Work: Varying forces and curved trajectories

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Lecture 14

• Work Varying forces and curved
  trajectories
• Power

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ACT Sign of work
A block slides down a rough inclined surface. The
forces acting on the block are depicted below.
The work done by the frictional force is
A. Positive B. Negative C. Zero
W fk ?x cos(180) fk ?x lt 0
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Work done by the normal force
N
WN N ?x cos(90) 0
90
?x
?x
N
fk
mg
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Work done by weight
Wmg mg ?x cos(? ) gt 0
?x
?x
?
N
fk
mg
?
mg
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ACT Zero net work
The system of pulleys shown below is used to lift
a bag of mass M at constant speed a distance h
from the floor. What is the work done by the
person?
A. Mgh B. ½Mgh C. 2Mgh
The force by the person is Mg/2 (see lecture 10),
but he needs to pull on a length of rope of 2h.
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The story so far
Kinetic energy
Work by a constant force, along a straight path
Work/Kinetic energy theorem
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Work by non-constant force, with straight line
trajectory
An object moves along the x-axis from point x1 to
x2. A non-constant force is applied on the
object. What is the work done by this force?
x1
x2
The journey is divided up into a series of
segments over which the force is constant.
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Workarea
Fx
FCx
FBx
FDx
FAx
FEx
x
?xC
?xA
?xB
?xD
?xE
This is still an approximation, because the force
is not really constant in each interval
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The calculation is really good in the limit where
the intervals are very small, the sum becomes an
integral
Work by non-constant force, with straight line
trajectory
Fx
WorkArea under F(x) curve
x
x1
x2
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Springs
Hookes law The force exerted by a spring is
proportional to the distance the spring is
compressed/stretched from the relaxed position.
Fx -k ?x k spring constant
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Hookes law is the basis of traditional scales.
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Stretching a Spring
What is the work done by the spring on the block
as the tip is pulled from x1 to x2?
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What is the work done by a spring as the tip is
pulled from x1 to x2?
What is the work done on the spring as the tip is
pulled from x1 to x2?
If x2 gt x1 (stretch), Wexternal gt 0
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Example Box and spring
A box of mass m 25 kg slides on a horizontal
frictionless surface with an initial speed v0
10 m/s. How far will it compress the spring
before coming to rest if k 3000 N/m?
m 25 kg
k 3000 N/m
v 10 m/s
x

• 0.50 m B. 0.63 m C. 0.75 m
• D. 0.82 m E. 0.91 m

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Use the work-kinetic energy theorem
W ?KE
Answer E
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Work on curved trajectories
A particle moves from A to B along this path
while a varying force acts upon it.

B x
Again, we can approximate the work by considering
breaking it into small displacements .
x A
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The line integral

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Example Spherical bowl
Find the work done by gravity on a pebble of mass
m as it rolls from the rim to the bottom of a
spherical bowl of radius R.
rim
R d?
bottom
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ACT Two elevators

• Two elevators A and B carry each a load of mass
  m from the first floor to the third floor of a
  building at constant speeds, but A is twice as
  fast as B. The work done by the cable tension
  (ie, the energy produced by the engine) is

A. Larger for A B. Larger for B C. The same for
both
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W net ?KE 0
T
WT Wg 0
?x
WT - Wg - (-mg?x)
mg
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Power
So, whats the difference between the two
elevators?
The machinery in elevator A has more power it is
doing the same work in less time (or more work
per unit time).
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Units of power
SI Watt 1 W 1 J/s Other Horsepower 1
hp 746 W
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Example Bullet and wood block

• A bullet of mass 103 kg traveling at a speed of
  300 m/s strikes a block of wood. It embeds a
  distance of 1 cm. How much force does the wood
  exert on the bullet as it slows it down?

• Identify
• Let us assume that the force is constant (or that
  we will find the average force)
• The force must do enough negative work on the
  bullet to render its kinetic energy 0.

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A bullet of mass 103 kg traveling at a speed of
300 m/s strikes a block of wood. It embeds a
distance of 1 cm. How much force does the wood
exert on the bullet as it slows it down?
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EXAMPLE Hammer
A hammer slides along 10 m down a 30? inclined
roof and off into the yard, which is 7 m below
the roof edge. Right before it hits the ground,
its speed is 14.5 m/s. What is the coefficient of
kinetic friction between the hammer and the roof?
This can be solved using Newtons laws and
kinematics, but its looooooooooooooooooooooooong.
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A hammer slides along 10 m down a 30? inclined
roof and off into the yard, which is 7 m below
the roof edge. Right before it hits the ground,
its speed is 14.5 m/s. What is the coefficient of
kinetic friction between the hammer and the roof?
?x 10 m
?
h 7 m
v 14.5 m/s
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A hammer slides along 10 m down a 30? inclined
roof and off into the yard, which is 7 m below
the roof edge. Right before it hits the ground,
its speed is 14.5 m/s. What is the coefficient of
kinetic friction between the hammer and the roof?