Chapter 7Conservation of Energy

Recap – Work & Energy

2 21 12 2f iW mv mv

The total work done on a particle is equal to the change in its kinetic energy

Potential Energy

The total work done on an object equals the change in its kinetic energy

But the total work done on a system of objects may or may not change its total kinetic energy. The energy may be stored as potential energy.

Potential Energy – A Spring

Both forces do work on the spring. Butthe kinetic energy of the spring is unchanged. The energy is stored as

potential energy

Conservative Forces

If the ski lift takes youup a displacement h, thework done on you, bygravity, is –mgh.

But when you ski downhill the work done by gravity is +mgh, independent of the path you take

Conservative Forces

The work done ona particle bya conservative force is independent of the path takenbetween any two points

Potential-Energy Function

2

12 1

s

sU U U F ds

If a force is conservative, then we candefine a potential-energy function as thenegative of the work done on the particle

Potential-Energy Function

0

0

0

0

0

( ˆˆ ˆ) ( )

( )

ˆ

s

s

s

s

y

y

i

U U U F ds

mg dx dy dz

mg dy

m

k

g y y

j j

potential-energy function associatedwith gravity (taking +y to be up)

0 0( )U U mg y y

The valueof U0 = U(y0)can be set to any convenientvalue

Potential-Energy Function of a Spring

210 2U U kx By convention,

one choosesU0 =U(0) = 0

Force & Potential-Energy Function

dUF

dx

In 1-D, given the potential energy function associated with a force one can compute the latter using:

Example:

212

dUU kx F kx

dx

7-1Conservation of Energy

Conservation of Energy

inE 0sysE

Energy can be neither created nor destroyed

outE

0sysE Closed System

Open System

Conservation of Mechanical Energy

constantmechE K U

If the forces acting are conservativethen the mechanical energy is conserved

Example 7-3 (1)

How high does the block go?

Initial mechanical energy of system

212iE kx

Final mechanical energy of system

fE mgh

Example 7-3 (2)

Forces are conservative, therefore,mechanical energy is conserved

212 kx mgh

Height reached2

2

kxh

mg

Example 7-4 (1)

How far does the mass drop?

2 21 12 2i i i iE mgy ky mv

Final mech. energy2 21 1

2 2f f ff mgy ky vE m

Initial mech. energy

Example 7-4 (2)

2 21 12 2

2 21 12 2

( ) ( ) (0

(0)

)

(0) (0)mg m

mg d d m

Final mech. energy = Initial mech. energy

2 21 12

2 2

2

2 21 1f f f

i i imgy ky mv

mgy ky mv

Example 7-4 (3)

2mgd

k

Solve for d

12( ) 0kd mg d

2mg

Since d ≠ 0

Example 7-4 (4)

gravE mgd

Note21

2springE kd

2mg

is equal to loss in gravitational potentialenergy

Conservation of Energy & Kinetic Friction

Non-conservative forces, such as kinetic friction, cause mechanical energy to be transformed into other forms of energy, such as thermal energy.

Work-Energy Theorem

Work done, on a system, by external forces is equal to the change in energyof the system

ext sysW EThe energy in a system can be distributed in many different ways

Example 7-11 (1)

ext sysW E

Find speed of blocks after spring isreleased. Consider spring & blocks as system. Write down initial energy.Write down final energy.Subtract initialfrom final

ext sysW E

Example 7-11 (2)

212i iE kx

Initial Energy

ext sysW E

Take potentialenergy of system to be zero initially

Kinetic energy of system is zeroinitially

Example 7-11 (3)

21

2112 2 1

212 20

othermf s m

k

E

m

E

m v m g

E E

gv

E

ms s

Final Energy

ext sysW E

Kinetic and potential energies of system have changed

Example 7-11 (4)

ext f iW E E Subtract initial energy from final energy

ext sysW E

But since noexternal forcesact, Wext = 0, soEf = Ei

Example 7-11 (5)

22 1

1 2

2 2i kkx m g s m g sv

m m

And the answer is…

ext sysW ETry to derivethis.

E = mc2

In a brief paper in 1905Albert Einstein wrotedown the most famousequation in science

E = mc2

Sun’s Power Output

Power1 Watt = 1 Joule/second100 Watt light bulb = 100 Joules/second

Sun’s power output3.826 x 103.826 x 102626 Watts Watts

Sun’s Power Output

Mass to Energy Kg/s = 3.826 x 10 3.826 x 102626 Watts Watts / (3 x 108 m/s)2

The Sun destroys mass at~ 4 billion kg / s

Problems

To go…

Ch. 7, Problem 19

Ch. 7, Problem 29

Ch. 7, Problem 74