#### Transcript Work, Energy and Power

```Work, Energy and
Power
Ms Houts
AP Physics C
Chapters 7 & 8
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Definition of Work

The work done by a constant force F applied
to an object that moves through a
displacement Dx is defined as
W  F cos  Dx  Fx Dx

Work is a scalar quantity that can be positive
or negative.
◦ Forces applied in the direction of motion is positive.
◦ Forces applied opposite the direction of motion
fk
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Definition of Kinetic Energy

An object moving with velocity v and
mass m has a kinetic energy given by:
KE  mv
1
2
2
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Work-Kinetic Energy Theorem

The total work done on a particle is equal
to the change in its kinetic energy.
Wtotal  DKE  mv  mv
1
2
2
f
1
2
2
i
Work-Kinetic Energy Example 1

A 6-kg box is raised from rest a distance
of 3 meter by a vertical force of 80 N.
Find the work done by the force, the work
done by gravity, and the final speed of the
box.
Graphical Interpretation of Work
Work is the area under the Force-position graph.
Work with a variable force
Work with a variable force


The work done by a force F that varies
with position x is given by
W 
x2
 F dx
x
x1

This integral equals the area under the Fversus-x curve.
Work-Kinetic Energy Example 2

A spring that obeys Hooke’s Law rests on a
frictionless surface.
◦ Hooke’s Law gives the force of a spring as F=-kx

Find the work done by the spring force when
it is stretched from x = 0 to x = xf.
Practice Exercises with Work

Further practice- pages 160-162
Work in three dimensions



Only the component of the force in the
direction of the displacement does work.
W  F cos  Dx  Fx Dx
If a force and displacement have
components
F  Fx ˆi  Fy ˆj  Fz kˆ and Ds  Δxˆi  Dyˆj  Dzkˆ
then the work done by the force is given by
W  Fx Dx  Fy Dy  Fz Dz
Dot Product

The dot product is the product of two
vectors A and B, where we consider only
the part of A that lies in the direction of B,
or the part of B that lies in the direction of
A.
A  B  AB cos  where  is the angle between A and B.

A  B  Ax Bx  Ay B y  Az Bz

Work as the Dot Product

The work done by a constant force F over
a displacement ∆s is
W  F  Ds

If the force varies with position, then the
work is given by
s2
W   F  ds
s1
Properties of Scalar Products
Power

Power is the rate at which work is done.
DW
dW
P
or in derivative form P 
Dt
dt
 Since W  Fx Dx
DW
Dx
P
 Fx
 Fv
Dt
Dt

or in derivative form
dx
P  Fx
dt
Power in 3 dimensions
A force F acting on an object moving with velo city v supplies
a power of
P  Fv

Practice problems, p. 162
Conservative Forces

A force is
conservative if the
total work it does
on a particle is
zero when the
particle moves
along any closed
path returning to
its initial position.
Potential Energy Functions

Doing work against a conservative force
stores energy. When the conservative
force does work on a particle, that energy
is released.
Let U be the potential energy associated with a conservati ve force.
 Let W be the work done by that conservati ve force.
W  DU
x2
DU  W    Fx dx if the force varies with position.
s2
x1
DU    F  ds
s1
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Gravitation Potential Energy

The work done by the gravitational force
in lifting up a weight a distance y is
negative, since the force is opposite the
displacement.
Fg   mgˆj
Ds  yj
W   mgy

The change in gravitational potential
energy is positive.
DU  W  (mgy)  mgy
Spring Potential Energy
s2
x1
s1
0
DU    F  ds    Fx dx
x1
DU    (kx)dx
0
Using the Potential Energy to Find
the Force

Since the potential energy is the negative
of the integral of the force function:
x2
DU  U f  U i    Fx dx
x1


Then the negative of the derivative of the
potential energy function is the force
function.
dU
Fx  
dx
Equilbrium

A particle is in equilibrium if the net force
acting on it is zero.

Since the force is the derivative of the
potential energy function, the equilibrium
points can be found graphically from a
potential energy graph by finding places
where the slope of the graph is zero.
In stable equilibrium, a small displacement results in a restoring force
That accelerates the particle back toward its equilibrium position.