Physics 201: Lecture 13, Pg 1 Lecture 13 l Goals Introduce concepts of Kinetic and Potential energy...

Physics 201: Lecture 13, Pg 1

Lecture 13 GoalsGoals

Introduce concepts of Kinetic and Potential energy

Develop Energy diagrams

Relate Potential energy to the external net force

Discuss Energy Transfer and Energy Conservation

Define and introduce power (energy per time)


Conservative vs. Non-Conservative forces

For a spring one can perform negative work but then reverse this process and recover all of this energy.

A compressed spring has the ability to do work For a Hooke’s law spring the work done is

independent of path The spring is said to be a conservative force

In the case of friction there is no immediate way to back transfer the energy of motion

In this case the work done can be shown to be dependent on path

Friction is said to be a non-conservative force


Hidden energy is Potential Energy (U)

For the compressed spring the energy is “hidden” but still has the ability to do work (i.e., allow for energy transfer)

This kind of “energy” is called “Potential Energy”

221 )()( eqspring xxkxU

The gravitation force, if constant, has the same properties.

)()( 0yymgyU gravity


Mechanical Energy (Kinetic + Potential)

If only “conservative” forces, then total mechanical energy If only “conservative” forces, then total mechanical energy

((potential U plus kinetic K energy) of a system) of a system is is conservedconserved

For an object in a gravitational “field”

K and U may change, but Emech = K + U remains a fixed value.

Emech = K + U = constant

Emech is called “mechanical energy”

Emech = K + U

K ≡ ½ mv2 U ≡ mgy

½ m vyi2 + mgyi = ½ m vyf

2 + mgyf = constant


Example of a conservative system: The simple pendulum.

Suppose we release a mass m from rest a distance h1 above its lowest possible point.

What is the maximum speed of the mass and where does this happen ?

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

v

h1 h2

m


Example: The simple pendulum.

y

y=0

y=h1


E = K + U = constant and so K is maximum when U is a minimum.



v

h1

y

y=h1

y=0


E = K + U = constant and so K is maximum when U is a minimum

E = mgh1 at top

E = mgh1 = ½ mv2 at bottom of the swing



y

y=h1=h2

y=0

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

E = K + U = constant and so when U is maximum again (when K = 0) it will be at its highest point.

E = mgh1 = mgh2 or h1 = h2


Potential Energy, Energy Transfer and Path

A ball of mass m, initially at rest, is released and follows three difference paths. All surfaces are frictionless

1. The ball is dropped

2. The ball slides down a straight incline

3. The ball slides down a curved incline

After traveling a vertical distance h, how do the three speeds compare?

(A) 1 > 2 > 3 (B) 3 > 2 > 1 (C) 3 = 2 = 1 (D) Can’t tell

h

1 32


Energy diagrams In general:

Ene

rgy

K

y

U

Emech

Ene

rgy

K

u = x - xeq

U

Emech

Spring/Mass systemBall falling

00


Conservative Forces & Potential Energy

For any conservative force F we can define a potential energy function U in the following way:

The work done by a conservative force is equal and opposite to the change in the potential energy function.

This can be written as:

W = F ·dr = - U

U = Uf - Ui = - W = - F • drri

rf

ri

rf Uf

Ui


Conservative Forces and Potential Energy

So we can also describe work and changes in potential energy (for conservative forces)

U = - W Recalling (if 1D)

W = Fx x

Combining these two,

U = - Fx x

Letting small quantities go to infinitesimals,

dU = - Fx dx

Or,

Fx = -dU / dx


Equilibrium

Example Spring: Fx = 0 => dU / dx = 0 for x=xeq

The spring is in equilibrium position

In general: dU / dx = 0 for ANY function establishes equilibrium

stable equilibrium unstable equilibrium

U U


Chapter 8 Conservation of Energy

Examples of energy transfer Work (by conservative or non-conservative forces) Heat (thermal energy transfer)

ansfer)(energy tr system TE

intsystem EUKE A “system” depends on situation Example: A can with internal non-conservative forces

In general, if just one external force acting on a system

WK


“Mechanical” Energy of a System…agai n

UKE mech

Isolated system without non-conservative forces

0 ifif UUKK

0mech UKE

constant iiff UKUK


ExampleThe Loop-the-Loop … again

To complete the loop the loop, how high do we have to let the release the car?

Condition for completing the loop the loop: Circular motion at the top of the loop (ac = v2 / R)

Exploit the fact that E = U + K = constant ! (frictionless)

(A) 2R (B) 3R (C) 5/2 R (D) 23/2 R

h ?

R

Car has mass mRecall that “g” is the source of

the centripetal acceleration and N just goes to zero is the limiting case.

Also recall the minimum speed at the top is gRv

Ub=mgh

U=mg2R

y=0U=0


ExampleThe Loop-the-Loop … again

Use E = K + U = constant mgh + 0 = mg 2R + ½ mv2

mgh = mg 2R + ½ mgR = 5/2 mgR

h = 5/2 R

R

gRv

h ?


With non-conservative forces

Mechanical energy is not conserved.

CNCmech WWUKE


An experimentTwo blocks are connected on the table as shown. The table

has a kinetic friction coefficient of k. The masses start at

rest and m1 falls a distance d. How fast is m2 going?

T

m1

m2

m2g

N

m1g

T

fk

Mass 1

Fy = m1ay = T – m1g

Mass 2

Fx = m2ax = -T + fk = -T + k N

Fy = 0 = N – m2g

| ay | = | ay | = a =(km2 - m1) / (m1 + m2)

2ad = v2 =2(km2 - m1) g / (m1 + m2)

K= - km2gd – Td + Td + m1gd = ½ m1v2+ ½ m2v2

v2 =2(km2 - m1) g / (m1 + m2)


Energy conservation for a Hooke’s Law spring

Associate ½ kx2 with the “potential energy” of the spring m

2212

212

212

21 ffii mvkxmvkx

fsfisiU K UK

Ideal Hooke’s Law springs are conservative so the mechanical energy is constant


Energy (with spring & gravity)

Emech = constant (only conservative forces)

At 1: y1 = h ; v1y = 0 At 2: y2 = 0 ; v2y = ? At 3: y3 = -x ; v3 = 0

Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2

Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0

1

32

h

0-x

mass: m

Given m, g, h & k,

how much does the spring compress?



Emech = constant (only conservative forces) At 1: y1 = h ; v1y = 0 At 2: y2 = 0 ; v2y = ? At 3: y3 = -x ; v3 = 0 Em1 = Ug1 + Us1 + K1 = mgh + 0 + 0 Em2 = Ug2 + Us2 + K2 = 0 + 0 + ½ mv2

Em3 = Ug3 + Us3 + K3 = -mgx + ½ kx2 + 0

Given m, g, h & k, how much does the spring compress? Em1 = Em3 = mgh = -mgx + ½ kx2 Solve ½ kx2 – mgx +mgh = 0

1

32

h

0-x

mass: m

Given m, g, h & k, how much does the spring compress?



When is the child’s speed greatest?

(A) At y1 (top of jump)

(B) Between y1 & y2

(C) At y2 (child first contacts spring)

(D) Between y2 & y3

(E) At y3 (maximum spring compression)

1

32

h

0-x

mass: m


Work & Power:

Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the same mass.

Assuming identical friction, both engines do the same amount of work to get up the hill.

Are the cars essentially the same ? NO. The Corvette can get up the hill quicker It has a more powerful engine.


Work & Power: Power is the rate at which work is done. Average Power is,

Instantaneous Power is,

If force constant in 1D, W= F x = F (v0 t + ½ at2)

and P = W / t = F (v0 + at)

dtdE

P

t

WP

avg

vFdtrd

FdtdW

P

1 W = 1 J / 1s


Exercise Work & Power

A. Top

B. Middle

C. Bottom

Starting from rest, a car drives up a hill at constant acceleration and then suddenly stops at the top.

The instantaneous power delivered by the engine during this drive looks like which of the following,

Z3

timePow

erP

owe r

Pow

e r

time

time


Exercise Work & Power

P = dW / dt and W = F d = ( mg cos mg sin dand d = ½ a t2 (constant accelation)So W = F ½ a t2 P = F a t = F v

(A)

(B)

(C)

Z3

timePow

erP

owe r

Pow

e r

time

time


Work & Power:

Power is the rate at which work is done.

A person of mass 80.0 kg walks up to 3rd floor (12.0m). If he/she climbs in 20.0 sec what is the average power used.

Pavg = F h / t = mgh / t = 80.0 x 9.80 x 12.0 / 20.0 W P = 470. W

Example:

tW

P

avg


Recap

Read through Chap 9.1-9.4

Physics 201: Lecture 13, Pg 1 Lecture 13 l Goals Introduce concepts of Kinetic and Potential energy...

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Transcript of Physics 201: Lecture 13, Pg 1 Lecture 13 l Goals Introduce concepts of Kinetic and Potential energy...