Chapter 5: Energy

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Chapter 5 Energy

• Energy

• Energy is present in a variety of forms
• mechanical, chemical, electromagnetic,
  nuclear, mass, etc.

• Energy can be transformed from one from to
  another.

• The total amount of energy in the Universe never
  changes.

• If a collection of objects can exchange energy
  with
• each other but not with the rest of the
  Universe (an
• isolated system), the total energy of the
  system is
• constant.
• If one form of energy in an isolated system
  decreases,
• another form of energy must increase.

• In this chapter, we focus on mechanical energy
  kinetic
• energy and potential energy.

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• Work

• The work W done on an object by a constant
  force F
• when the object is displaced by Dx by the
  force

SI unit joule (J) newton-meter (N m) kg m2/s2

• Work is a scalar quantity.
• If the force exerted on an
• object is not in the same
• direction as the displacement

component of the force along the direction of the
displacement
dot product or inner product
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• Work

• If an object is displaced vertical to the
  direction of
• a force exerted, then no work is done.

• If an object is displaced in opposite
• direction to that of an exerted force,
• the work done by the
• force is negative (if FltFg).

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• Work

• Work and dissipative forces

• The friction force between two objects in
  contact and in relative
• to each other always dissipate energy in
  complex ways.

• Friction is a complex process caused by numerous
  microscopic
• interactions over the entire area of the
  surfaces in contact.

• The dissipated energy above is converted to heat
  and other forms
• of energy.

• Frictional work is extremely important without
  it Eskimos cant
• pull sled, cars cant move, etc.

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• Work

• Examples

• Example 5.1 Sledding through the Yukon

m50.0 kg F 1.20x102 N Dx5.00 m
(a) How much work is done if q0?
(b) How much work is done if q30o?
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• Work

• Examples

• Example 5.2 Sledding through the Yukon (with
  friction)

m50.0 kg F 1.20x102 N Dx5.00 m
(a) How much work is done if q0?
fk0.200
(b) How much work is done if q30o?
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• Kinetic Energy

• Kinetic energy (energy associated with motion)

• Consider an object of mass m moving to the right
  under action of
• a constant net force Fnet directed to the
  right.

(constant acceleration)
Define the kinetic energy KE as
SI unit J
work-energy theorem
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• Kinetic Energy

• An example

• Example 5.3 Collision analysis

m1.00x103 kg vi 35.0 m/s -gt 0
8.00x103 N
(a) The minimum necessary stopping distance?
(b) If Dx30.0 m what is the speed at impact?
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• Kinetic Energy

• Conservative and non-conservative forces

• Two kinds of forces conservative and
  non-conservative forces

• Conservative forces gravity, electric force,
  spring force, etc.

• A force is conservative if the work it does
  moving an object
• between two points is the same no matter what
  path is taken.
• It can be derived from potential energy.

• Non-conservative forces friction, air drag,
  propulsive force, etc.

• In general dissipative it tends to randomly
  disperse the energy
• of bodies on which it acts.
• The dispersal of energy often takes the form of
  heat or sound.
• The work done by a non-conservative force
  depends on what
• path of an object that it acts on is taken.
• It cannot be derived from potential energy.

• Work-energy theorem in terms of works by
  conservative and non-
• conservative force

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• Gravitational Potential Energy

• Gravitational work and potential energy

• Gravity is a conservative force and can be
  derived from a potential
• energy.

Work done by gravity on the book
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• Gravitational Potential Energy

• Gravitational work and potential energy

• Gravity is a conservative force and can be
  derived from a potential
• energy.

• Lets define the gravitational potential energy
  of a system consisting
• of an object of mass m located near the surface
  of Earth and Earth
• as

y the vertical position of the mass to a
reference point ( often at y0 ) g the
acceleration of gravity
SI unit J
where
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• Gravitational Potential Energy

• Reference levels for gravitational potential
  energy

• As far as the gravitational potential is
  concerned, the important
• quantity is not y (vertical coordinate) but the
  difference Dy between
• two positions.

• You are free to choose a reference point at any
  level (but usually
• at y0).

yi
yf
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• Gravitational Potential Energy

• Gravity and the conservation of mechanical
  energy

• When a physical quantity is conserved the
  numeric value of
• the quantity remains the same throughout the
  physical process.

• When there is no non-conservative force involved,

• Define the total mechanical energy as

• The total mechanical energy is conserved.

• In general, in any isolated system of objects
  interacting only
• conservative forces, the total mechanical
  energy of the system
• remains the same at all times.

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• Gravitational Potential Energy

• Examples

• Example 5.5 Platform diver

(a) Find the divers speed at y5.00 m.
(b) Find the divers speed at y0.0 m.
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• Gravitational Potential Energy

• Examples

• Example 5.8 Hit the ski slopes

(a) Find the skiers speed at the bottom (B).
(b) Find the distance traveled on the
horizontal rough surface.
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• Spring Potential Energy

• Spring and Hookes law

• Force exerted by a spring Fs

Hookes law
If x gt 0, Fs lt0 If x lt 0, Fs gt0
Fs to the left
xgt0
Fs to the right
k a constant of proportionality called
spring constant. SI unit N/m
Fs

• The spring always exerts its
• force in a direction opposite
• the displacement of its end
• and tries to restore the attached
• object to its original position.
• Restoring force

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• Spring Potential Energy

• Potential due to a spring

• The spring Fs is associated with elastic
  potential energy.

Between xi -1/2Dx and xi1/2Dx the work exerted
by the spring is approximately
Between x0 and x, the total work exerted by the
spring is approximately
-Fs
width Dx
xi-1/2Dx
xi1/2Dx
-Fi
In general when the spring is stretched from xi
to xf, the work done by the spring is
x
xi1
xi
-Ws,i areai
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• Spring Potential Energy

• Potential due to a spring (contd)

• The energy-work theorem including a spring and
  gravity

elastic potential energy
Extended form of conservation of mechanical energy
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• Spring Potential Energy

• Examples

• Example 5.9 A horizontal spring

(a) Find the speed at x0 without friction.
m5.00 kg k4.00x102 N/m xi0.0500 m
mk0
(b) Find the speed at xxi/2.
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• Spring Potential Energy

• Examples

• Example 5.9 A horizontal spring (contd)

(c) Find the speed at x0 with friction
m5.00 kg k4.00x102 N/m xi0.0500 m
mk 0.150
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• Spring Potential Energy

• Examples

• Example 5.10 Circus acrobat

What is the max. compression of the spring d?
m50.0 kg h 2.00 m k 8.00 x 103 N/m
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• Spring Potential Energy

• Examples

• Example 5.11 A block projected up a
  frictionless incline

m0.500 kg xi10.0 cm k625 N/m q30.0o

• Find the max. distance
• d the block travels up the
• incline.

(b) Find the velocity at hafl height h/2.
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• Spring Potential Energy

• Systems and energy conservation

• Work-energy theorem

• Consider changes in potential

The work done on a system by all non-conservative
forces is equal to the change in mechanical
energy of the system. If the mechanical energy
is changing, it has to be going somewhere. The
energy either leaves the system and goes into the
surrounding environment, or stays in the system
and is converted into non- mechanical form(s).
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• Systems and Energy Conservation

• Forms of energy

• Forms of energy stored

kinetic, potential, internal energy

• Forms of energy transfer between a non-isolated
  system and its
• environment

• Mechanical work
• transfers energy to a system by
  displacing it with a force.
• Heat transfers energy through microscopic
  collisions between
• atoms or molecules.
• Mechanical waves
• transfers energy by creating a
  disturbance that propagates
• through a medium (air etc.).
• Electrical transmission
• transfers energy through electric
  currents.
• Electromagnetic radiation
• transfers energy in the form of electromagnetic
  waves such
• as light, microwaves, and radio waves.

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• Systems and Energy Conservation

• Energy conservation

• Principle of energy conservation

Energy cannot be created or destroyed, only
transferred from one form to another.

• The principle of conservation of energy is not
  only true in physics
• but also in other fields such as biology,
  chemistry, etc.

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• Power

• Power

• The rate at which energy is transferred is
  important in the design
• and use of practical devices such as electrical
  appliances and engines.

• If an external force is applied to an object and
  if the work done by this
• force is W in time interval Dt, then the
  average power delivered to the
• object during this interval is the work done
  divided by the time interval

SI unit watt (W) J/s 1 kg m2/s3
WFDt
More general definition
U.S. customary unit 1 hp 550 ft lb/s
746 W 1 kWh
(103 W)(3600 s) 3.60 x 105 J
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• Power

• Examples

• Example 5.12 Power delivered by an elevator

What is the min. power to lift the elevator
with the max. load?
M1.00x103 kg m8.00x102 kg f 4.00x103 N v
3.00 m/s
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• Power

• Examples

• Example 5.14 Speedboat power

How much power would a 1.00x103 kg speed boat
need to go from rest to 20.0 m/s in 5.00 s,
assuming the water exerts a constant drag force
of magnitude fd5.00x102 N and the acceleration
is constant?
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• Power

• Energy and power in a vertical jump

• Center of mass (CM)

The point in an object at which all the may be
considered to be concentrated.
h0.40 m depth of crouch Dt0.25 s time for
extension m68 kg

• Stationary jump

Two phases (1) Extension, (2) free flight