Goals:

1
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 Hookes Law spring potential energies
• Use energy diagrams

• Assignment
• HW6 due Tuesday Oct. 25th
• For Monday Read Ch. 11

2
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

3
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 )

4
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
5
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)

6
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.

7
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
8
Inelastic collision in 1-D Example 1

• What is the momentum of the bullet with speed v
  ?
• What is the initial energy of the system ?
• What is the final energy of the system ?
• Is momentum conserved (yes)?
• Is energy conserved? Examine Ebefore-Eafter

v
No!
V
x
before
after
9
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.

10
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 is called mechanical energy
K and U may change, K U remains a fixed value.
11
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 ?

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.

y
yh1
y0
13
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
yh1
h1
y0
v
14
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
yh1h2
y0
15
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
• The ball is dropped
• The ball slides down a straight incline
• The ball slides down a curved incline
• After traveling a vertical distance h, how do the
  three speeds compare?

(A) 1 gt 2 gt 3 (B) 3 gt 2 gt 1 (C) 3 2 1
(D) Cant tell
16
ExampleThe 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)

(A) 2R (B) 3R (C) 5/2 R (D) 23/2 R
y0
U0
17
ExampleThe Loop-the-Loop again

• Use E K U constant
• mgh 0 mg 2R ½ mv2
• mgh mg 2R ½ mgR 5/2 mgR
• h 5/2 R

h ?
R
18
Example Fully Elastic Collision

• Suppose I have 2 identical bumper cars.
• One is motionless and the other is approaching it
  with velocity v1. If they collide elastically,
  what is the final velocity of each car ?
• Identical means m1 m2 m
• Initially vGreen v1 and vRed 0

• COM ? mv1 0 mv1f mv2f ? v1 v1f
  v2f
• COE ? ½ mv12 ½ mv1f2 ½ mv2f2 ? v12 v1f2
  v2f2
• v12 (v1f v2f)2 v1f2 2v1fv2f v2f2 ? 2
  v1f v2f 0
• Soln 1 v1f 0 and v2f v1 Soln 2 v2f
  0 and v1f v1

19
Variable force devices Hookes Law Springs

• Springs are everywhere,
• The magnitude of the force increases as the
  spring is further compressed (a displacement).
• Hookes Law,
• Fs - k Ds
• Ds is the amount the spring is stretched or
  compressed from it resting position.

Rest or equilibrium position
F
Ds
20
Hookes Law Spring

• For a spring we know that Fx -k s.

21
Exercise Hookes Law
(A) 50 N/m (B) 100 N/m (C) 400 N/m (D) 500 N/m
22
F vs. Dx relation for a foot arch
Force (N)
Displacement (mm)
23
Force vs. Energy for a Hookes 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 ?

24
Energy for a Hookes Law spring

• Associate ½ ku2 with the potential energy of
  the spring

• Ideal Hookes Law springs are conservative so the
  mechanical energy is constant

25
Energy diagrams

• In general

Ball falling
Spring/Mass system
Emech
K
Energy
U
0
0
u x - xeq
26
Equilibrium

• Example
• Spring Fx 0 gt dU / dx 0 for xxeq
• The spring is in equilibrium position
• In general dU / dx 0 ? for ANY function
  establishes equilibrium

stable equilibrium
unstable equilibrium
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)
• 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.

28
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.