Download lecture 18 mechanical energy

LECTURE 18
MECHANICAL ENERGY
Instructor: Kazumi Tolich
Lecture 18
2
¨ 
Reading chapter 8-3
¤  Mechanical
energy
¤  Conservation of mechanical energy
Mechanical energy
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¨ 
The mechanical energy of a system is the sum of
kinetic energy and potential energy of the system.
Emech ≡ K sys + U sys
Conservation of mechanical energy
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¨ 
If there is no external work done on the system, and
there is no work done by non-conservative force,
mechanical energy is conserved.
From work energy theorem, Wtotal = ΔK
If only internal conservaive forces are involved, Wtotal = Wc
From the definition of potential energy, Wc = −ΔU
ΔK = −ΔU
ΔK + ΔU = 0
K i + Ui = K f + U f
Conservation of mechanical energy, graphically
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Emech
Emech
Demo 1
6
¨ 
x2 Energy Dependence of a Spring
¤  Demonstration
of elastic potential energy converted
into kinetic energy and gravitational potential energy.
¤  Conservation of mechanical energy
U spring
1 2
= kx
2
1 2
→ U gravity + K = mgh + mv
2
Demo 2
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Energy Well Track
¨  Triple Track
¨ 
¤  Demonstration
of conservation of mechanical energy
Pendulum & conservation of mechanical energy
8
¨ 
Swinging pendulum (neglecting air resistance and
friction at the pivot)
¤  The
bob-Earth system has the maximum gravitational
potential energy, and has zero kinetic energy at the
highest points.
¤  It has the minimum gravitational potential energy, and
the maximum kinetic energy at the lowest point.
Problems solving
9
¨ 
Remember that you can set gravitational potential
energy to be zero anywhere you want!
Clicker question: 1
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Example: 1
11
¨ 
A block of mass m = 1.6 kg slides
with an initial speed of
v0 = 0.950 m/s on a frictionless,
horizontal surface until it encounters
a spring with a force constant of
k = 902 N/m. The block comes to
rest after compressing the spring by
x = 4.00 cm. Find the spring
potential energy U, kinetic energy of
the block K, and the total mechanical
energy of the block-spring system E
for compressions of 0.00 cm,
1.00 cm, and 4.00 cm.
Example: 2
12
¨ 
At an amusement park, a
swimmer uses a water slide
to enter the main pool. If the
swimmer starts at rest, slides
without friction, and
descends through a vertical
height of h = 2.31 m, what
is her speed at the bottom
of the slide?
Example: 3
13
¨ 
Suppose the pendulum bob in the
figure has a mass of m = 0.33 kg
and is moving to the right at point B
with a speed of vB = 2.4 m/s. Air
resistance is negligible.
a) 
What is the change in gravitational
potential energy of the bob-Earth
system when the bob moves from
point B to point A?
b) 
What is the speed of the bob at
point A?
Demo: 3
14
¨ 
Galileo's Pendulum
¤  Demonstration
of mechanical energy
Clicker question: 2
15
Example: 4
16
¨ 
The two masses (m1 and m2,
m1 < m2) in the Atwood’s
machine are initially at the same
height and moving with a speed
of v0 with m2 moving upward.
How high does m2 rise above its
initial position before
momentarily coming to rest?
Evaluate the answer for
m1 = 3.7 kg, m2 = 4.1 kg, and
v0 = 0.20 m/s.
Example: 5
17
¨ 
The length of the string, L, is
120 cm. The distance to the fixed
peg, R, is 45 cm. When the ball is
released from rest in the position
shown, it will follow the arc shown in
the figure. How fast will it be going
a) 
when it reaches the lowest point in
its swinging and
b) 
when it reaches the highest point
after the string catches on the peg?