Conservation of Energy

Forms of Energy

Mechanical Energy

Ug UsK

Thermal Energy

Eth

Other forms include

Echem Enuclear

Law of Conservation of Energy

What you put in is what you get out

Total energy is conserved

Practical Applications

Gasoline converts to energy which moves the car

A battery converts stored chemical energy to electrical energy

Dams convert the kinetic energy of falling water into electrical energy

Example : step 1

Example : step 2

Example : step 3

Example : last step

Conservation of Mechanical Energy

Emghmv 2

2

1

Total Energy

Potential Energy

Kinetic Energy

m = mass

v = velocity

g = gravitational acceleration

h = height

ILYA, did you know that

even though it was a

bumpy ride, our energy remained constant!

Conservation of Mechanical Energy We denote the total mechanical energy by

Since

The total mechanical energy is conserved and remains the same at all times

PEKEE

iiff KEPEPEKE

ffii mghmvmghmv 22

2

1

2

1

A block projected up a incline Point A (initial state): Point B (final state):

mcmxyv iii 1.010,0,0

m

smkg

mmN

mg

kxd i

28.1

30sin)/8.9)(5.0(

)1.0)(/625(5.0

sin

2

2

221

2222

2

1

2

1

2

1

2

1fffiii kxmgymvkxmgymv

0,sin,0 fff xdhyv

sin2

1 2 mgdmgykx fi

Trucks with the noted masses moving at the noted speeds crash into barriers that bring them to rest with a constant force. Which truck compresses the barrier by the largest distance?

Questions :

2 Types of Forces Conservative forces

Work and energy associated with the force can be recovered

Examples: Gravity, Spring Force, EM forces

Nonconservative forces The forces are generally

dissipative and work done against it cannot easily be recovered

Examples: Kinetic friction, air drag forces, normal forces, tension forces, applied forces …

The work done by a conservative force is independent of the path, and depends only on the starting and ending points.

Closed path, W=0. Pick any starting and ending points.

A

B

A

B

W1

W2

W1 = WAB

W2 = WBA

W1 + W2 = 0W3

W1 + W3 = 0

So, all paths from B to A take the same amount of work.

W1

Path doesn’t matter!

gHv

gHv

MvMgH

2

2

02

10

2

2

Initial Final

Conservative Forces A force is conservative if the work it does on an

object moving between two points is independent of the path the objects take between the points The work depends only upon the initial and final

positions of the object Any conservative force can have a potential energy

function associated with it Work done by gravity Work done by spring force

fifig mgymgyPEPEW

22

2

1

2

1fisfsis kxkxPEPEW

Ex n.1: Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA = 1.2

m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Assuming the surface to be frictionless, calculate the maximum compression of the spring after the collision.

msmmN

kgvk

mx A 15.0)/2.1(

/50

8.0max

002

100

2

1 22max Amvmv

2222

2

1

2

1

2

1

2

1iiifff kxmghmvkxmghmv

Nonconservative Forces A force is nonconservative if the work it does

on an object depends on the path taken by the object between its final and starting points. The work depends upon the movement path For a non-conservative force, potential energy can

NOT be defined Work done by a nonconservative force

It is generally dissipative. The dispersal of energy takes the form of heat or sound

sotherforceknc WdfdFW

Problem-Solving Strategy Define the system to see if it includes non-conservative

forces (especially friction, drag force …) Without non-conservative forces With non-conservative forces

Select the location of zero potential energy Do not change this location while solving the problem

Identify two points the object of interest moves between One point should be where information is given The other point should be where you want to find out something

2222

2

1

2

1

2

1

2

1iiifff kxmghmvkxmghmv

)()( iiffnc PEKEPEKEW

)2

1

2

1()

2

1

2

1( 2222

iiifffsotherforce kxmghmvkxmghmvWfd

Ex n 2:Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA = 1.2

m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Suppose a constant force of kinetic friction acts between the block and the surface, with µk = 0.5, what is the maximum compression xc in the spring.

)002

1()

2

100(0 22 Ack mvkxNd

)2

1

2

1()

2

1

2

1( 2222

iiifffsotherforce kxmgymvkxmgymvWfd

ckAc mgxmvkx 22

2

1

2

1

cxdmgN and

058.09.325 2 cc xx mxc 093.0

Connected Blocks in Motion Two blocks are connected by a light string that passes over a

frictionless pulley. The block of mass m1 lies on a horizontal surface and is connected to a spring of force constant k. The system is released from rest when the spring is unstretched. If the hanging block of mass m2 falls a distance h before coming to rest, calculate the coefficient of kinetic friction between the block of mass m1 and the surface.

22 2

10 kxghmNxk

PEKEWfd sotherforce

hxmgN and

)02

1()0( 2

2 kxghmPEPEPE sg

221 2

1khghmghmk gm

khgm

k1

2 21

gRv min

jmggmW ˆ

jR

vmFc ˆ

2

To stay on loop, the normal force, N, must be greater than zero.

mgR

vm

mgR

vmN

jmgNjR

vm

jmgjNWNjR

vmFc

2min

2

2

2

0

ˆ)(ˆ

ˆˆˆ

jNN ˆ

gRv min

Ex n. 3 :Block on wheel of death

RRRh

mghRmgmgR

mghRmgmv

2

52

2

1

)2(2

1

)2(2

1 2min

Mass must start higher than top of loop

h

gRRgghv

mghmgmv

52

522

)0(2

1 2

gRRgghv

mgRRmgmv

mghRmgmv

32

322

2

5)(

2

1

)(2

1

2

2

h

h = 5/2 R

Vmax = gR5

gRv min

Vmax

Vfin

gR3Vfin =

Potential Energy vs. Force