of these special forces (let us label it
  Fc) does some work (Wc) by changing the system
  configuration, the force derives the energy from
  the stored potential energy associated with that
  force

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• Conservative and Nonconservative forces
• Let us say the work done by a force F when a
  system is changed from configuration 1 to
  configuration 2 is W12. We now reverse the
  process, i.e. take the system back to 1 from 2
  and let us say that we measure the work done by
  the same force now to be W21. It is obvious that
  we can define a potential energy for this force
  by the equation DU-W only if W12 - W21.
• A force for which W12 - W21 is called a
  conservative forces. This is same as saying that
  the net work done by a conservative force around
  any closed path (return back to the initial
  configuration) is zero. A force that is not
  conservative is called a nonconservative force.
  We cannot define potential energy associated with
  a nonconservative forces.

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• The gravitational force and the spring force are
  examples of conservative forces. The frictional
  force and fluid drag force are examples of
  nonconservative forces.

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Non-conservative forces

• A force is non-conservative if it causes a change
  in mechanical energy mechanical energy is the
  sum of kinetic and potential energy.
• Example Frictional force.
• This energy cannot be converted back into other
  forms of energy (irreversible).
• Work does depend on path.

Sliding a book on a table
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• Determining Potential Energy Values

• In general we can write

• Gravitational potential energy, DUg

DO SP 8-2
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• Elastic potential energy, DUs

Work done by a spring
xi
xf
Elastic potential energy stored in a spring
The spring is stretched or compresses from its
equilibrium position by x
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• SP 8-2

• (a)  What is the gravitational potential energy U
  of the slothEarth system if we take the
  reference point y 0 to be (1) at the ground,
  (2) at a balcony floor that is 3.0 m above the
  ground, (3) at the limb, and (4) 1.0 m above the
  limb? Take the gravitational potential energy to
  be zero at y 0.
• (b) The sloth drops to the ground. For each
  choice of reference point, what is the change DU
  in the potential energy of the sloth-Earth system
  due to the fall?

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• Conservation of Mechanical Energy

If we deal only with conservative forces and If
we deal with an isolated system (no energy added
or removed) The total mechanical energy
of a system remains constant!!!!
The final and initial energy of a system remain
the same Ei Ef
Thus
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• Figure 8-7

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• SP 8-6

A 61.0 kg bungee-cord jumper is on a bridge 45.0
m above a river. The elastic bungee cord has a
relaxed length of L 25.0 m. Assume that the
cord obeys Hooke's law, with a spring constant of
160 N/m. If the jumper stops before reaching the
water, what is the height h of her feet above the
water at her lowest point?
What is your system?
Jumper Earth Cord
Initial configuration
Is your system isolated?
YES
Is friction or Drag present?
No
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• SP 8-6

A 61.0 kg bungee-cord jumper is on a bridge 45.0
m above a river. The elastic bungee cord has a
relaxed length of L 25.0 m. Assume that the
cord obeys Hooke's law, with a spring constant of
160 N/m. If the jumper stops before reaching the
water, what is the height h of her feet above the
water at her lowest point?
Write the equation for conservation of mechanical
energy
final configuration
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• SP 8-6

A 61.0 kg bungee-cord jumper is on a bridge 45.0
m above a river. The elastic bungee cord has a
relaxed length of L 25.0 m. Assume that the
cord obeys Hooke's law, with a spring constant of
160 N/m. If the jumper stops before reaching the
water, what is the height h of her feet above the
water at her lowest point?
Substitute the values
final configuration
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• Work done on a System by an External Force (and
  friction)
• When we stated the conservation of mechanical
  energy for a system in the previous section, we
  specified two conditions
• Isolated system (no external forces)
• Only conservative forces in the system.
• Let us now introduce external forces doing work
  on the system, then
• And also add nonconservative forces (friction
  involved) in the system

(work done on the system, friction involved)
(increase in thermal energy by sliding)
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• Conservation of Energy
• The total energy of a system can change only by
  amounts of energy Wex that are transferred to or
  from the system.
• DEint acknowledges the fact that thermal energy
  is not the only other form of energy that a
  system can have which is not mechanical energy,
  e.g. chemical energy in your muscles or in a
  battery, or nuclear energy.
• The total energy of an isolated system cannot
  change.
• Power as the rate at which energy is transferred
  from one form to another

Average power
Instantaneous power
The rate at which the work is done is a special
case of energy being transferred to (or from)
kinetic energy (one form of energy).
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4E

• A frictionless roller coaster with an initial
  speed of v0 10.00 m/s, at the initial height h
  100.0 m, has a mass m 1000.0 kg
• What is the speed at point A?
• What is the speed at point B
• How high will it move up on the last hill?

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• SP 8-7

In the figure, a 2.0 kg package of tamale slides
along a floor with speed v1 4.0 m/s. It then
runs into and compresses a spring, until the
package momentarily stops. Its path to the
initially relaxed spring is frictionless, but as
it compresses the spring, a kinetic frictional
force from the floor, of magnitude 15 N, acts on
it. The spring constant is 10,000 N/m. By what
distance d is the spring compressed when the
package stops?