potential energy - umasspeople.umass.edu/kastor/energy/pdflectures/lecture-3.pdf · physics 190e:...

Physics 190E: Energy & Society

Fall 2007

Physics of Energy II - 1

Now, let’s talk about a second form of energy

Potential energy

Imagine you are standing on top of half dome in

Yosemite valley, holding a rock in your hand.

The rock has no kinetic energy, but if you threw

it off the cliff it would have quite a bit of kinetic

energy by the time it hit the valley floor.

We say that the rock has potential energy. If m

is the mass of the rock and h the height above

ground, the potential energy of the rock is…

PE = mghWhat is g here?

Let’s pick up where we left off last time…..the topic was gravitational potential

energy


Fall 2007


Reading assignment in

textbook - chapter 3 - work,

energy & power

Recall also the reading assignment….


Fall 2007


g is known as the gravitational constant. It measures the strength of

the Earth’s gravitational pull on falling objects.

Galileo demonstrated that all objects fall the same way.

If two objects are dropped from the same height at the

same time, then they will hit the ground at the same time

(as long as other forces like air resistance are negligible).

Falling objects accelerate downwards at a rate of …

g = 9.8m / s2

Acceleration is the rate of change of velocity with time.

So, the units of acceleration are the units for velocity

divided by another factor of time.


Fall 2007


More on acceleration & related physics…

2007 Ferrari F430

Weight: 3196 lb (1450 kg)

Fuel Economy city/highway 11/16 mpg

Acceleration: 0-62 mph in 4.0s

Top Speed:>196 mph (>315 km/h)

Let’s calculate its acceleration in meters/(second)2

2007 Toyota Prius

0-60 mph in 10s

60mpg(city), 50mpg(hway)


Fall 2007


Basic physics result - if an object starts at rest at time t=0 and accelerates with a

constant acceleration, its velocity increases linearly with time….

�

v = a ! t

acceleration

If we want to figure out the acceleration, we can rewrite this as

�

a = v / t

The car accelerates, reaching a velocity of v=62 mph = 28 m/s in t=4 s,

which gives

�

a = (28ms!1) /(4s) = 7ms

!2A little bit smaller than the

gravitational acceleration of

g=9.8m/s2


Fall 2007


While we’re talking about acceleration, let’s introduce another piece

of basic physics … Newton’s 2nd law.

�

F = m ! a

Force equals mass times acceleration. If there is a net

force on an object, it will accelerate. Conversely, if

something is accelerating, there must be a force on it.

Back to gravity…the gravitational force (at the

earth’s surface) is

�

F = m ! gSetting these two expressions equal, we see that the mass cancels giving

�

a = g

independent of the mass of the object.

This is your weight. The force

that a scale pushes up on your feet

with to counterbalance gravity.


Fall 2007


The fact that the masses are the same in these

two equations has very deep significance in

physics. This “equivalence principle” led

Einstein to his theory of gravity - general

relativity - in which the gravitational force is a

manifestation of the curvature of spacetime.

The mass in Newton’s 2nd law (F=ma) is

known as the inertial mass, while the mass

in the gravitational force law (F=mg) is

known as the gravitational mass.

The equivalence principle has been demonstrated

experimentally to one part in a trillion…


Fall 2007


We can check that potential energy indeed has the units of energy…

�

PE = mgh

If the mass is measured in kilograms and the height in meters

then the units of potential energy work out to be…

�

kg ! (ms"2) !m = kg !m

2! s

"2= Joules

Finally, back to gravitational potential energy

Recall these units came out naturally from

the formula for Kinetic energy 1/2 mv2


Fall 2007


We can also check that falling objects satisfy conservation of energy.

If we drop something from a height D at time t=0, then it’s position and

velocities as functions of time are given by

�

h(t) = D!1

2gt

2

�

v(t) = !gt

Now, let’s calculate the total energy as a function of time.

�

E = KE + PE =1

2mv(t)

2+ mgh(t)

�

E =1

2(!gt)

2+ mg(D!

1

2gt

2) = mgD

The result is actually independent of time and equal to the initial potential energy,

demonstrating conservation of energy.


Fall 2007


A practical application of gravitational potential energy …… How to store

energy without a battery?

We’ll see that one problem with electricity is that it’s difficult to

store. Batteries are only practical for relatively small amounts of

energy. How do you store more massive quantities? One way

is to use it to lift up water and convert the electrical energy to

gravitational potential energy. This is called pumped storage

hydroelectricity.

The Northfield Mountain pumped storage hydroelectric plant - operated by First

Light Power Resources - is located in Northfield, MA about 20 minutes north of

campus (up route 63).


Fall 2007


Info from http://en.wikipedia.org/wiki/Northfield_Mountain

The 1080 MegaWatt plant at

Northfield Mountain facility

opened in 1972 and was the

largest in the world at that time.

During periods of low demand,

water is pumped 5.5 miles from

the Connecticut river to a 300

acre reservoir, 800 feet above the

river, which holds 5.6 billion

gallons of water.

In generating mode, water flows

downhill through 4 turbine

generators at a rate of 20,000

gallons per second.

Possible paper topic


Fall 2007


So far, we’ve talked about two forms of energy - kinetic energy

and gravitational potential energy. Now, we’ll introduce a 3rd -

thermal energy. We will try to understand what it means for

something to be hot and how much energy it takes to heat

something up? We’ll see that for a gas, like the air in this room,

thermal energy is just the sum of the kinetic energies of the

individual gas molecules.

Understanding the mechanical equivalence

of heat - that mechanical energy could be

transformed into heat and vice-versa - was a

major achievement of 19th century physics.

Thermal Energy

This effort was closely tied to the industrial

revolution and the need to understand how

things like steam engines (which convert heat

into mechanical energy) work …

Read Chapter 7


Fall 2007


A good way to start into this subject is to talk

about the amount of energy it takes to heat

something up?

One way to talk about this is just to give it a name …..

The British Thermal Unit (or BTU) is defined as

1 BTU = amount of energy required to raise the

temperature of 1 pound of water by 1 degree

farenheit.

We already have another unit of energy - the

Joule. We need to know how many Joules

does a BTU correspond to? This is a question

for experimentalists?

Frigidaire 6000 BTU Air Conditioner

Really this means BTU/hour - a

measure of the cooling capacity of the

air conditioner


Fall 2007


James Joule, 1818-1889

Of course, at the time the mechanical unit of energy in use was

not the Joule.

It was the foot-pound. Before coming back to Joule’s work,

let’s take yet another detour into units and talk about the foot-

pound as a measure of energy…. This will allow us to bring up

another important point about energy.

This was a question that interested James Joule, himself….


Fall 2007


The usual definition of energy given in introductory physics textbooks

is ….

energy = capacity to do work

Of course, to complete the definition we need to ask what physicists

mean by work?

If you sit in the library reading a book for a

course, are you doing work in the physics

sense?

No

In physics work means very specifically exerting a force through a distance, with

the direction of motion in the same direction as the force.


Fall 2007


This part about directions is important ….

An elevator does work when it takes us between

floors, because the force it exerts is in the same

direction as its motion - up.

However, if someone is walking along carrying

something, they are not doing any work (at least on

the object they are carrying) in the physics sense -

there is no force in the direction of motion. The

force is upwards, while the direction of motion is

forwards.

This makes sense because in the first case, the elevator goes up and the work it does

increases the potential energy of itself and whoever is inside. However, in carrying

water, the water is always staying at the same height.


Fall 2007


So long as the force and motion are in the same direction, the formula

for work is

W = (Force)(Distance) = F D

The foot-pound combines a unit of force - a pound -with a unit of

distance - a foot - and is thereby a unit of work or energy. 1 foot-pound

is the amount of work that must be done to raise a 1 pound weight by 1

foot. This also gives the change in potential energy of the 1 pound

weight.

Note: Pounds are used both as a measure of force and of mass, which

can be confusing. A pound-mass is the amount of mass that weighs 1

pound on the surface of the Earth. However, on the surface of the moon

it would weigh something less than a pound…. When we make a

conversion 1 pound = 2.2 kilograms, we are really talking about the

pound-mass.


Fall 2007


Yet another unit…..the SI unit for force is called the Newton.

1Newton = 1N = 1(ki log ram)(meter) / (second)2

This makes sense, based on the equation

�

F = m ! a

The unit of force is the unit of mass times the unit of acceleration.

Let’s check that work has the same dimensions as energy. Work equals force

times distance. So a unit of work is a Newton-meter.

1 Newton-meter = 1 (kg m s-2) m = 1 kg m2/s2=1 Joule


Fall 2007


Back to Joule and the mechanical

equivalent of heat

Joule built an apparatus in which

water was heated by mechanical

agitation. He could measure both

the temperature change in the

water and the amount of work

done by the agitator.

Recall that 1 BTU is the amount of energy needed to raise the temperature

of a pound of water by 1 degree farenheit. Joule found that

1 BTU = 773 foot-pounds.

The modern measurement is 1 BTU = 778.3 foot pounds. So Joule’s

measurement was quite good.


Fall 2007


Let’s focus on this relation between heat and mechanical work

We see that this same amount of energy with lift a 1 pound weight nearly 800 feet

in the air, or equivalently a 100 pound weight up to a height of 8 feet.

Indeed, the fact that burning fossil fuels yields quite

useful amounts of mechanical energy is what made the

industrial revolution possible….. We’ll return to this

in much more detail later.

1 BTU = 778.3 foot-pounds

and note that 1 BTU is approximately the amount of energy

released by burning a match. Burning releases the stored

chemical energy in the wood.

It is quite remarkable that the chemical energy stored in

such a small piece of material could accomplish such a feat!


Fall 2007


Not only are the capacities of refrigerators usually stated in BTU’s. Total

annual world energy usage is often stated in terms of “Quads”.

1 Quad = 1 quadrillion BTU = 1 x 1015 BTU

In 2004, total world energy

consumption was 447 Quads…


Fall 2007


Back to thermal energy

Thanks to Joule we can measure the amount of energy that it takes to heat

something up. Can we also understand the nature of the energy contained in a

hot object via the basic laws of physics? What is thermal energy?

Matter comes in 3 basic phases -

solid, liquid & gas. The easiest to

understand are gases and that’s

where we’ll start.

In the case of simple gases, there is a

simple formula relating the thermal

energy of the gas to its temperature.

Solids and liquids are more complicated


Fall 2007


We’ll consider a gas that’s made up of single

atoms in a container. The atoms travel around

in straight lines colliding occasionally with

each other and with the walls of the container.

The number of atoms in a gas is immense

(approximately 3x1022 in a liter container at room

temperature and pressure).

The pressure of a gas comes from collisions of the atoms

with the walls of the container. The faster the atoms in the

gas are moving, the higher the pressure.

The speed of the atoms is in turn related to

temperature.


Fall 2007


m = mass of atoms

T = temperature

The average kinetic energy of gas atoms is …

�

1

2mv

2=3

2kBT

where kB is known as Boltzmann’s constant and is given by

�

kB

=1.4 !10"23J /K

and temperature is measured using degrees Kelvin.

(This is the K in the units of Boltzmann’s constant)

The total energy of the gas is just the sum for all the gas atoms.

�

Egas =3

2NkBT N = number of gas atoms


Fall 2007


8 quantitative problems

1 short answer problem

To be handed in in class

Be sure to show your

work on the quantitative

problems. Don’t just

write down the answer.

potential energy - umasspeople.umass.edu/kastor/energy/pdflectures/lecture-3.pdf · physics 190e:...

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