12 - potential energy.pptx

7/29/2019 12 - Potential Energy.pptx

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Potential Energy and Energy

Conservation

Ms. Mikaela Fudolig

Physics 71


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Work and Energy

Recall:

Nonzero total work done change in kinetic

energy

This lesson:

Conservative forces

Potential Energy

Work and Potential Energy


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Potential Energy

f

i

x

F

x

U W F dx

stored energy due to interactions

associated with conservative forces

The potential energy associated with a

conservative force F is


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Potential Energy

FW U

stored energy due to interactions

associated with conservative forces

The potential energy associated with a

conservative force F is


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Conservative Forces

Only conservative forces have

associated potential energies.

A force is conservative ifthe work done on a particle starting from

point A and ending at point B is independent

of the path taken;the work done on a particle starting and

ending at the same point A (aka return trip)

is zero.


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An example of a nonconservative

force: Sliding friction

A box of mass m is pushed across a rough

horizontal surface, and then pulled back to

its original position. If the coefficient of

kinetic friction is k, what is the work doneby friction on the box?


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An example of a conservative

force: Weight

A box slides a distance lon a frictionless

table before it slides off the table and falls

a vertical distance h before reaching the

floor.

What is the work done by gravity on the box?


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An example of a conservative

force: Weight

If the box had instead slid down a

frictionless ramp with length s and inclined

at an angle =tan-1(h/l)

What is the work done by gravity on the box?


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Gravitational potential energy

Potential energy associated with weight

0

U mgy U


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Gravitational potential energy

U0 is usually unimportant because we are

mostly concerned with the CHANGE in

potential energy.

We usually set U=0 at y=0.

U mgySIGN of y isIMPORTANT!


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Exercise 1

An object is thrown upwards at t=0 and

reaches a height hat time t=t. What is the

change in the gravitational potential

energy of the object-earth system?


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Exercise 2

An object is thrown from the ground at an

angle =30 above the horizontal. At a

later time t, it reaches a height h above the

ground. What is the change in thegravitational potential energy of the object-

earth system?


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Conceptual Exercise 1

Two mountaineers wish to reach the

summit of Mt. Everest using two different

paths. Edmund uses the steeper path,

while Leo uses the longer but less steeppath. If they both started at ground level,

who gains more gravitational potential

energy upon reaching the summit?Assume that Edmund and Leo have the

same mass.


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Conceptual Exercise 2

Two friends are going down a flight of

stairs to the foodcourt. They both start

from the same level, but Mark goes down

the stairs twice as fast as Ron. Once bothMark and Ron are on the foodcourt, who

has gained more gravitational potential

energy? Assume that Mark and Ron havethe same mass.


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Potential Energy

The change in gravitational potential

energy is INDEPENDENT OF THE PATH

TAKEN as long as the initial and final

elevations are the same.


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Exercise 3

The gravitational potential energy of an

object changes by6J.

Does the elevation of the object increase,

decrease, or remain the same?

What is the work done by the gravitational force

on the object?

FW U


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Another example of a

conservative force: springs

Recall Hookes law: the force exerted by a

spring on an object is

xF kx

This force is also conservative.


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Elastic potential energy

potential energy associated with spring-

mass systems

unless otherwise indicated, we assume that the

mass is attached to the spring


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Elastic potential energy

The elastic potential energy of a spring-mass

system is

21

2U kx

x= length by whichthe spring iscompressed (x0)x=0 -> unstretchedposition

k= spring constant


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Work done by a spring

The work done by the spring on the object is

then given by:

FW U

2 21 12 2

F f iW kx kx


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Work done by a spring

The work done by the spring on the object is

then given by:

2 21 1

2 2F i fW kx kx


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Exercise 4

A force of 720N stretches a certain spring

a distance of 0.150m. What is the potential

energy of the spring when a 60.0-kg mass

hangs vertically from it?


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Work-Energy theorem

a statement of the conservation ofTOTAL

energy

is NOT really different from the work-KE

theorem


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Work-Energy theorem

Suppose that the following forces act on an

object:

weight

restoring force of a spring

F1, F2, , Fn[collectively called others]

Therefore, the total work done on an object

can be written as

1 2...

ntot grav el F F F W W W W W W


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Work-Energy theorem

Defining the quantity Wothers as

tot grav el othersW W W W

1 2 ... nothers F F F W W W W We get


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Work-Energy theorem

From the definition of potential energy:

el el W U grav gravW U

tot grav el othersW U U W

So the total work on the object can bewritten as:


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Work-Energy theorem

tot grav el othersW U U W

But the work-KE theorem gives us

totW K When we combine this with


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Work-Energy theorem

grav el othersK U U W

We get the work-energy theorem


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Work-Energy theorem


Rewriting:

work done by all otherforces, conservative ornonconservative


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Steps in solving problems involving

work and energy

1. Identify the initial and final states of the

object.

2. Get Ugrav,i, Ugrav,f, U el,i, U el,fGet U

3. Get Kf, Ki.

Get K.


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Steps in solving problems

involving work and energy

4. Are there other forces (F1,F2, , Fn)

aside from weight and the restoring

force of a spring?Get Wothers=WF1 + WF2 + ... + WFn

5. Use the work-energy theorem to find

the unknowns.



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Exercise 5

A skier starts at rest from a height h abovethe bottom of a hill. Neglecting friction:Find the speed of the skier upon reaching the

bottom of the hill.If the skier had undergone freefall, initially from

rest at a height h above the ground, what is hisspeed upon reaching the bottom of the hill?


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Exercise 6

A box rests on a ramp inclined at an angle

. Initially, it is pushed against a spring

with spring constant k. The spring is

compressed by a distance s. At thisconfiguration, the box, which is NOT

attached to the spring, is at a height h

above the ground.What is the speed of the box upon reaching the

ground?


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Exercise 6

A box with mass m=12kg slides up a

rough ramp inclined at an angle =30

above the horizontal. It is given an initial

speed v0=5m/s at the bottom of the

ramp. If the box slides s=1.6m up the

ramp before it comes to a stop,

what is the work done by friction on the box?

what is the magnitude of the frictional force

acting on the box?


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Exercise 7


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Exercise 7

A 5.00-kg block is moving at v0=6.00m/s alonga frictionless horizontal surface. It then hits anunstretched spring with spring constantk=500N/m that is attached to a wall. The spring

thus compresses until the block stopsmomentarily (afterwards it will move in theopposite direction away from the spring, andthe spring will return to its unstretched

position). By how much did the spring compress?


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Exercise 8


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Exercise 8

A system of two paint buckets connected

by a lightweight rope is released from rest

with the 12.0-kg bucket 2.00m above the

floor. Use the principle of conservation ofenergy to find the speed with which this

bucket strikes the floor. Ignore friction and

the inertia of the pulley.


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Some tips

It might be helpful to split the motion of the

object into two parts, in case

nonconservative forces appear in between

the initial time ti and the final time tf.


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Force and Potential Energy

Consider a particle moving along the x-axisacted upon by a single force with x-

component Fx(x).

The work done by Fx(x) on the particle thatmoves with displacement x is:

( )xW F x x

But if the force Fx is conservative:

W U ( )xF x x U


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( )xF x x U

( )xU

F xx

( )( )x

dU xF x

dx


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Assume that only F(x) acts on the ibject,and F(x) = FX(x).Equilibrium occurs when

( ) 0xF x Equilibrium is achieved when U(x) is anyof the following:

a minimum a maximum

an inflection point


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Stable Equilibrium

U(x) is concave UP.

where will the particle

be in equilibrium?

U(x)

x

In stable equilibrium,a smalldisplacementresults in a restoring

force that brings backthe particle toequilibrium.


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Stable Equilibrium

U(x)

x

Any MINIMUM in aU vs. x graph is a

STABLE equilibriumpoint.


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Unstable Equilibrium

U(x) is concave DOWN.

where will the particle be

in equilibrium?

U(x)

xIn unstableequilibrium, a smalldisplacement results in

a force that pushes theparticle away fromequilibrium.


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U(x)

x

Unstable Equilibrium

Any MAXIMUM in aU vs. x graph is an

UNSTABLEequilibrium point.


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Neutral Equilibrium

Neither a maximum nor

a minimum, but the

SLOPE of U(x) is ZERO.

Where is the equilibrium

point?

A smalldisplacement

results in zero force andthe particle remains in

equilibrium.

U(x)

x


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Exercise 7: 1D motion (x-axis)

Identify theequilibrium points inthe potential energy

curve shown.Classify them as

stable, unstable, orneutral.

What is the forceexerted on theparticle at theequilibrium points?

12 - potential energy.pptx

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