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Potential energy , and conservation of energy Chap 8


Conservation of mechanical energy ??? ... The change in mechanical work must equal to the total change in kinetic and potential energy. ... – PowerPoint PPT presentation

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Title: Potential energy , and conservation of energy Chap 8

Potential energy ??, ?? and conservation of
energy (Chap 8)
Potential energy is the energy associated with
the arrangement of the objects in a system. It is
like the energy stored in the system.
  • When throwing a tomato upward, the gravitational
    force does a negative work, i.e. it gains energy.
    Here it means the tomato-Earth system.
  • Then energy, gravitational potential energy, is
    stored in the tomato-Earth system (or it can be
    said that the energy is stored in the
    gravitational field).
  • kinetic energy of tomato reduced
  • when the tomato falls back, gravitational force
    does positive work, the system loses energy
  • the tomato gains energy and speeds up

Another example
  • (a) the work done by the spring is negative.
  • therefore energy is stored in the block-spring
    system, elastic ?? potential energy.
  • when the block moves to the left, energy
  • energy from potential energy to kinetic energy.

Conservative and nonconservative forces
  • in the previous examples, the work done
    (negative) by the force is stored in the terms of
    potential energy and is able to transfer back to
    the kinetic energy
  • in this situation, the force is called
    conservative force ??? gravitational force,
    spring force.
  • other situations where certain amount of energy
    is lost and cannot be transferred back, the force
    is nonconservative, like frictional force and
    drag force.

Path independence of conservative force
(1) For a conservative force, the total work done
around a closed path is zero.
  • on the other hand, since the force is
    conservative, then

(2) the work done by a conservative force is
independent of the path taken
No, F is nonconservative because the work done
depends on the path.
Because the gravitational force is conservative,
its work done on any paths are the same.
Relation between potential energy and
conservative force.
The potential energy gain is simply the negative
of work done by the conservative force
Gravitational potential energy
Note that only the difference in U is important,
i.e. the energy change relative to a reference
Elastic potential energy
take xi 0
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(3), (1), (2)
Conservation of mechanical energy ???
Mechanical energy the sum of kinetic energy and
potential energy.
  • Assumptions
  • the system concerned is isolated, i.e. no
    external force acting on the system
  • there is only conservative force.

Since we have
and by definition the potential energy is the
negative of work
No change in mechanical energy
Or in other form
This is called the principle of conservation of
mechanical energy
A typical example
The mechanical energy of the pendulum is always
the same.
An experiment done by Galileo Galileo discovered
that when the swing of a pendulum was interrupted
by a peg, the bob still rose to its initial level.
It is because of the conservation of mechanical
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Fmax is determined using the conservation of
mechanical energy
Gravitation potential energy of the climber
elastic potential energy of the rope.

A quadratic equation of d
The solution of d is
Only take the positive solution
Fmax is then given by
The factor 2H/L is important here
Put in the numbers we have for (a) Fmax4.7x103N
and for (c) Fmax5.3x103N
The situation in (c) is more dangerous.
Potential energy curve
We have learned previously how to find the
potential energy by integrating a conservative
But we can also do reverse, find the force by
differentiate the potential
  • For example
  • Spring potential
  • Gravitational potential

Since K?0, so the particle cannot move beyond the
turning point where K0.
Consider a general potential
Given a mechanical energy Emec, the kinetic
energy K is just the difference between the line
of Emec and the curve U(x).
  • Equilibrium point is the location where the force
    is zero, i.e. the slope of U(x) is 0.
  • If it is a minimum in U(x), then it is a stable
    equilibrium point.
  • If it is a maximum, then it is a unstable
    equilibrium a small derivation will cause the
    particle moves away from the point.

For different mechanical energies
In this case, if Emec lt 4, the particle moves
back and forth within a region.
  • Another example a typical potential curve of a
    molecule (e.g. H2).
  • the molecule has an energy less than U0, then it
    is bounded (it forms a molecule), otherwise the
    two atoms can move apart.
  • the equilibrium point is located at r0.
  • the potential is huge when two atoms move too
    close to each other.

Work done by external force
The work done by the external force will be the
same as the change in mechanical work of the
The change in mechanical work must equal to the
total change in kinetic and potential energy.
If friction is involved, work done by the
frictional force should also be considered
Therefore total work done on the system is now
Work done by internal force
  • for an isolated system, the total energy must
    conserve, i.e. total work done on the system is

Change in internal energy
  • in this example, force F does no work on the
    skater. There is no energy transfer from the rail
    to the skater.
  • but the skaters kinetic energy does change
    (increases), the energy comes from her
    bio-chemical energy (internal energy).

Initial energy Final energy
Conservation of energy tells us that
Assume it stops at a distance d from the left
Initial potential energy
Heat lost due to friction
Since d is larger than L, so the particle has
enough energy to climb the right curved portion
Problems 5, 22, 23, 51, 135
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