ISNS 3371 - Phenomena of Nature Solar energy striking Earth’s surface per second = 2.5 x 10 17 J. Energy released by burning 1 liter of oil = solar energy

ISNS 3371 - Phenomena of Nature

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

Energy Comparisons


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

Fundamental Forces of Nature


Energy

Three basic categories:

Kinetic energy = energy of motion

KE = 1/2mv2

Potential energy = stored energy

gravitational, chemical, elastic,electrostatic, etc…

Radiative - energy carried by light

{MechanicalEnergy


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)

PE = mgh

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

m

h

g


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 = mcE = mc22

[ 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 weekA 1-megaton hydrogen bomb converts only about 3 ounces of mass into energy.


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.


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


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

CarChemical - thermal - mechanical


Power:Rate of change of energy

Power = work done/time interval = E/t

(remember: means a change in a quantity)

Power:1 watt = 1J/sThus 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.

Power


Applications of Conservation of Energy


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!


Levers

Fulcrum is in the center:d1 = d2

so

F1 = F2

Fulcrum is closer to one end:

d1 > d2

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).


Pulleys


€

Fnet = −mgsinθ

For small angles, sin =

€

Fnet = −mgθ = ma

€

−mgθ = mαl

vt, at

r

= vt/r is the angularvelocity

= at/r is the angular acceleration

so = r

This becomes the differential equation:

€

d2θ

dt 2+g

lθ = 0

Pendulum solution (you are not expected to know this)

With solution

€

=max cosg

lt

For a complete oscillation:

€

g

lP = 2π so

€

P = 2πl

g


For a small pendulum clock, P = 1s

So

€

P = 2πl

g

€

l =P 2g

4π 2

€

l =g

4π 2=

9.8

4π 2= 24.8cm

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.


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.


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(r1

2 + 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 mr1

2/2 + mr2/2 or larger than the moment of inertia of the solid cylinder by mr1

2/2.

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.


€

m1v1 + m2v2 = m1V1 + m2V2

1

2m1v1

2 +1

2m2v2

2 =1

2m1V1

2 +1

2m2V2

2

m1v1 −m1V1 = m2v2 −m2V2

m1(v12 −V1

2) = m2(v22 −V2

2)m1(v1 −V1)(v1 +V1) = m2(v2 −V2)(v2 +V2)

v1 +V1 = v2 +V2

V1 =V2 + v2 − v1

Using Conservation of Energy and Momentum to Calculate the Velocity of Two Bodies After a Collision

Conservation of momentum says

Conservation of energy says

(1)

(2)

(1) Can be written

(1a)

(2a)

(2) Can be written

From (1a) we see that m1(v1 - V1) cancels out m2(V2 - v2) so that

Which can be rewritten as


€

m1v1 + m2v2 = m1(V2 + v2 − v1) + m2V2

m1v1 + m2v2 = m1v2 + m1V2 −m1v1 + m2V2

2m1v1 + m2v2 −m1v2 = m1V2 + m2V2

2m1v1 + (m2 −m1)v2 = (m1 + m2)V2

V2 =2m1v1

m1 + m2

+m2 −m1

m1 + m2

v2

V1 =m1 −m2

m1 + m2

v1 +2m2v2

m1 + m2

Substitute V1 = V2 + v2 - v1 into (1)

This leaves us with one equation with one unknown, V2

Similarly

ISNS 3371 - Phenomena of NatureThe 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 NatureConservation of momentum

(1)

b = before collision- mb and vb are for the ball/bulleta = 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:

h = height of pendulum at end of swing

Substitute into (1):

€

mbvb = mava

vb =mavamb

€

1

2mava

2 = magh

€

va2 = 2gh

€

va = 2ghmamb


Alternate Way Using Projectile Motion and g

h

x

Fire ball from top of table. Measure initial height of ball (h) and horizontal distance traveled (x).

€

h =1

2gt 2 ⇒ t =

2h

g

x = vt ⇒ v =x

t

v = xg

2h

Vertical motion

Horizontal motion

(1)

Substitute from (1)

Documents

ISNS 3371 - Phenomena of Nature Solar energy striking Earth’s surface per second = 2.5 x 10 17 J. Energy released by burning 1 liter of oil = solar energy