Potential Energy and Conservation of Mechanical Energy Conservative - - PDF document
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Potential Energy and Conservation of Mechanical Energy Conservative and Nonconservative Forces Potential Energy Conservation of Mechanical Energy Homework 1 First Definition of Conservative and Nonconservative Forces A force
SLIDE 1
Potential Energy and Conservation of Mechanical Energy
Conservative and Nonconservative Forces
Potential Energy
Conservation of Mechanical Energy
Homework
1
SLIDE 2
First Definition of Conservative and Nonconservative Forces
A force is conservative if the kinetic energy of a parti-
cle on which it acts returns to its initial value after any round trip. A force is nonconservative if the kinetic energy changes.
An example of a conservative force is the force ex-
erted by a spring
An example of a nonconservative force is friction
2
SLIDE 3
Second Definition of Conservative and Nonconservative Forces
A force is conservative if the work done by the force
n a particle that moves through any round trip is
zero. A force is nonconservative if the work done by
the force on a particle that moves through any round trip is not zero.
First and second definitions are equivalent from the
Work-Energy Theorem (W = ∆K)
3
SLIDE 4
Third Definition of Conservative and Nonconservative Forces
A force is conservative if the work done by it on a
particle that moves between two points depends only
n these points and not on the path followed. A force
is nonconservative if the work done by it on a particle that moves between two points depends on the path taken between these two points.
The work done against friction depends on the path
taken since it is a nonconservative force
4
SLIDE 5
Potential Energy
When conservative forces are acting (e.g. spring force,
gravity) ∆K + ∆U = 0 where U = potential energy (has meaning only for conservative forces)
This means that the sum of the kinetic and potential
energies is a constant U + K = a constant
From the Work-Energy Theorem
W = ∆K = −∆U
For one dimensional motion
∆U = −W = −
xf
xi F(x)dx
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SLIDE 6
Conservation of Mechanical Energy
Since the sum of the kinetic and potential energies is
a constant when only conservative forces are acting, we can write Ki + Ui = Kf + Uf
We can also calculate the force from the potential en-
ergy function F(x) = −dU(x) dx
Check
∆U = −
xf
xi F(x)dx = −
xf
xi
−dU(x)
dx
dx
∆U =
xf
xi dU(x)dx = U(xf) − U(xi) = Uf − Ui
6
SLIDE 7
Conservation of Mechanical Energy (Cont’d)
Consider a particle moving from A to B along the x-axis with a single conservative force acting on it ∆U = UB − UA UB = ∆U + UA = −
xB
xA F (x) dx + UA
We cannot assign a value to UB until we assign one to
UA. Usually we choose UA = 0.
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SLIDE 8
Gravitational Potential Energy
Fg = −mg Ug(y) = −
y
0 Fgdy + Ug(0) = −
y
0 (−mg) dy + Ug(0)
Ug(y) = mgy + Ug(0) Let Ug(0) = 0 ⇒ Ug(y) = mgy Note : Fg = −dUg(y) dy = −d (mgy) dy = −mg
What is the change in gravitational potential energy when a 720-kg elevator moves from street level to the top of the Empire State building, 380 m above street level?
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SLIDE 11
Example 1 Solution
What is the change in gravitational potential energy when a 720-kg elevator moves from street level to the top of the Empire State building, 380 m above street level? ∆U = Uf − Ui = mgyf − mgyi = mg(yf − yi) ∆U = (720 kg)
9.8 m/s2
(380 m) = 2.7 MJ
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SLIDE 12
Example 2
The spring in a spring gun has a force constant k = 700 N/m. It is compressed 3.0 cm from its natural length, and a 0.012-kg ball is put in the barrel against it. Assuming no friction and a horizontal gun barrel, with what speed will the ball leave the gun when released?
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SLIDE 13
Example 2 Solution
The spring in a spring gun has a force constant k = 700 N/m. It is compressed 3.0 cm from its natural length, and a 0.012-kg ball is put in the barrel against it. Assuming no friction and a horizontal gun barrel, with what speed will the ball leave the gun when released? Ki + Ui = Kf + Uf 0 + 1 2kx2 = 1 2mv2 + 0 v =
