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Chapter 5
Work and Energy
Forms of Energy
- (^) Mechanical
- Thermal
- (^) Electromagnetic
- (^) Nuclear Energy is conserved!
Work
- (^) Relates force to change in energy
- Scalar quantity
- Independent of time
W =
F! (
x f "
xi )
= F # x cos $
Units of Work and Energy
SI unit = Joule 1 J = 1 N!m = 1 kg!m^2 /s^2
W = F! x
Work can be positive or negative
- Man does positive work lifting box
- (^) Man does negative work lowering box
- Gravity does positive work when box lowers
- (^) Gravity does negative work when box is raised
Kinetic Energy
Same units as work Remember the Eq. of motion Multiply both sides by m,
mv^2 f^!
mvi^2 = ma " x
KE f! KEi = F " x
v^2 f
vi^2
= a " x
KE =
mv^2
Example 5.
A skater of mass 60 kg has an initial velocity of 12 m/s. He slides on ice where the frictional force is 36 N. How far will the skater slide before he stops? 120 m
Potential Energy
If force depends on distance, For gravity (near Earth’s surface)
! PE = " F! x
! PE = mgh
Conservation of Energy
Conservative forces:
- Gravity, electrical, QCD… Non-conservative forces:
- Friction, air resistance… Non-conservative forces still conserve energy! Energy just transfers to thermal energy
PE f + KE f = PEi + KEi
! KE = "! PE
Pendulum and Track Demos
Example 5.
A diver of mass m drops from a board 10.0 m above the water surface, as in the Figure. Find his speed 5.00 m above the water surface. Neglect air resistance. 9.9 m/s
Example 5.
A skier slides down the frictionless slope as shown. What is the skier’s speed at the bottom? H=40 m start finish 28.0 m/s
"Energy"
conservation
$1.00 toll $1.00 credit Conservative! (Potential Money) $
Springs (Hooke’s Law)
Proportional to displacement from equilibrium
F =! kx
Potential Energy of Spring
"PE=-F"x "x
F
"! PE =^
( kx ) x
PE =
kx^2 x
Example 5.
b) To what height h does the block rise when moving up the incline? A 0.50-kg block rests on a horizontal, frictionless surface as in the figure; it is pressed against a light spring having a spring constant of k = 800 N/m, with an initial compression of 2.0 cm. 3.2 cm
Graphical connection betweenFandPE
F x x 1 "x x 2
! PE = " F! x
PE 2! PE 1 = !Area under curve
Graphical connection betweenFandPE
PE x F = -slope, points down hill
! PE = " F! x
F = "
! PE
! x
Graphs ofFandPEfor spring
PE=(1/2)kx^2 x x F=-kx Force pushes you to bottom of potential well
Example 5.9a
PE (J)
x (m) 10 20 30 40 50 60 0 1.0 2.0 3.0 4. 0 Release point A At point 'A', which are zero? a) force b) acceleration c) force and acceleration d) velocity
Example 5.9b
PE (J)
x (m) 10 20 30 40 50 60 0 1.0 2.0 3.0 4. 0 Release point B At point 'B', which are zero? a) force b) acceleration c) force and acceleration d) velocity e) kinetic energy
Example 5.9c
PE (J)
x (m) 10 20 30 40 50 60 0 1.0 2.0 3.0 4. 0 Release point
I
All points for which force is negative (to the left): a) C, E and G b) B and F c) A and I d) D and H e) D, H and I
B
C
D E
F G
H
A
Example 5.9d
PE (J)
x (m) 10 20 30 40 50 60 0 1.0 2.0 3.0 4. 0 Release point D At point 'D', which are zero? a) force b) acceleration c) force and acceleration d) velocity e) Velocity and kinetic energy
Example 5.
PE (J)
x (m) 10 20 30 40 50 60 0 1.0 2.0 3.0 4. 0 A particle of mass m = 0.5 kg is at a position x = 1.0 m and has a velocity of -10.0 m/s. What is the furthest points to the left and right it will reach as it oscillates back and forth? 0.125 and 3.75 m Etot
Example 5.
If the power required to accelerate the air is 40% of the answer from the last problem due to the professor’s sleek aerodynamic shape, a) what is the power required to accelerate the air? b) If the professor has an efficiency of 20%, how many kilocalories will he burn in three hours? DATA: 1 kcal=4187 J a) 52.4 W b) 676 kcal Since mass swept out is proportional to v, and KE ~ (1/2)mv^2 , Power scales as v^3! If one goes from 35 km/hr to 50 km/r, power required would rise by 2.91.
Power ~ v^3
Ergometer Demo
Example 5.
A dam wishes to produce 50 MW of power. If the height of the dam is 75 m, what flow of water is required? (in m^3 /s) 68.9 m^3 /s = 1.80x10^4 gallons/s
Example 5.
2001 cost of electricity How much money does it cost to run a 100-W light bulb for one year if the cost of electricity is 8. cents/kW!hr? $ 70.
Some energy facts
http://css.snre.umich.edu
- US consumes 24% of Worlds energy (5% of population)
- Each day, each of us consumes:
- 3 gallons of oil
- 20 lbs of coal
- (^) 221 cubic feet of natural gas
- (^) In 2000 the US consumed 9.9x10^16 BTUs= 1.05x10^20 J 1 BTU is energy required to raise 1 lb of H 2 0 1 degree F 1BTU = 1055 J
Einstein...
c is velocity of light “Rest” energy
E = mc^2
For small velocities,
E = mc^2 +
mv^2
For any v,
E = mc^2 1!
v^2
c^2
Example 5.
Suppose one had a supply of anti-matter which one could mix with matter to produce energy. What mass of antimatter would be required to satisfy the U.S. energy consumption in 2000? (9.9x10^16 BTUs) 574 kg
