Lecture 14
Goals:
• Chapter 10
Understand the relationship between motion and energy
Define Kinetic Energy
Define Potential Energy
Define Mechanical Energy
 Exploit Conservation of energy principle in problem solving
 Understand Hooke’s Law spring potential energies
 Use energy diagrams
Assignment:
 HW6 due Tuesday Oct. 25th
 For Monday: Read Ch. 11
Physics 207: Lecture 14, Pg 1
Kinetic & Potential energies

Kinetic energy, K = ½ mv2, is defined to be the
large scale collective motion of one or a set of
masses

Potential energy, U, is defined to be the “hidden”
energy in an object which, in principle, can be
converted back to kinetic energy

Mechanical energy, EMech, is defined to be the sum
of U and K

Others forms of energy can be constructed
Physics 207: Lecture 14, Pg 2
Recall if a constant force over time then

y(t) = yi + vyi t + ½ ay t2

v(t) = vyi + ay t
Eliminating t gives
 2 ay ( y- yi ) = vx2 - vyi2

m ay ( y- yi ) = ½ m ( vx2 - vyi2 )
Physics 207: Lecture 14, Pg 3
Energy (dropping a ball)
-mg (yfinal – yinit) = ½ m ( vy_final2 –vy_init2 )
A relationship between
y- displacement and change in the y-speed squared
Rearranging to give initial on the left and final on the right
½ m vyi2 + mgyi = ½ m vyf2 + mgyf
We now define mgy ≡ U as the “gravitational potential energy”
Physics 207: Lecture 14, Pg 4
Energy (throwing a ball)

Notice that if we only consider gravity as the external force then
the x and z velocities remain constant

To
½ m vyi2 + mgyi = ½ m vyf2 + mgyf

Add
½ m vxi2 + ½ m vzi2
and
½ m vxf2 + ½ m vzf2
½ m vi2 + mgyi = ½ m vf2 + mgyf

where
vi2 ≡ vxi2 +vyi2 + vzi2
½ m v2 ≡ K terms are defined to be kinetic energies
(A scalar quantity of motion)
Physics 207: Lecture 14, Pg 5
When is mechanical energy not conserved
 Mechanical
energy is not conserved when
there is a process which can be shown to
transfer energy out of a system and that
energy cannot be transferred back.
Physics 207: Lecture 14, Pg 6
Inelastic collision in 1-D: Example 1

A block of mass M is initially at rest on a frictionless
horizontal surface. A bullet of mass m is fired at the block
with a muzzle velocity (speed) v. The bullet lodges in the
block, and the block ends up with a speed V.
 What is the initial energy of the system ?
 What is the final energy of the system ?
 Is energy conserved?
x
v
V
before
after
Physics 207: Lecture 14, Pg 7
Inelastic collision in 1-D: Example 1

mv
What is the momentum of the bullet with speed v ?
 What is the initial energy of the system ?
1
 What is the final energy of the system ?
2
1
2
(m  M )V
2
mv
2
 Is momentum conserved (yes)?
mv  M 0  (m  M )V
 Is energy conserved? Examine Ebefore-Eafter
2
mv 
2
1
[( m  M )V]V 
2
1
2
v
mv 
2
1
( mv)
2
m
mM
v
1
mv
2
No!
before
2
(1 
m
mM
)
V
after
x
1
Physics 207: Lecture 14, Pg 8
Elastic vs. Inelastic Collisions


A collision is said to be inelastic when “mechanical” energy
( K +U ) is not conserved before and after the collision.
How, if no net Force then momentum will be conserved.
Kbefore + U  Kafter + U
 E.g. car crashes on ice: Collisions where objects stick
together

A collision is said to be perfectly elastic when both energy &
momentum are conserved before and after the collision.
Kbefore + U = Kafter + U
 Carts colliding with a perfect spring, billiard balls, etc.
Physics 207: Lecture 14, Pg 9
Energy

If only “conservative” forces are present, then the
mechanical energy of a system is conserved
For an object acted on by gravity
½ m vyi2 + mgyi = ½ m vyf2 + mgyf
Emech = K + U = constant
Emech is called “mechanical energy”
K and U may change, K + U remains a fixed value.
Physics 207: Lecture 14, Pg 10
Example of a conservative system:
The simple pendulum.

Suppose we release a mass m from rest a distance h1
above its lowest possible point.
 What is the maximum speed of the mass and where does
this happen ?
 To what height h2 does it rise on the other side ?
m
h1
h2
v
Physics 207: Lecture 14, Pg 11
Example: The simple pendulum.
 What is the maximum speed of the mass and where does
this happen ?
E = K + U = constant and so K is maximum when U is a
minimum.
y
y=h1
y=
0
Physics 207: Lecture 14, Pg 12
Example: The simple pendulum.
 What is the maximum speed of the mass and where does
this happen ?
E = K + U = constant and so K is maximum when U is a
minimum
E = mgh1 at top
E = mgh1 = ½ mv2 at bottom of the swing
y
y=h1
y=0
h1
v
Physics 207: Lecture 14, Pg 13
Example: The simple pendulum.
To what height h2 does it rise on the other side?
E = K + U = constant and so when U is maximum
again (when K = 0) it will be at its highest point.
E = mgh1 = mgh2 or h1 = h2
y
y=h1=h2
y=0
Physics 207: Lecture 14, Pg 14
Potential Energy, Energy Transfer and Path
A ball of mass m, initially at rest, is released and follows
three difference paths. All surfaces are frictionless
1. The ball is dropped
2. The ball slides down a straight incline
3. The ball slides down a curved incline
After traveling a vertical distance h, how do the three speeds
compare?

1
2
3
h
(A) 1 > 2 > 3
(B) 3 > 2 > 1
(C) 3 = 2 = 1 (D) Can’t tell
Physics 207: Lecture 14, Pg 15
Example
The Loop-the-Loop … again



To complete the loop the loop, how high do we have to let
the release the car?
Condition for completing the loop the loop: Circular motion
at the top of the loop (ac = v2 / R)
Exploit the fact that E = U + K = constant ! (frictionless)
Ub=mgh
U=mg2R
h?
y=0(A) 2R
U=0
Recall that “g” is the source of
Car has mass m the centripetal acceleration
and N just goes to zero is
the limiting case.
Also recall the minimum speed
at the top is
R
(B) 3R
(C) 5/2 R
(D)
23/2
R
v

gR
Physics 207: Lecture 14, Pg 16
Example
The Loop-the-Loop … again


Use E = K + U = constant
mgh + 0 = mg 2R + ½ mv2
mgh = mg 2R + ½ mgR = 5/2 mgR
v

gR
h = 5/2 R
h?
R
Physics 207: Lecture 14, Pg 17
Variable force devices: Hooke’s Law Springs

Springs are everywhere,
Rest or equilibrium position
F
Ds


The magnitude of the force increases as the spring is
further compressed (a displacement).
Hooke’s Law,
Fs = - k Ds
Ds is the amount the spring is stretched or compressed
from it resting position.
Physics 207: Lecture 14, Pg 19
Exercise Hooke’s Law
8m
9m
What is the spring constant “k” ?
50 kg
(A) 50 N/m
(B) 100 N/m (C) 400 N/m (D) 500 N/m
Physics 207: Lecture 14, Pg 21
Force (N)
F vs. Dx relation for a foot arch:
Displacement (mm)
Physics 207: Lecture 14, Pg 22
Force vs. Energy for a Hooke’s Law spring




F = - k (x – xequilibrium)
F = ma = m dv/dt
= m (dv/dx dx/dt)
= m dv/dx v
= mv dv/dx
So - k (x – xequilibrium) dx = mv dv
Let u = x – xeq. & du = dx 
m
uf
 ku du   mv dv
ui


1
2
1
2
vf
1
2
vi
ku |u  2 mv |v
2 uf
2 vf
1
i
i
ku f  2 kui  2 mv f  2 mvi
2
1
2
1
2
1
kui  2 mvi  2 ku f  2 mv f
2
1
2
1
2
1
2
Physics 207: Lecture 14, Pg 23
2
Energy for a Hooke’s Law spring
ku

mv

ku

mv
i
i
f
f
2
2
2
2
1

2
1
2
1
2
Associate ½ ku2 with the
“potential energy” of the spring
1
2
m
U si  K i  U sf  K f

Ideal Hooke’s Law springs are conservative so
the mechanical energy is constant
Physics 207: Lecture 14, Pg 24
Energy diagrams

In general:
Ball falling
Spring/Mass system
Emech
Emech
K
U
Energy
Energy
K
0
y
U
0
u = x - xeq
Physics 207: Lecture 14, Pg 25
Equilibrium

Example
 Spring: Fx = 0 => dU / dx = 0 for x=xeq
The spring is in equilibrium position

In general: dU / dx = 0  for ANY function establishes
equilibrium
U
stable equilibrium
U
unstable equilibrium
Physics 207: Lecture 14, Pg 26
Comment on Energy Conservation

We have seen that the total kinetic energy of a system
undergoing an inelastic collision is not conserved.
 Mechanical energy is lost:
 Heat (friction)
 Bending of metal and deformation

Kinetic energy is not conserved by these non-conservative
forces occurring during the collision !

Momentum along a specific direction is conserved when
there are no external forces acting in this direction.
 In general, easier to satisfy conservation of momentum
than energy conservation.
Physics 207: Lecture 14, Pg 27
Comment on Energy Conservation

We have seen that the total kinetic energy of a system
undergoing an inelastic collision is not conserved.
 Mechanical energy is lost:
 Heat (friction)
 Deformation (bending of metal)

Mechanical energy is not conserved when non-conservative
forces are present !

Momentum along a specific direction is conserved when there
are no external forces acting in this direction.
 Conservation of momentum is a more general result than
mechanical energy conservation.
Physics 207: Lecture 14, Pg 28