Summary Lecture 7

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Summary Lecture 7
7.1-7.6 Work and Kinetic energy 8.2 Potential
energy 8.3 Conservative Forces and Potential
energy 8.5 Conservation of Mech.
Energy 8.6 Potential-energy curves 8.8 Conservat
ion of Energy Systems of Particles 9.2 Centre of
mass
Problems Chap. 8 5, 8, 22, 29, 36, 71, 51
Chap. 9 1, 6, 82
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Outline Lecture 7
Work and Kinetic energy Work done by a net force
results in kinetic energy Some examples
gravity, spring, friction Potential energy Work
done by some (conservative) forces can be
retrieved. This leads to the principle that
energy is conserved Conservation of
Energy Potential-energy curves The dependence of
the conservative force on position is related to
the position dependence of the PE F(x) -d(U)/dx
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Kinetic Energy
Work-Kinetic Energy Theorem Change in KE? work
done by all forces
DK ? Dw
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Work-Kinetic Energy Theorem
1/2mvf2 1/2mvi2 Kf - Ki DK
Work done by net force change in KE
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Gravitation and work
Work done by me (take down as ve) F.(-h)
-mg(-h) mgh
Work done by gravity mg.(-h) -mgh
________ Total work by ALL forces
(?W)
0
DK
Lift mass m with constant velocity
Work done by ALL forces change in KE DW DK
What happens if I let go?
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Compressing a spring
Compress a spring by an amount x
Work done by me ?Fdx ?kxdx 1/2kx2
Work done by spring ?-kxdx -1/2kx2
0
Total work done (DW)
DK
What happens if I let go?
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Moving a block against friction at constant
velocity
Work done by me F.d Work done by
friction -f.d -F.d Total work done
0
What happens if I let go?
NOTHING!!
Gravity and spring forces are Conservative Frictio
n is NOT!!
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Conservative Forces
A force is conservative if the work it does on a
particle that moves through a round trip is zero
otherwise the force is non-conservative
A force is conservative if the work done by it on
a particle that moves between two points is the
same for all paths connecting these points
otherwise the force is non-conservative.
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Conservative Forces
A force is conservative if the work it does on a
particle that moves through a round trip is zero
otherwise the force is non-conservative
Consider throwing a mass up a height h
work done by gravity for round trip
On way up work done by gravity -mgh On
way down work done by gravity mgh Total work
done 0
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Conservative Forces
A force is conservative if the work done by it on
a particle that moves between two points is the
same for all paths connecting these points
otherwise the force is non-conservative.
Work done by gravity w -mgDh1 -mgDh2-mgDh3
Each step heightDh
-mg(Dh1Dh2Dh3 ) -mgh Same as
direct path (-mgh)
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Conservation of Energy
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Potential Energy
The change in potential energy is equal to minus
the work done BY the conservative force ON the
body.
DU -Dw
Work done by gravity mg.(-h) -mgh
Therefore change in PE is DU -Dw
?Ugrav mgh
Lift mass m with constant velocity
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Potential Energy
The change in potential energy is equal to minus
the work done BY the conservative force ON the
body.
Compress a spring by an amount x
Work done by spring is Dw ?-kx dx - ½ kx2
Therefore the change in PE is DU - Dw
?Uspring ½ kx2
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Potential Energy
The change in potential energy is equal to minus
the work done BY the conservative force ON the
body.
DU -Dw but recall that Dw DK so that DU
-DK or
DU DK 0
Any increase in PE results from a decrease in KE
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DU DK 0
Lets check this for a body of mass m moving
under gravity.
Dw DK Kf - Ki DK ½ mvf2 ½ mvi2
vf
DK ½ mvi2 -mgh ½ mvi2 DK -mgh
Dw -DU
vi
For motion under gravity you know v2
u2 2as ? vf2
vi2 - 2gh mult by ½ m ? ½ m vf2 ½ mvi2 -mgh
so DU DK 0
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DU DK 0
In a system of conservative forces, any change in
Potential energy is compensated for by an inverse
change in Kinetic energy
U K E
In a system of conservative forces, the
mechanical energy remains constant
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Potential-energy diagrams
Dw - DU
The force is the negative gradient of the PE
curve
If we know how the PE varies with position, we
can find the conservative force as a function of
position
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PE of a spring
here U ½ kx2
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Potential energy U ½ kx2
At any position x PE KE E U K E K E -
U ½ kA2 ½ kx2 ½ k(A2 -x2)
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Roller Coaster
Fnet mg R R mg - Fnet
Fnet-dU/dt
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Fnet mg R R mg - Fnet
Fnet-dU/dx
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Conservation of Energy
We said when conservative forces act on a
body DU DK 0 ?U K E (const)
This would mean that a pendulum would swing for
ever. In the real world this does not happen.
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Conservation of Energy
When non-conservative forces are involved, energy
can appear in forms other than PE and KE (e.g.
heat from friction)
DU DK DUint 0 ?Ki Ui Kf Uf Uint
Energy may be transformed from one kind to
another in an isolated system, but it cannot be
created or destroyed. The total energy of the
system always remains constant.
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Stone thrown into air, with air resistance. How
high does it go?
Ei Ef Eloss
Ki Ui Kf Uf Eloss
½mvo2 0 0 mgh fh
½mvo2 h(mg f)
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Stone thrown into air, with air resistance. What
is the final velocity ?
Ei Ef Eloss
Ki Ui Kf Uf Eloss
0 mgh ½mvf2 0 fh
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System of Particles
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That's all folks
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Centre of Mass (1D)
M m1 m2
M xcm m1 x1 m2 x2
In general