Lecture 9
Potential Energy and Conservation
of Energy
(HR&W, Chapter 8)
http://web.njit.edu/~sirenko/
Physics 105 Summer 2006
Lecture 9 Andrei Sirenko, NJIT 1 Lecture 9 Andrei Sirenko, NJIT 2
Power
Average Power
Sample Problem 7- 7-10:
10: Two constant forces F1 and
Units: Watts F2 acting on a box as the box slides rightward across
a frictionless floor. Force F1 is horizontal, with
magnitude 2.0 N, N, force F2 is angled upward by 60º
60º
Instantaneous Power to the floor and has a magnitude of 4.0 N.N. The speed
v of the box at a certain instant is 3.0 m/s.
m/s.
a) What is the power due to each force acting
on the box? Is the net power changing at that
instant?
b) If the magnitude F2 is, instead, 6.0 N,
N, what is
P= ½*60kg*(5m/s)2 now the net power, and is it changing?
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Work and Potential Energy
• Potential Energy and Conservation of
Energy Potential Energy
• Conservative Forces
General Form:
• Gravitational and Elastic Potential Energy
• Conservation of (Mechanical) Energy
• Potential Energy Curve
• External and Internal Forces Gravitational Potential Energy
Elastic Potential Energy
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Energy and Work Conservation of Mechanical Energy
Kinetic energy Mechanical Energy
Conservation of
Units of Work and Energy: Joule
Mechanical Energy
Work done by a constant force
In an isolated system where
only conservative forces cause
energy changes, the kinetic
Work–kinetic energy theorem and potential energy can
change, but their sum, the
mechanical energy Emec of the
system, cannot change.
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Kinetic Energy: Kinetic Energy:
Potential Energy: Potential Energy:
• Gravitation: • Gravitation:
• Elastic (due to spring force): • Elastic (due to spring force):
1 K=0
1 UÆK UÅÆK
y
U=0
2
mg Conservation of Conservation of
2 Mechanical Energy Mechanical Energy
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Path Independence of Sample Problem
A circus beagle of mass
Conservative Forces m = 6.0 kg runs onto the left
end of a curved ramp with
Sample Problem 8-1: A 2.0 kg block slides speed v0 = 7.8 m/s at height
along a frictionless track from a to point b. The y0 = 8.5 m above the floor. It
block travels through a total distance of 2.0 m, then slides to the right and
and a net vertical distance of 0.8 m. How much
work is done on the block by the gravitational comes to a momentary stop
force? when it reaches a height
y = 11.1 m from the floor.
The ramp is not frictionless.
What is the increase ∆Eth in
the thermal energy of the
beagle and the ramp because
of the sliding?
Lecture 9 Andrei Sirenko, NJIT 11 Lecture 9 Andrei Sirenko, NJIT Answer: 30J 12
Examples for Energy Conservation Problems:
Kinetic Energy changes
+ Gravitational Potential Energy
+ Elastic Potential Energy
__________________________________
Total Mechanical Energy = Const.
Kf - Ki = W = mgy - |Wfriction|
Wfriction
Ef - Ei = Kf –(Ki + mgy) = -|Wfriction|=fk⋅d ⋅cos180 ° =
= - fk⋅d =-mg µ⋅ d ⋅ cos18°
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Example of the 3rd Common Exam
V1 =4.0 m/s Kf = 0
1 V2 =8.6 m/s
10 m
What minimum coefficient of Kf - Ki = Wfriction = -mgµkd
kinetic friction µk is required to 0- ½mv2 = -mgµkd
bring the crate to a stop over a
distance of ½mv2 = mgµkd; µk= v2/2gd =
10 m along the lower surface ? (8.6m/s)2/(2·10m/s2 ·10m) = 0.37
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Perpetual Motion and “Free Energy”
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Perpetuum Mobile
“Machine, which works itself forever” Example 1
English: Perpetual Motion Balance of Forces:
Balance of Torques:
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More Examples: More Examples:
Iron Disk
Water
Magnet
All parts of the cylinder that fall in the greater gravity (magnetic)
(magnetic) level must be pushet out, as well. All the work, that a part of Put an innertube on a wheel. Fill it two thirds with wather.
wather. Put an axle through it so it can spin. Now make another one like
like it. Now hold
the axels and push the wheel up against each other so that they can squeez each others wather to the outside. The results are that one
cylinder gets when it's moving toward the greater gravity (magnetic)
(magnetic) level is needed when it is pushed back out of it.. side of each wheel is lighter than its other side. That is why the
the wheel spins.
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More Examples: More Examples:
Iron Ball Iron Ball
Magnets
Pendulum
Magnet
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More Examples: More Examples:
Water
Iron Ball
Magnets buoyant force of
Archimedes'
Pendulum principle: "A body
immersed in liquid
experiences and
upward buoyant
force equal to the
weight of the
displaced liquid."
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Sample Problem Sample Problem
A 61 kg bungee-
bungee-cord jumper is on a 45 m bridge A 61 kg bungee-
bungee-cord jumper is on a 45 m bridge
above a river. The elastic bungee cord has a above a river. The elastic bungee cord has a
relaxed length of L = 25 m.
m. Assume that the cord relaxed length of L = 25 m.
m. Assume that the cord
obeys Hooke’s law, with a spring constant of 160 obeys Hooke’s law, with a spring constant of 160
N/m.
N/m. If the jumper stops before reaching the water, N/m.
N/m. If the jumper stops before reaching the water,
what is the height h of her feet above the water at what is the height h of her feet above the water at
her lowest point?
point? -10.5 m her lowest point?
point?
L + d + h = 45m L + d + h = 45m
(∆ K =0)
E = K + Ug + Ue = Const; (∆ (∆ K =0)
E = K + Ug + Ue = Const; (∆
∆Ug = – mgy = – 61 kg · 9.8m/s2 · * (L+d) ∆Ug = – mgh = – 61 kg · 9.8m/s2 · (L+d)
∆Ue = kd2/2 = 160 N/m · d2/2 ∆Ue = kd2/2 = 160 N/m · d2/2
80d2 – 600d – 15000 = 0; (d2 – 7.5d – 187.5 = 0) 80d2 – 600d – 15000 = 0; (d2 – 7.5d – 187.5 = 0)
d = 18 m and d = -10.5 m d = 18 m and d = -10.5 m
h = 45m – 25m – 18m = 2m h = 45m – 25m – 18m = 2m
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+H/2
Sample Problem Sample Problem
A 61 kg bungee-
bungee-cord jumper is on a 45 m bridge
above a river. The elastic bungee cord has a
relaxed length of L = 25 m.
m. Assume that the cord
An elevator cab of mass m = 500 kg is descending with
obeys Hooke’s law, with a spring constant of 160 speed vi = 4.0 m/s when its supporting cable begins to
N/m.
N/m. If the jumper stops before reaching the water, slip, allowing it to fall with constant acceleration a = g/5.
what is the height h of her feet above the water at
her lowest point?
point? During the 12 m fall, what is the work WT done by
L + d + h = 45m the upward pull T of the elevator cab?
(∆ K =0)
E = K + Ug + Ue = Const; (∆ T – mg = – ma = – mg/5
∆Ug = – mgh = – 61 kg · 9.8m/s2 · (L+d+H) T=0.8·
T=0.8·mg;
∆Ue = kd2/2 = 160 N/m · d2/2 W= – 0.8mgd = – 0.8 · 500kg · 9.8m/s2 · 12m =
80d2 – 600d – 15000 = 0; (d2 – 7.5d – 202.5 = 0) = – 47,000 J
– H/2
d = 18.5 m and d = – 11 m
h = 45m – 25m – 18.5m = 1.5m but H=2m …
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QZ#9:
Sample Problem 1. An elevator cab of mass m = 500 kg is descending with
speed vi = 4.0 m/s when its supporting cable begins to slip,
An elevator cab of mass m = 500 kg is descending with allowing it to fall with constant acceleration a = g/5,
/5, d=12 m
speed vi = 4.0 m/s when its supporting cable begins to What is the elevator’s kinetic energy at the end of the
slip, allowing it to fall with constant acceleration a = g/5. fall? Hint: Wmg = 59000 J ; WT = – 47000 J
During the fall through a distance d = 12 m, what is
the work Wg done on the cab by the gravitational
force Fg?
magnet iron
Wmg= +mgd
+mgd = 500kg · 9.8m/s2 · 12m = 59000J 2 We are not using this type of vehicle because
a) perpetual motion is forbidden by the
Newton's Laws
b) police does not allow it
c) sitting next to a strong magnet is not good for the driver’s health
d) there are no such strong magnets so far
e) this vehicle is not going to start moving by itself, so it is not very
practical for plane roads. Can be only used to go down the hill.
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