Physics II Today’s Agenda

Honors Physics, Pg 1

Physics IIPhysics II

Today’s AgendaToday’s Agenda Work & Energy.

Discussion.Definition.

Work of a constant force. Power Work kinetic-energy theorem. Work of a sum of constant forces. Work for a sum of displacements with constant force. Work done by a spring Conservation of Energy Comments.


Work & EnergyWork & Energy

One of the most important concepts in physics.Alternative approach to mechanics.

Many applications beyond mechanics.Thermodynamics (movement of heat).Quantum mechanics...

Very useful tools.You will learn new (sometimes much easier) ways to

solve problems.

See text


Forms of EnergyForms of Energy

KineticKinetic: Energy of motion.A car on the highway has kinetic energy.

We have to remove this energy to stop it.The breaks of a car get HOT !This is an example of turning one form of energy into

another. (More about this soon)...


Forms of EnergyForms of Energy

PotentialPotential: Stored, “potentially” ready to use.Gravitational.

» Hydro-electric dams etc...Electromagnetic

» Atomic (springs, chemical...)Nuclear

» Sun, power stations, bombs...


Mass = Energy Mass = Energy

Particle Physics:

+ 5,000,000,000 V

e-

- 5,000,000,000 V

e+(a)

(b)

(c)

E = 1010 eV

M E = MC2

( poof ! )


Energy ConservationEnergy Conservation Energy cannot be destroyed or created.

Just changed from one form to another.

We say energy is conservedenergy is conserved !True for any isolated system. i.e when we put on the brakes, the kinetic energy of the car is turned into heat using friction in the brakes. The total energy of the “car-breaks-road-atmosphere” system is the same.The energy of the car “alone” is not conserved...

» It is reduced by the braking.

Doing “workwork” on a system will change it’s “energyenergy”...


Definition of Work:Definition of Work:

Ingredients: Ingredients: Force ( FF ), displacement ( SS )

Work, W, of a constant force FF

acting through a displacement SS

is:

W = FF..SS = FScos() = FS S

FF

SS

displace

ment

FS

“Dot Product”


Work: 1-D Example Work: 1-D Example (constant force)(constant force)

A force FF = 10N pushes a box across a frictionless floor for a distance x x = 5m.

xx

FF

Work done byby F F onon box :

WF = FF.xx = F x (since FF is parallel to xx)

WF = (10 N)x(5m) = 50N-m.

See example 7.1


Units:Units:

N-m (Joule) Dyne-cm (erg)

= 10-7 J

BTU = 1054 J

calorie = 4.184 J

foot-lb = 1.356 J

eV = 1.6x10-19 J

cgs othermks

Force x Distance = Work

Newton x

[M][L] / [T]2

Meter = Joule

[L] [M][L]2 / [T]2


PowerPower We have seen that W = FF. SS

This does not depend on time !

Power is the “rate of doing work”:

If the force does not depend on

time: W/ t = FF. SS/ t = FF.v v P = FF.vv

Units of power: J/sec = Nm/sec = Watts

t

WP

FFSS

vv


Comments:Comments:

Time interval not relevant.Run up the stairs quickly or slowly...same W.

Since W = FF.SS

No work is done if: FF = 0 or SS = 0 or = 90o


Comments...Comments...W = FF.SS No work done if = 90o.

No work done by TT.

No work done by N.

TT

v v

vvNN


Work & Kinetic Energy:Work & Kinetic Energy:

A force FF = 10N pushes a box across a frictionlessfloor for a distance x x = 5m. The speed of the box is v1 before the push, and v2 after the push.

xx

FFv1 v2

ii

m


Work & Kinetic Energy...Work & Kinetic Energy...

Since the force FF is constant, acceleration a a will be constant. We have shown that for constant a:v2

2 - v12 = 2a(x2-x1 ) = 2ax.

multiply by 1/2m: 1/2mv22 - 1/2mv1

2 = ma x

But F = ma 1/2mv22 - 1/2mv1

2 = Fx

xx

FFv1 v2

aa

ii

m


Work & Kinetic Energy...Work & Kinetic Energy...

So we find that1/2mv2

2 - 1/2mv12 = Fx = WF

Define Kinetic Energy K: K = 1/2mv2

K2 - K1 = WF

WF = K (Work kinetic-energy theorem)(Work kinetic-energy theorem)

xx

FFv1 v2

aa

ii

m


Work Kinetic-Energy Theorem:Work Kinetic-Energy Theorem:

{NetNet WorkWork done on object}

=

{changechange in kinetic energy kinetic energy of object}

This is true in general:

K1

K2

FFnet

dSdS

W K K K mv mvnet 2 1 22

121

2

1

2


Work done by Variable Force: (1D)Work done by Variable Force: (1D)

When the force was constant, we wrote W = Fxarea under F vs x plot:

For variable force, we find the areaby integrating:dW = F(x) dx. F

x

Wg

x

W F x dxx

x

( )

1

2

F(x)

x1 x2 dx


A simple application:A simple application:Work done by gravity on a falling objectWork done by gravity on a falling object

What is the speed of an object after falling a distance H, assuming it starts at rest ?

Wg = FF.S S = mgScos(0) = mgH

Wg = mgH

Work Kinetic-Energy Theorem:Work Kinetic-Energy Theorem:

Wg = mgH = 1/2mv2

SSmg g

H

j j

v0 = 0

v v gH 2


Conservation of EnergyConservation of Energy

If only conservative forces are present, the total energy If only conservative forces are present, the total energy (sum of potential and kinetic energies) of a system(sum of potential and kinetic energies) of a system is is conserved (i.e. constant).conserved (i.e. constant).

E = K + U is constantconstant !!!!!!

Both K and U can change as long as E = K + U is constant.


Example: The simple pendulum.Example: The simple pendulum.

Suppose we release a bob or mass m from rest a distance h1 above it’s lowest possible point.What is the maximum speed of the bob and where

does this happen ?To what height h2 does it rise on the other side ?

v

h1 h2

m

See example A Pendulum



Energy is conserved since gravity is a conservative force (E = K + U is constant)

Choose y = 0 at the original position of the bob, and U = 0 at y = 0 (arbitrary choice).

E = 1/2mv2 + mgy.

v

h1 h2

y

y=0

See example , A Pendulum



E = 1/2mv2 + mgy.Initially, y = 0 and v = 0, so E = 0.Since E = 0 initially, E = 0 always since energy is conserved.

y

y=0




E = 1/2mv2 + mgy. So at y = -h, E = 1/2mv2 - mgh = 0. 1/2mv2 = mgh 1/2mv2 will be maximum when mgh is minimum. 1/2mv2 will be maximum at the bottom of the swing !

y

y=0y=-h

h




1/2mv2 will be maximum at the bottom of the swing ! So at y = -h1 1/2mv2 = mgh1 v2 = 2gh1

v

h1

y

y=0

y=-h1

v gh 2 1




Since 1/2mv2 - mgh = 0 it is clear that the maximum height on the other side will be at y = 0 and v = 0.

The ball returns to it’s original height.

y

y=0




The ball will oscillate back and forth. The limits on it’s height and speed are a consequence of the sharing of energy between K and U.

E = 1/2mv2 + mgy = K + U = 0.

y

See example A Pendulum


Vertical Springs and HOOKE’S LAWVertical Springs and HOOKE’S LAW

A spring is hung vertically, it’s relaxed position at y=0 (a). When a mass m is hung from it’s end, the new equilibrium position is yE (b).

Hook’s Law relates the force exerted by the spring with the elongation of the spring

Force exerted by the spring is directly proportional to it’s elongation from it’s resting position

F=-kx(negative sign shows that the force is in the opposite direction of the force)

F=mg when spring is elongated and nonmoving so that mg=kx

x = 0

X=xf

j j

k

m

(a) (b)

mg


Vertical SpringsVertical Springs

If we choose x = 0 to be at the equilibrium position of the mass hanging on the spring, we can define the potential in the simple form.

Notice that g does not appear in this expression !!By choosing our coordinates and constants cleverly, we can hide

the effects of gravity.

x = 0

j j

k

m

(a) (b)

2

2

1.. kxEP


1-D Variable Force Example: Spring1-D Variable Force Example: Spring

For a spring we know that Fx = -kx.

F(x) x2

x

x1

-kxequilibrium

F = - k x1

F = - k x2


Spring...Spring...

The work done by the spring Ws during a displacement from x1 to x2 is the area under the F(x) vs x plot between x1 and x2.

Ws

F(x) x2

x

x1

-kxequilibrium


Spring...Spring...

21

22

2

2

1

2

1

)(

)(

2

1

2

1

2

1

xxk

kx

dxkx

dxxFW

x

x

x

x

x

x

s

F(x) x2

Ws

x

x1

-kx

The work done by the spring Ws during a displacement from x1 to x2 is the area under the F(x) vs x plot between x1 and x2.


Non-conservative Forces:Non-conservative Forces:

If the work done does not depend on the path taken, the force involved is said to be conservative.

If the work done does depend on the path taken, the force involved is said to be non-conservative.

An example of a non-conservative force is friction:

Pushing a box across the floor, the amount of work that is done by friction depends on the path taken.Work done is proportional to the length of the path !


Non-conservative Forces: FrictionNon-conservative Forces: Friction

Suppose you are pushing a box across a flat floor. The mass of the box is m and the kinetic coefficient of friction is . The work done in pushing it a distance D is given by:

Wf = FFf ..DD = -mgD.

D

Ff = -mg


Non-conservative Forces: FrictionNon-conservative Forces: Friction

Since the force is constant in magnitude, and opposite in direction to the displacement, the work done in pushing the box through an arbitrary path of length L is just Wf = -mgL.

Clearly, the work done depends on the path taken.

Wpath 2 > Wpath 1.

A

B

path 1

path 2

See text: 8-6


Generalized Work Energy Theorem:Generalized Work Energy Theorem:

Suppose FNET = FC + FNC (sum of conservative and non-conservative forces).

The total work done is: WTOT = WC + WNC

The Work Kinetic-Energy theorem says that: WTOT = K.

WTOT = WC + WNC = K

But WC = -U

So WNC = K + U = E or WNC = Ei - Ef


Generalized Work Energy Theorem:Generalized Work Energy Theorem:

The change in total energy of a system is equal to the work done on it by non-conservative forces. E of system not conserved ! Or the Potential Energy + Kinetic Energy + Internal Energy is a constant equal to the Total Energy

If all the forces are conservative, we know that energy is conserved: K + U = E = 0 which says that WNC = 0,which makes sense.

If some non-conservative force (like friction) does work,energy will not be conserved by an amount equal to this work, which also makes sense.

WNC = K + U = E

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Physics II Today’s Agenda