Conservation of Energy

1
Conservation of Energy

• Chapter 11

2
Conservation of Energy

• The Law of Conservation of Energy simply states
  that
• The energy of a system is constant.
• Energy cannot be created nor destroyed.
• Energy can only change form (e.g. electrical to
  mechanical to potential, etc).
• True for any system with no external forces.
• ET KE PE Q
• KE Kinetic Energy
• PE Potential Energy
• Q Internal Energy kinetic energy due to the
  motion of molecules (translational, rotational,
  vibrational)

3
Conservation of Energy
Energy
Mechanical
Nonmechanical
Potential
Kinetic
Gravitational
Elastic
4
Conservation of Mechanical Energy

• Mechanical Energy
• If Internal Energy is ignored
• ME KE ?PE
• PE could be a combination of gravitational and
  elastic potential energy, or any other form of
  potential energy.
• The equation implies that the mechanical energy
  of a system is always constant.
• If the Potential Energy is at a maximum, then the
  system will have no Kinetic Energy.
• If the Kinetic Energy is at a maximum, then the
  system will not have any Potential Energy.

5
Conservation of Mechanical Energy

• ME KE ?PE
• KEinitial PEinitial KEfinal PEfinal

6
Example 4

• A student with a mass of 55 kg goes down a
  frictionless slide that is 3 meters high. What
  is the students speed at the bottom of the
  slide?
• KEinitial PEinitial KEfinal PEfinal
• KEinitial 0 because v is 0 at top of slide.
• PEinitial mgh
• KEfinal ½ mv2
• PEfinal 0 at bottom of slide.
• Therefore
• PEinitial KEfinal
• mgh ½ mv2
• v 2gh
• V (2)(9.81 m/s2)(3 m) 7.67 m/s

7
Example 5

• A student with a mass of 55 kg goes goes down a
  non-frictionless slide that is 3 meters high.
• Compared to a frictionless slide the students
  speed will be
• the same.
• less than.
• more than.
• Why?
• Because energy is lost to the environment in the
  form of heat (Q) due to friction.

8
Example 5 (cont.)

• Does this example reflect conservation of
  mechanical energy?
• No, because of friction.
• Is the law of conservation of energy violated?
• No, some of the mechanical energy is lost to
  the environment in the form of heat.

9
Energy of Collisions

• While momentum is conserved in all collisions,
  mechanical energy may not.
• Elastic Collisions Collisions where the kinetic
  energy both before and after are the same.
• Inelastic Collisions Collisions where the
  kinetic energy after a collision is less than
  before.
• If energy is lost, where does it go?
• Thermal energy, sound.

10
Collisions

• Two types
• Elastic collisions objects may deform but after
  the collision end up unchanged
• Objects separate after the collision
• Example Billiard balls
• Kinetic energy is conserved (no loss to internal
  energy or heat)
• Inelastic collisions objects permanently deform
  and / or stick together after collision
• Kinetic energy is transformed into internal
  energy or heat
• Examples Spitballs, railroad cars, automobile
  accident

11
Example 4

• Cart A approaches cart B, which is initially at
  rest, with an initial velocity of 30 m/s. After
  the collision, cart A stops and cart B continues
  on with what velocity? Cart A has a mass of 50 kg
  while cart B has a mass of 100kg.

B
A
12
Diagram the Problem
B
A
Before Collision
pB1 mvB1 0
After Collision
pA2 mvA2 0
13
Solve the Problem

• pbefore pafter
• mAvA1 mBvB1 mAvA2 mBvB2
• mAvA1 mBvB2
• (50 kg)(30 m/s) (100 kg)(vB2)
• vB2 15 m/s
• Is kinetic energy conserved?

14
Example 5
Per 7

• Cart A approaches cart B, which is initially at
  rest, with an initial velocity of 30 m/s. After
  the collision, cart A and cart B continue on
  together with what velocity? Cart A has a mass of
  50 kg while cart B has a mass of 100kg.

B
A
15
Diagram the Problem
B
A
Before Collision
pB1 mvB1 0
After Collision
Note Since the carts stick together after the
collision, vA2 vB2 v2.
16
Solve the Problem

• pbefore pafter
• mAvA1 mBvB1 mAvA2 mBvB2
• mAvA1 (mA mB)v2
• (50 kg)(30 m/s) (50 kg 100 kg)(v2)
• v2 10 m/s
• Is kinetic energy conserved?

17
Key Ideas

• Gravitational Potential Energy is the energy that
  an object has due to its vertical position
  relative to the Earths surface.
• Elastic Potential Energy is the energy stored in
  a spring or other elastic material.
• Hookes Law The displacement of a spring from
  its unstretched position is proportional the
  force applied.
• Conservation of energy Energy can be converted
  from one form to another, but it is always
  conserved.

18
Simple Harmonic Motion Springs

• Simple Harmonic Motion
• An oscillation around an equilibrium position in
  which a restoring force is proportional the the
  displacement.
• For a spring, the restoring force F -kx.
• The spring is at equilibrium when it is at its
  relaxed length.
• Otherwise, when in tension or compression, a
  restoring force will exist.

19
Simple Harmonic Motion Springs

• At maximum displacement ( x)
• The Elastic Potential Energy will be at a maximum
• The force will be at a maximum.
• The acceleration will be at a maximum.
• At equilibrium (x 0)
• The Elastic Potential Energy will be zero
• Velocity will be at a maximum.
• Kinetic Energy will be at a maximum

20
Harmonic Motion The Pendulum

• Pendulum Consists of a massive object called a
  bob suspended by a string.
• Like a spring, pendulums go through simple
  harmonic motion as follows.
• T 2pvl/g
• Where
• T period
• l length of pendulum string
• g acceleration of gravity
• Note
• This formula is true for only small angles of ?.
• The period of a pendulum is independent of its
  mass.

21
Conservation of ME The Pendulum

• In a pendulum, Potential Energy is converted into
  Kinetic Energy and vise-versa in a continuous
  repeating pattern.
• PE mgh
• KE ½ mv2
• MET PE KE
• MET Constant
• Note
• Maximum kinetic energy is achieved at the lowest
  point of the pendulum swing.
• The maximum potential energy is achieved at the
  top of the swing.
• When PE is max, KE 0, and when KE is max, PE
  0.