Download Work and power

Work and power
•  Tutorial homework due in recitation Thursday/Friday
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Clicker question 1
Set frequency to BA
A hockey puck sliding on an ice rink is at 1 m/s slides onto a rug
someone left on the ice. The puck comes to rest after moving 1 m
on the carpet. How far along the rug would the puck go, if its
initial speed was 2 m/s?
Apply the Work-Energy
A.  1.5 m
Theorem. If the puck
B.  2 m
slides twice as far, the
C.  3 m
friction does twice as
D.  4 m
much (negative) work.
E.  Impossible to determine
Doubling the speed quadruples the kinetic energy
(K = 12 mv2 )
! !
So the work W = F ⋅ Δr by friction to reduce the kinetic energy
(
)
to 0 (make it stop) must also increase by a factor of 4. Force is
constant so the distance must increase by a factor of 4.
Wnet = f ⋅ d = ΔK = 12 mv 2f − 12 mvi2
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Power
Work is a force applied over a distance.
Isn’t it strange that walking up 10 flights of stairs is the
same amount of work as running up 10 flights of stairs?
New concept: Power is the rate at which work is done.
Average power is Pavg =
ΔW
Δt
Instantaneous power is P =
dW
dt
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Power continued
Remember work is force times distance so for a constant
force F acting in the same direction as the displacement
FΔx
P
=
Δx the average power is avg Δt = Fvavg
If we allow the force to be in any direction
! ! and let t → 0
then the instantaneous power is P = F ⋅ v
So we have two ways to determine power:
ΔW
which for a
Δt
force along the velocity vector is Pavg = Fvavg
Average power: Pavg =
dW ! !
Instantaneous power: P = dt = F ⋅ v
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Power units
Since power is work divided by time, the unit is joules/second
(J/s) which is called a watt (W). So 1 W = 1 J/s.
Be careful not to confuse work with watt (both are W)
The other common power unit is
horsepower (hp). 1 hp = 746 W
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Clicker question 2
Set frequency to BA
A sports car accelerates from zero to 30 mph in 2 seconds.
How long does it take to accelerate from zero to 60 mph,
assuming that the power output of the engine is independent of
velocity and neglecting friction and air resistance? Consider
ΔW
how power is related to work Pavg =
and how work is related
Δt
to kinetic energy Wnet = ΔK.
A.  4 seconds
B.  6 seconds
Doubling the speed quadruples
2
1
C.  8 seconds
K
=
mv
the kinetic energy
2
D.  16 seconds
E.  None of the above
(
)
Therefore, 4 times more work is needed to accelerate from
0 to 60 mph than 0 to 30 mph. Since the power is constant
it will take 4 times longer to get 4 times more work
Since P = ΔW/Δt, we can write ΔW = PŸΔt or Δt = ΔW/P
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Work done by gravity
A block moves from
m
the top to the bottom
of an inclined plane
(which has friction).
h
φ
θ
m
h
θ
What is the work done by gravity? Displacement is d = h / sin θ
! !
Three ways to calculate work done by gravity: Wg = Fg ⋅ d
Force times displacement
times cosine of angle between Wg = mgd cos ϕ = mgd sin θ = mgh
Component of force in displacement
direction times displacement
Wg = mgsin θ ⋅ d = mgh
Force times displacement in the force direction
Wg = mg ⋅ h
Same result no matter the angle or the method.
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Gravity is a conservative force
The reason the work done by gravity is independent of the
path taken is because gravity is a conservative force.
Some properties of conservative forces
Work done by a conservative force is independent of path taken
Work done by a conservative force over a closed loop (starting
and ending at the same place) is 0
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Clicker question 3
Set frequency to BA
Work done by a conservative force is independent of path taken
Work done by a conservative force over a closed loop (starting
and ending at the same place) is 0
Which of the following is NOT a conservative force?
A.  Gravity (W = −mg Δy)
B.  Kinetic friction (W = −µknd)
C.  Spring W = − 1 k "# x22 − x12 $%
2
D.  All of the above are conservative forces
E.  More than one of the above are not conservative forces
Work done by gravity and spring force just depend on the starting
and ending position (not how you get there).
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Work done by friction depends on the path taken (how far you go).
Conservative and non-conservative forces
Conservative forces:
Gravity: Work done by gravity is determined by the force
of gravity and the change in height (W = −mg Δy). All
that matters is the starting and ending height.
Spring: Work done by spring is determined by the force of the
spring and the amount of displacement from the equilibrium
2$
1 " 2
W
=
−
k
x
−
x
point of the spring
1 % . All that matters is the
2 # 2
initial and final distance from the equilibrium position.
(
)
Non-conservative forces:
Friction: Work doesn’t depend on the starting and
ending point, but depends on the distance traveled.
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Potential energy
If we do work against gravity, by raising an object a height h,
we can get that work back when the object is lowered.
This is a consequence of being a conservative force.
One way to think of this is that when we do work against
gravity, this work is stored in some way for retrieval later.
This idea is called potential energy and is labeled U.
For any conservative force, there is an
associated potential energy, labeled U.
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Potential energy
The change in potential energy associated with a conservative
force is the negative of the work done by that force.
For gravity, the work done by gravity is W = −(mg)(Δy)
Therefore, the change in potential energy is ΔUg = (mg)(Δy)
Remember Δy is change in height (final – initial) so Δy = yf − yi
Therefore you gain potential energy when you move up (yf>yi)
and lose potential energy when you move down (yf<yi).
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Clicker question 4
Set frequency to BA
How does my potential energy change when I drive from western
Kansas (1000 m above sea level) to Silverthorne (3000 m above
sea level)? My mass is 100 kg and can use g = 10 m/s2.
A.  0 MJ
B.  1 MJ
C.  2 MJ
D.  3 MJ
E.  Depends on how far it is from western Kansas to Silverthorne.
ΔU = mgΔy = (100 kg)(10 m/s2)(3000 m − 1000 m)
= 2000000 J = 2 MJ
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