Work and Energy

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Work and Energy
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Work
?
?
Work by a constant force F during a displacement
s
Work F s Fscos(?)
Units N m joule (J)
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Example
2.5 m
Fp 120
w 100 N
5
3
fk 50 N
4
n
The block is dragged 2.5 m along the slope. Find

1. work done by Fp
2. work done by fk
3. work done by gravity
4. work done by normal force
5. Total work on the block

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1. Wp (120 N)(2.5 m) 300 J
2. Wf (- 50 N)(2.5 m) -125 J
3. Wg
4. Wn 0 (? motion)

s
mg
Total 300 (- 125) (- 150) 25 J
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Forces which are not constant
Example How much work is done to stretch a
spring scale from zero to the 20-N mark (a
distance of 10 cm)? We cant just multiply force
times distance because the force changes during
the motion. Our definition of work is not
complete.
Varying force split displacement into short
segments over which F is nearly constant.
F(x)
x
For each small displacement Dx, the work done is
approximately F(x) Dx, which is the area of the
rectangle.
F
D x
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Work is the area (A) under a graph of force vs.
distance
We get the total work by adding up the work done
in all the small steps. As we let Dx become
small, this becomes the area under the curve, and
the sum becomes an integral.
F(x)
F(x)
A
x
x
Split displacement into short steps Dx over which
F is nearly constant...
Take the limit as Dx? 0 and the number of steps ?
?
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In 1D (motion along the x-axis)
Another way to look at it Suppose W(x) is the
total work done in moving a particle to position
x. The extra work to move it an additional small
distance Dx is, approximately, DW ? F(x) Dx.
Rearrange to get
In the limit as Dx goes to zero,
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Determine the work done by a force as the
particle moves from x0 to x6m
Variable Force
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Example An Ideal Spring.
Hookes Law The tension in a spring is
proportional to the distance stretched.
or, F ks
The spring constant k has units of N/m
Directions The force exerted by the spring when
it is stretched in the x direction is opposite
the direction of the stretch (it is a restoring
force) F -kx
and E½kx2
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Example Work by a Spring
Fs
Fs - kx and Es½kx2
When x0, Energy0, when sprig is at xA,
thenE½kA2.
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Concept Quiz
A spring is hanging vertically. A student
attaches a 0.100-kg mass to the end, and releases
it from rest. The mass falls 50 cm, stretching
the spring, before stopping and bouncing
back. During the 50-cm descent, the total work
done on the mass was

1. zero
2. 0.49 J
3. -0.49 J
4. none of the above

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Concept Quiz
A physicist uses a spring cannon to shoot a ball
at a gorilla. The cannon is loaded by compressing
the spring 20 cm. The first 10 cm of compression
requires work W. The work required for the next
10 cm (to increase the compression from 10 cm to
20 cm) would be

1. W
2. 2W
3. 3W
4. 4W

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Kinetic Energy
Definition for a particle moving with speed v,
the kinetic energy is
K ½ mv2
(a SCALAR)
Then the Work-Energy Theorem says
The total work done by all external forces acting
on a particle is equal to the increase in its
kinetic energy.
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Example
A bartender slides a 1-kg glass 3 m along the
bar to a customer. The glass is moving at 4 m/s
when the bartender lets go, and at 2 m/s when the
customer catches it. Find the work done by
friction, and calculate the force of friction.