# Uniform Circular Motion and Gravitation

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```					      Dr. Baron
Physics
Ch. 11: Energy and it’s
Conservation
Work is Exchange of Energy
• Energy is the capacity to do work
• Two main categories of energy
– Kinetic Energy: Energy of motion
• A moving baseball can do work
• A falling anvil can do work
– Potential Energy: Stored (latent) capacity to do work
• Gravitational potential energy (perched on cliff)
• Mechanical potential energy (like in compressed spring)
• Chemical potential energy (stored in bonds)
• Nuclear potential energy (in nuclear bonds)
• Energy can be converted between types
Kinetic Energy
• The kinetic energy for a mass in linear
motion is
K.E. = ½mv2
• Example: 1 kg at 10 m/s has 50 J of kinetic energy
• The kinetic energy for a mass in rotational
motion is
K.E. = ½Iω2
Kinetic Energy Units
• One joule (J) is the work done, or energy
expended, by a force of one newton
moving one meter along the direction of
the force.
• Also denoted as a newton meter with the
symbol N·m.
• In basic SI Units: mv2
Gravitational Potential Energy
• Gravitational Potential Energy near the
surface of the Earth:
Work = Force  Distance

m

h       DW = mg  h

m
Potential Energy
• PE = mgh (m is mass, g acceleration, h is height).
• Ball dropped from rest at a height h hits
the ground with speed v.
• Ball has converted its available
gravitational potential energy into kinetic
energy: the energy of motion
• ½mv2 = mgh
Potential Energy (cont)
• Reference height is zero and you can choose!
• PE = mgh (m is mass, g acceleration, h is height.
• Ball dropped from rest at a height h hits
the ground with speed v.
• Ball has converted its available
gravitational potential energy into kinetic
energy: the energy of motion
• ½mv2 = mgh
Examples “Worked” Out
• How much work does it take to lift a 30 kg suitcase onto
the table, 1 meter high?
W = PE = mgh
= (30 kg)  (9.8 m/s2)  (1 m) = 294 J
• Unit of work (energy) is the N·m, or Joule (J)
• Pushing a crate 10 m across a floor with a force of 250 N
requires 2,500 J (2.5 kJ) of work
• Gravity does 20 J of work on a 1 kg (~10 N) book that it
has pulled off a 2 meter shelf
Elastic Potential Energy
• The energy stored in strings, rubber
bands, slingshots, and pole vaults…..
“REST” Energy
• The energy buried in mass
• E = mc2
Law of Conversion of Energy
• Energy can neither be created nor
destroyed, but it can be transformed into
another form of energy
• This is one of nature’s “conservation laws”
– Conservation applies to:
•   Energy (includes mass via E = mc2)
•   Momentum
•   Angular Momentum
•   Electric Charge
Conversion of Energy
• Falling object converts gravitational potential energy into
kinetic energy
• Friction converts kinetic energy into vibrational (thermal)
energy
– makes things hot (rub your hands together)
– irretrievable energy
• Doing work on something changes that object’s energy
by amount of work done, transferring energy from the
agent doing the work
Energy Conservation
Demonstrated

•   Roller coaster car lifted to initial height (energy in)
•   Converts gravitational potential energy to motion
•   Fastest at bottom of track
•   Re-converts kinetic energy back into potential as it
climbs the next hill
Mechanical Exchange
• THE MECHANICAL ENERGY OF A SYSTEM IS THE
SUM OF THE KINETIC ENERGY AND
POTENTIALENERGY
• ME = KE + PE

• ME = ½mv2 + mgh

• THE MECHANICAL ENERGY OF A SYSTEM IS
ALWAYS CONSERVED
Energy Exchange Example
• Though the total energy of a system is constant, the form of the
energy can change
• A simple example is that of a simple pendulum, in which a continual
exchange goes on between kinetic and potential energy

pivot

K.E. = 0; P. E. = mgh                          K.E. = 0; P. E. = mgh
h
height reference
P.E. = 0; K.E. = mgh
Perpetual Motion
• Why won’t the pendulum swing forever?
• It’s hard to design a system free of energy paths
• The pendulum slows down by several mechanisms
– Friction at the contact point: requires force to oppose;
force acts through distance  work is done
– Air resistance: must push through air with a force
(through a distance)  work is done
– Gets some air swirling: puts kinetic energy into air
(not really fair to separate these last two)
• Perpetual motion means no loss of energy
– solar system orbits come very close
Some Energy Chains:
• A coffee mug with some gravitational potential energy is
dropped
• potential energy turns into kinetic energy
• kinetic energy of the mug goes into:
– ripping the mug apart (chemical: breaking bonds)
– sending the pieces flying (kinetic)
– into sound
– into heating the floor and pieces through friction as
the pieces slide to a stop
• In the end, the room is slightly warmer
Gasoline Example
• Put gas in your car, containing 9 Cal/g
• Combust gas, turning 9 Cal/g into kinetic energy of
explosion
• Transfer kinetic energy of gas to piston to crankshaft to
drive shaft to wheel to car as a whole
• That which doesn’t go into kinetic energy of the car goes
into heating the engine block (and radiator water and
surrounding air), and friction of transmission system
(heat)
• Much of energy goes into stirring the air (ends up as
heat)
• Apply the brakes and convert kinetic energy into heat
• It all ends up as waste heat, ultimately
Bouncing Ball
• Superball has gravitational potential energy
• Drop the ball and this becomes kinetic energy
• Ball hits ground and compresses (force times
distance), storing energy in the spring
• Ball releases this mechanically stored energy
and it goes back into kinetic form (bounces up)
• Inefficiencies in “spring” end up heating the ball
and the floor, and stirring the air a bit
• In the end, all is heat
Why don’t we get hotter and
hotter
• If all these processes end up as heat, why aren’t we
continually getting hotter?
• If earth retained all its heat, we would get hotter
• All of earth’s heat is radiated away
F = T4
• If we dump more power, the temperature goes up, the
– comes to equilibrium: power dumped = power
– stable against perturbation: T tracks power budget
• No matter what, you can’t create energy out of nothing: it
has to come from somewhere
• We can transform energy from one form to another; we
can store energy, we can utilize energy being conveyed
from natural sources
• The net energy of the entire Universe is constant
• The best we can do is scrape up some useful crumbs
Shift Gears: The Global
Energy Scene
• Global energy production is about 400 QBtu/yr
– a QBtu is a quadrillion Btu, or 1015 Btu
– so about 41020 J per year
• U.S. share is about one fourth of this (1020 J)
– 1996 value in book is 93 QBtu/year
• 1020 J/yr = 31012 W
– divided by 300 million people (3108) = 104 W per
person (10 kW)
Kinetic Energy, cont.

• Kinetic energy is proportional to v2…
• Watch out for fast things!
– Damage to car in collision is proportional to v2
– Trauma to head from falling anvil is proportional to
v2, or to mgh (how high it started from)
– Hurricane with 120 m.p.h. packs four times the
punch of gale with 60 m.p.h. winds
Energy Conversion/Conservation Example
10 m   P.E. = 98 J
K.E. = 0 J      • Drop 1 kg ball from 10 m
– starts out with mgh = (1 kg)(9.8 m/s2)(10
8m
P.E. = 73.5 J        m) = 98 J of gravitational potential energy
K.E. = 24.5 J
– halfway down (5 m from floor), has given up
6m                         half its potential energy (49 J) to kinetic
P.E. = 49 J          energy
K.E. = 49 J
4m                          • ½mv2 = 49 J  v2 = 98 m2/s2  v  10
m/s
P.E. = 24.5 J
2m    K.E. = 73.5 J      – at floor (0 m), all potential energy is given
up to kinetic energy
P.E. = 0 J            • ½mv2 = 98 J  v2 = 196 m2/s2  v = 14
0m
K.E. = 98 J              m/s
Momentum and Kinetic Energy

Perfectly inelastic collision: After colliding, particles stick
together. There is a loss of energy (deformation).
Elastic collision: Particles bounce off each other without loss of
energy.
Inelastic collision: Particles collide with some loss of energy,
but don’t stick together.


P
Elastic Collision: Billiard Balls
• Whack stationary ball with identical ball moving at velocity vcue

8
To conserve both energy and momentum, cue ball stops dead,
and 8-ball takes off with vcue

8               8

Momentum conservation: mvcue = mvcue, after + mv8-ball
Energy conservation: ½mv2cue = ½mv2cue, after + ½mv28-ball

The only way v0 = v1 + v2 and v20 = v21 + v22 is if either v1 or v2 is 0.
Since cue ball can’t move through 8-ball, cue ball gets stopped.
Desk Toy Physics

• The same principle applies to the suspended-ball
desk toy, which eerily “knows” how many balls you let
go…
• Only way to simultaneously satisfy energy and
momentum conservation
• Relies on balls to all have same mass

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 views: 11 posted: 5/25/2010 language: English pages: 27