Chapter 7: Potential energy and energy conservation
•  Two types of Potential energy
–  gravitational and elastic potential energy
•  Conservation of total mechanical energy
–  When
–  What: Kinetic energy+Potential energy
•  Sometimes total mechanical energy is not conserved
•  Conservative forces VS nonconservative forces
•  The law of energy conservation
•  Force can be determined if the potential energy is known
Gravitational potential energy
•  When one drops a basketball, it
hits the ground with certain
kinetic energy. Where does it
come from?
W grav = Fs = w(y1 − y 2 ) = mgy1 − mgy 2
U grav = mgy
€
–  Gravitational force did the
work!
–  Gravitational potential
energy is transformed into
kinetic energy
Conservation of energy
•  (Mechanical) Energy is conserved, and we can prove it!
–  Gravitational potential energy
kinetic energy
W tot = ΔK (where did this come from?)
W tot = W grav = U1 − U 2 = −ΔU
⇒ ΔK = −ΔU
or
K 2 − K1 = U1 − U 2
⇒
E ≡ K1 + U1 = K 2 + U 2
•  If only gravity does work, sum of kinetic energy and
gravitational potential energy remain constant!
–  Does zero height matter? It’s always the difference of height!
Application of energy conservation.
•  We can determine how high a baseball can reach:
•  Does the mass matter?
•  Does free fall motion formula gives the same result?
1
E1 = mv12
2
E 2 = mgy 2
E1 = E 2 ⇒
1 2
mv1 = mgy 2
2
v12
⇒ y2 =
2g
€
Other forces doing work will change the total energy
1 2
1
mv1 + mgy1 + W other = mv 22 + mgy 2
2
2
€
When throwing a baseball,
we apply the “other” force
before releasing it.
What is v3 and the force we
applied on it?
F(y 2 − y1 ) = E 2 − E1
1
E 2 − E1 = mv 22 − mgy1
2
F =?
E2 = E3
1 2
1
mv 2 + mgy 2 = mv 32 + mgy 3
2
2
€
Can v3 be negative?
Q7.1
A piece of fruit falls straight down. As it falls,
A. the gravitational force does positive work on it and the
gravitational potential energy increases.
B. the gravitational force does positive work on it and the
gravitational potential energy decreases. C. the gravitational force does negative work on it and the
gravitational potential energy increases.
D. the gravitational force does negative work on it and the
gravitational potential energy decreases.
Q7.2
You toss a 0.150-kg baseball
straight upward so that it leaves
your hand moving at 20.0 m/s. The
ball reaches a maximum height y2.
What is the speed of the ball when
it is at a height of y2/2? Ignore air
resistance.
v2 = 0
y2
v1 = 20.0 m/s
m = 0.150 kg
A. 10.0 m/s
B. less than 10.0 m/s but more than zero C. more than 10.0 m/s
D. not enough information given to decide
y1 = 0
Work and energy along a curved path
•  For gravitational
energy, only the
height matters, not
the path!
•  For a curved path,
Ugrav is solely
dependent on the
height!
Q7.3
As a rock slides from A to B along
the inside of this frictionless
hemispherical bowl, mechanical
energy is conserved. Why?
(Ignore air resistance.) A. The bowl is hemispherical.
B. The normal force is balanced by centrifugal force.
C. The normal force is balanced by centripetal force.
D. The normal force acts perpendicular to the bowl’s surface.
E. The rock’s acceleration is perpendicular to the bowl’s surface.
Q7.4
The two ramps shown are both frictionless. The heights y1 and y2
are the same for each ramp. A block of mass m is released from rest
at the left-hand end of each ramp. Which block arrives at the righthand end with the greater speed?
A. the block on the curved track
B. the block on the straight track C. Both blocks arrive at the right-hand end with the same speed.
D. The answer depends on the shape of the curved track.
Consider projectile motion using energetics
•  Knowing that the
initial speeds are the
same (v0), what is the
speed at height=h?
Does it matter at what
angle it is released?
E1 = E 2
1 2
1 2
⇒ mv1 = mgh + mv 2
2
2
The direction of v1 or v2 has no effect of v2 magnitude!
€
What’s the speed in a vertical circle?
K1 = ?
U1 = ?
K2 = ?
U2 = ?
€
A particle moving in a full vertical frictionless circle.
It is released from the bottom at the speed of v 0 = 68gR
What is the speed on the top?
€
Real work always has friction, though
•  If the skater
reaches the
bottom at
6.00m/s instead
of 7.67m/s, due
to friction? How
much work did
friction do?
•  Where did those
missing energy
go?
One could also use Newton’s second law.
But it is difficult to apply.
Do you want the easy way?
The energy approach is the SHORTCUT
you should take!
The energy of a crate on an inclined plane
•  Not thinking about
friction, man pushes
a 12kg crate at
5.0m/s, only to see
that it slid 1.6m, and
slid back down.
•  What is the
magnitude of the
friction?
•  How fast will the
crate be at the
bottom?
Is E1-E2=E3-E2 ?
Work and energy in the motion of a mass on a spring
•  Work done on elastic (spring force) is stored (in the spring), as
elastic potential energy. U = 1 kx
2
el
2
1
1
W el = −W = kx12 − kx 2 2 = −ΔU el
2
2
•  If only elastic force does work:
€
€
1
1
1
1
mv12 + kx12 = mv 2 2 + kx 2 2
2
2
2
2
Potential energy curve
•  It’s symmetric! And parabolic!
•  Both stretching and compressing the spring will increase the
elastic potential energy!
•  When do we have minimum elastic potential energy?
1 2
U el = kx
2
€
“Bungee jumping: two potential energies in play!
•  Gravitational
energy kinetic
energy+elastic
potential
energyelastic
potential energy
…
•  Will this go on
forever?
Motion with elastic potential energy
•  What is v2x?
•  If we apply F=+0.610N, and move it to x1=0.1m, what is v1x, and
what is v2x when it comes back to x2, and x0=0?
k=?
Q7.5
A block is released from rest on a
frictionless incline as shown. When the
moving block is in contact with the spring
and compressing it, what is happening to
the gravitational potential energy Ugrav
and the elastic potential energy Uel?
A. Ugrav and Uel are both increasing.
B. Ugrav and Uel are both decreasing.
C. Ugrav is increasing, Uel is decreasing.
D. Ugrav is decreasing, Uel is increasing.
E. The answer depends on how the block’s speed is changing.
Worst case scenario: two potential energies and friction
•  A) In order to stop a falling
elevator using friction and a
cushioning spring, how
strong should the spring be
(k=?)
W f = E 2 − E1 = − f (−y 2 )
1
E 2 = K 2 + U 2 = 0 + ( ky 22 + mgy 2 )
2
1
E1 = K1 = mv12 ⇒
2
2
mv + 2(mg − f )(−y 2 )
k= 1
y 22
€
y1=0
y2= -2.0m
•  B) What if there is no
friction available?
•  Friction must be larger than
k(-y2)-mg? Otherwise what
would happen?
Conservative forces
•  The work by a conservative force like gravity does
not depend on the path your hiking team chooses,
only how high you climb.
Force as the derivative of potential energy
 ∂
∂ ˆ ∂ ˆ
∇ = iˆ +
j+ k
∂x
∂y
∂z
and
∂U(x, y,z)
Fx = −
∂x
∂U(x, y,z)
Fy = −
∂y
∂U(x, y,z)
Fz = −
∂z
W = −ΔU
W = FΔx ⇒
FΔx = −ΔU ⇒
F =−
ΔU
dU(x)
⇒ F(x) = −
Δx
dx
and


F (x, y,z) = −∇U(x, y,z)
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€
Energy diagrams give us insight
•  Figure 7.23 shows the
situation and the
resulting energy
diagram. Information
may be discerned
about limits and zeros
for the physical
properties involved.
The potential energy curve for motion of a particle
•
Refer the potential energy function and its corresponding components of force.
Q7.8
The graph shows the potential
energy U for a particle that moves
along the x-axis. At which of the
labeled x-coordinates is there zero
force on the particle?
A. at x = a and x = c B. at x = b only
C. at x = d only
D. at x = b and d
E. misleading question — there is a force at all values of x.
Q7.10
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx = 0 at x = a. Which statement
about the associated potential energy function
U at x = a is correct?
Fx
0
A. U = 0 at x = a
B. U is a maximum at x = a.
C. U is a minimum at x = a.
D. U is neither a minimum or a maximum at x = a, and
its value at x = a need not be zero.
x
a
Q7.11
The graph shows a conservative force Fx as a
function of x in the vicinity of x = a. As the
graph shows, Fx > 0 and dFx/dx < 0 at x = a.
Which statement about the associated potential
energy function U at x = a is correct?
A. dU/dx > 0 at x = a
B. dU/dx < 0 at x = a
C. dU/dx = 0 at x = a
D. Any of the above could be correct.
Fx
0
x
a
Summary of CH7: Potential energy
Two kinds of potential energies:
Gravitational: U gr = mgh
Elastic:
U el =
Corresponding forces:
Fx (x) = −
1
kx 2
2 €
dU
dx
w = −mg
F = −kx
Mechanical Energy conservation: K1 + U1 = K 2 + U 2
With other forces than gr and el: K1 + U1 + W other = K 2 + U 2
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(Friction Internal Energy)
Law of energy conservation:
K1 + U1 − ΔU int = K 2 + U 2
(ΔK + ΔU + ΔU int = 0)
From potential energy to force: (required)
(not-required)
Fx (x) = −
dU
dx


∂U
∂U ˆ ∂ U ˆ
F = −∇U = −( iˆ +
j+
k)
∂x
∂x
∂x
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