ISNS3371_013007_bw - The University of Texas at Dallas
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Transcript ISNS3371_013007_bw - The University of Texas at Dallas
ISNS 3371 - Phenomena of Nature
Energy Comparisons
Solar energy striking Earth’s surface per second = 2.5 x 1017 J.
Energy released by burning 1 liter of oil = solar energy striking square
100 m on a side in 1 second
ISNS 3371 - Phenomena of Nature
Fundamental Forces of Nature
Four Types of Forces:
•
Gravitational – holds the world together
•
Electromagnetic – attraction/repulsion of charged matter
•
Strong Nuclear – holds nucleus together
•
Weak Nuclear – involved in reactions between subatomic
particles
ISNS 3371 - Phenomena of Nature
Energy
Three basic categories:
Mechanical
Energy
{
Kinetic energy = energy of motion
KE = 1/2mv2
Potential energy = stored energy
gravitational, chemical, elastic,electrostatic, etc…
Radiative - energy carried by light
ISNS 3371 - Phenomena of Nature
Potential Energy
One form of potential energy is gravitational
potential energy - the energy which an object
stores due to its ability to fall
•It depends on:
– the object’s mass (m)
– the strength of gravity (g)
– the distance which it falls (h)
g
m
PE = mgh
h
Before the sun was formed - matter contained
in cloud diffuse gas cloud - most far from the
center - large gravitational energy. As cloud
contracted under its own gravity - gravitational
energy converted to thermal energy until hot
enough to ignite nuclear fusion
ISNS 3371 - Phenomena of Nature
Potential Energy
• energy is stored in matter itself
• this mass-energy is what would be released if an amount of
mass, m, were converted into energy
E = mc2
[ c = 3 x 108 m/s is the speed of light; m is in kg, then E is in joules]
The mass energy in a 1-kg rock is equal to as much energy as 7.5 billion
liters of oil = enough to run all the cars in the U.S. for a week
A 1-megaton hydrogen bomb converts only about 3 ounces of mass into
energy.
ISNS 3371 - Phenomena of Nature
Conservation of Energy
• Energy can be neither created nor destroyed.
• It merely changes it form or is exchanged between objects.
• This principle (or law) is fundamental to science.
• The total energy content of the Universe was determined in the
Big Bang and remains the same today.
ISNS 3371 - Phenomena of Nature
Types of Energy
Energy cannot be created or destroyed, only changed
–
Mechanical –
• Potential - stored energy
• Kinetic- energy of motion KE=1/2mv2
–
Electrical
–
Chemical
–
Elastic
–
Gravitational
–
Thermal
–
Radiant
–
Nuclear
ISNS 3371 - Phenomena of Nature
Conversion of Energy
Throwing a baseball
Nuclear energy (nuclear fusion on sun) - Radiative energy
(sunlight) - Chemical energy (photosynthesis) - Chemical energy in
pitcher’s body (from eating plants) - Mechanical kinetic energy
(motion of arm) - Mechanical kinetic energy (movement of the
baseball). Thus, ultimate source of KE in baseball is mass energy
stored in hydrogen of Sun - created in Big Bang.
Hydroelectric dam
Gravitational - mechanical - electrical
Nuclear reactor
Nuclear - thermal - mechanical - electrical
Car
Chemical - thermal - mechanical
ISNS 3371 - Phenomena of Nature
Power
Power:
Rate of change of energy
Power = work done/time interval = E/t
(remember: means a change in a quantity)
Power:
1 watt = 1J/s
Thus for every second a 100 W light bulb is on, the electric
company charges for 100 J of energy.
The average daily power requirement for a human is about
the same as for a 100-W light bulb.
ISNS 3371 - Phenomena of Nature
Applications of Conservation of Energy
ISNS 3371 - Phenomena of Nature
Machines
Machines can be used to multiply force:
(force X distance)input = (force X distance)output
Decrease the distance and the force will increase.
Work/Energy is not changed!
ISNS 3371 - Phenomena of Nature
Levers
Fulcrum is in the center:
d1 = d 2
so
F1 = F2
Fulcrum is closer to one
end:
d1 > d 2
So
F2 > F1
Give me a long enough lever and a place to put the fulcrum and I can
move the world (Archimedes, 250 BC).
ISNS 3371 - Phenomena of Nature
Pulleys
ISNS 3371 - Phenomena of Nature
Pendulum solution (you are not
expected to know this)
Fnet mgsin
For small angles, sin =
Fnet mg ma
mg ml
This becomes the differential equation:
d 2 g
0
2
dt
l
vt, at
r
= vt/r
is the angular
velocity
= at/r
is the angular
acceleration
so
= r
With solution
g
max cos t
l
For a complete oscillation:
g
P 2
l
so
l
P 2
g
ISNS 3371 - Phenomena of Nature
l
P 2
g
P 2g
l
4 2
For a small pendulum clock, P = 1s
So
g
9.8
l
24.8cm
2
2
4
4
If P = 2s, then l = 0.993 m
This is the length of the typical grandfather clock’s pendulum which
advances
each time the pendulum reaches its maximum displacement
or twice every period.
ISNS 3371 - Phenomena of Nature
Objects Moving Down an Inclined Plane
Compare the speed of an object rolling down an inclined plane without
slipping and one sliding without friction. Which gets to the bottom first?
We simulate this with two cylinders of the same mass - one is a solid
cylinder, and one has wheels on the sides which turn while the cylinder
itself doesn’t.
The sliding object will always reach the bottom first because all the initial
potential energy is converted into translational energy with none wasted in
rotation.
ISNS 3371 - Phenomena of Nature
Which will roll down the inclined plane faster - a solid cylinder or a hollow
cylinder (of the same mass and outer radius)?
As the object rolls down the plane, its initial potential energy is converted
into both translational energy of the center-of-mass and also into rotational
energy.
- ratio of rotational to translational energy is I / mr2 where I is the
moment of inertia, m is the mass and r is the radius of the object.
- moment of inertia is mr2/2 for the solid cylinder and m(r12 + r22)/2 for
the hollow cylinder.
- Since r2 of the hollow cylinder is equal to r of the hollow cylinder, and
the mass is the same, the moment of inertia of the hollow cylinder is
mr12/2 + mr2/2 or larger than the moment of inertia of the solid
cylinder by mr12/2.
Thus ratio of the rotational to the translational energy for the hollow cylinder
is greater than for the hollow cylinder. The hollow cylinder therefore
acquires the most rotational energy and the least translational energy (and
velocity) and thus takes the longest to get down the plane.
ISNS 3371 - Phenomena of Nature
Using Conservation of Energy and Momentum to Calculate
the Velocity of Two Bodies After a Collision
Conservation of momentum says
(1)
m1v1 m2v 2 m1V1 m2V2
Conservation of energy says
(2)
1
1
1
1
m1v12 m2v 2 2 m1V12 m2V2 2
2
2
2
2
(1) Can be written
(1a)
m1v1 m1V1 m2v 2 m2V2
(2) Can be written
m1 (v1 V1 ) m2 (v 2 V2 )
(2a) m1 (v1 V1 )(v1 V1 ) m 2 (v 2 V2 )(v 2 V2 )
2
2
2
2
From (1a) we see that m1(v1 - V1) cancels out m2(V2 - v2) so that
v1 V1 v 2 V2
Which can be rewritten as
V1 V2 v 2 v1
ISNS 3371 - Phenomena of Nature
Substitute V1 = V2 + v2 - v1 into (1)
m1v1 m2v 2 m1 (V2 v 2 v1 ) m 2V2
This leaves us with one equation with one unknown, V2
m1v1 m2v 2 m1v 2 m1V2 m1v1 m2V2
2m1v1 m2v 2 m1v 2 m1V2 m 2V2
2m1v1 (m2 m1 )v 2 (m1 m2 )V2
2m1v1
m 2 m1
V2
v2
m1 m2 m1 m2
Similarly
V1
m1 m 2
2m2v 2
v1
m1 m2
m1 m2
ISNS 3371 - Phenomena of Nature
The Ballistic Pendulum
The ballistic pendulum is used to determine the speed of a projectile.
Invented in the 18th century by Benjamin Robins to determine the
speed of a bullet.
A bullet of mass m is fired at a block of wood (mass M) hanging from a string.
The bullet embeds itself in the block, and causes the combined block plus
bullet system to swing up a height h. Conservation of momentum and
conservation of energy are used to determine the bullet’s speed.
ISNS 3371 - Phenomena of Nature
Conservation of momentum
mb v b ma v a
mav a
v
(1)
b
mb
b = before collision- mb and vb are for the ball/bullet
a = after collision- ma and va are for the ball/bullet and pendulum
Conservation of energy
Kinetic Energy of ball and pendulum just after collision = Potential
Energy of ball and pendulum at end of swing:
1
2
ma v a ma gh
2
h = height of pendulum at end of swing
va 2gh
2
Substitute into (1):
ma
v a 2gh
mb
ISNS 3371 - Phenomena of Nature
Alternate Way Using Projectile Motion and g
Fire ball from top of table. Measure initial height of ball (h) and horizontal
distance traveled (x).
h
x
Vertical motion
1 2
2h
h gt t
2
g
Horizontal motion
x
x vt v
t
Substitute from (1)
g
vx
2h
(1)