Physics 218, Lecture XI1 Physics 218 Lecture 11 Dr. David Toback

Physics 218, Lecture XI 1

Physics 218Lecture 11

Dr. David Toback


Checklist for Today•Things that were due Monday:

– Chaps. 3 and 4 HW on WebCT– Progress on 5&6 problems

•Things that were due Tuesday:– Reading for Chapter 7

•Things due for Wednesday’s Recitation:– Problems from Chap 5&6

•Things due for Today:– Read Chapters 7, 8 & 9

•Things due Monday– Chap 5&6 turned in on WebCT


Last time:• WorkThis time• More on Work• Work and Energy

– Work using the Work-Energy relationship

• Potential Energy• Conservation of Mechanical Energy

Chapters 7, 8 & 9 Cont




Work in Two DimensionsYou pull a crate of mass M a distance X along a horizontal floor with a constant force. Your pull has magnitude FP, and acts at an angle of . The floor is rough and has coefficient of friction .

Determine:•The work done by each force•The net work on the crate

X


What if the Force is changing direction?What if the Force is

changing magnitude?


What if the force or direction isn’t constant?

I exert a force over a distance for awhile, then exert a different force over a different distance (or direction) for awhile. Do this a number of times. How much work did I do?

Need to add up all the little

pieces of work!


Fancy sum notationIntegr

al

Find the work: CalculusTo find the total work, we must sum up all the little pieces of work (i.e., F.d). If the force is continually changing, then we have to take smaller and smaller lengths to add. In the limit, this sum becomes an integral.

b

a

xdF


Use an Integral for a Constant Force

Fd0FFd| FxFdx xdFW dx0x

d

0

d

0

Assume a constant Force, F, doing work in the same direction, starting at x=0 and continuing for a distance d. What is the work?

Region of integrationW=Fd


Non-Constant Force: Springs

•Springs are a good example of the types of problems we come back to over and over again!

•Hooke’s Law

•Force is NOT CONSTANT over a distance

Some constantDisplacement

xkF


Work done to stretch a Spring

How much work do you do to stretch a spring (spring constant k), at constant velocity, from x=0 to x=D?

D


Kinetic Energy and Work-Energy

• Energy is another big concept in physics

• If I do work, I’ve expended energy

– It takes energy to do work (I get tired)

• If net work is done on a stationary box it speeds up. It now has energy

• We say this box has “kinetic” energy! Think of it as Mechanical Energy or the Energy of Motion

Kinetic Energy = ½mV2


Work-Energy Relationship

•If net work has been done on an object, then it has a change in its kinetic energy (usually this means that the speed changes)

•Equivalent statement: If there is a change in kinetic energy then there has been net work on an objectCan use the change in energy

to calculate the work


Summary of equations

Kinetic Energy = ½mV2

W= KECan use change in speed to calculate the work, or

the work to calculate the speed


Multiple ways to calculate the work

doneMultiple ways to

calculate the velocity


Multiple ways to calculate work

1. If the force and direction is constant–F.d

2. If the force isn’t constant, or the angles change– Integrate

3. If we don’t know much about the forces–Use the change in kinetic energy


Multiple ways to calculate velocity

If we know the forces:•If the force is constant

–F=ma →V=V0+at, or V2-V02 = 2ad

•If the force isn’t constant

–Integrate the work, and look at the change in kinetic energy W= KE = KEf-KEi

= ½mVf2 -½mVi

2


Quick Problem

I can do work on an object and it doesn’t change the kinetic energy.

How? Example?


Problem Solving

How do you solve Work and Energy

problems? BEFORE and AFTER

Diagrams


Problem Solving

Before and After diagrams1.What’s going on

before work is done 2.What’s going on after

work is doneLook at the energy before and the energy after


Before…


After…


Compressing a SpringA horizontal spring has

spring constant k1.How much work must you

do to compress it from its uncompressed length (x=0) to a distance x=-D with no acceleration?

2.You then place a block of mass m against the compressed spring. Then you let go. Assuming no friction, what will be the speed of the block when it separates at x=0?


Potential Energy

• Things with potential: COULD do work–“This woman has great potential as an engineer!”

• Here we kinda mean the same thing

• E.g. Gravitation potential energy:

–If you lift up a brick it has the potential to do damage


Example: Gravity & Potential Energy

You lift up a brick (at rest) from the ground and then hold it at a height Z

•How much work has been done on the brick?

•How much work did you do?•If you let it go, how much work will be done by gravity by the time it hits the ground?

We say it has potential energy: U=mgZ

–Gravitational potential energy


Mechanical Energy

•We define the total mechanical energy in a system to be the kinetic energy plus the potential energy

•Define E≡K+U


Conservation of Mechanical Energy

• For some types of problems, Mechanical Energy is conserved (more on this next week)

• E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick

K2+U2 = K1+U1

Conservation of Mechanical EnergyE2=E1


Problem Solving• What are the types of examples

we’ll encounter?– Gravity– Things falling– Springs

• Converting their potential energy into kinetic energy and back again

E = K + U = ½mv2 + mgy


Problem Solving

For Conservation of Energy problems:

BEFORE and AFTER diagrams


Quick Problem

We drop a ball from a height D above the ground

Using Conservation of Energy, what is the speed just before it hits the ground?


Next Week• Reading for Next Time:

–Finish Chapters 7, 8 and 9 if you haven’t already

–Non-conservative forces & Energy

• Chapter 5&6 Due Monday on WebCT

• Start working on Chapter 7 for recitation next week



Compressing a SpringA horizontal spring has spring

constant k1.How much work must you do

to compress it from its uncompressed length (x=0) to a distance x= -D with no acceleration?

2.You then place a block of mass m against the compressed spring. Then you let go. Assuming no friction, what will be the speed of the block when it separates at x=0?

3.What is the speed if there is friction with coefficient ?


Roller CoasterA Roller Coaster of

mass M=1000kg starts at point A.

We set Y(A)=0. What is the potential energy at height A, U(A)?

What about at B and C?

What is the change in potential energy as we go from B to C?

If we set Y(C)=0, then what is the potential energy at A, B and C? Change from B to C


Kinetic EnergyTake a body at rest, with mass m, accelerate

for a while (say with constant force over a distance d). Do W=Fd=mad:

• V2- V02 = 2ad= V2

ad = ½V2

• W = F.d = (ma) .d= madad = ½V2

mad = ½ mV2

W = mad = ½ mV2

Kinetic Energy = ½ mV2


Work and Kinetic Energy

• If V0 not equal to 0 then

• V2 - V02 = 2ad

• W=F.d = mad = ½m (V2 - V02)

= ½mV2- ½mV02 = (Kinetic Energy)

W= KE

Net Work on an object (All forces)


A football is thrownA 145g football starts at rest and is

thrown with a speed of 25m/s.

1. What is the final kinetic energy?2. How much work was done to reach

this velocity?

We don’t know the forces exerted by the arm as a function of time, but this allows us to sum them all up to calculate the work


Example: Gravity

• Work by Gravity


Potential Energy in General

• Is the potential energy always equal to the work done on the object?– No, non-conservative forces– Other cases?

• What about for conservative forces?


Water Slide

Who hits the bottom with a faster speed?


Mechanical Energy

• Consider a Conservative System

• Wnet = K (work done ON an object)

UTotal = -Wnet Combine

K = Wnet = -UTotal

=> K + U = 0 Conservation of Energy


Conservation of Energy

• Define E=K+U

K + U = 0 => (K2-K1) +(U2-U1)=0

K2+U2 = K1+U1

Conservation of Mechanical EnergyE2=E1


Conservative vs. Non-Conservative Forces

• Nature likes to “conserve” certain types of things

• Keep them the same

• Kinda like conservative politicians

• Conservationists


Conservative Forces• Physics has the same meaning. Except

nature ENFORCES the conservation. It’s not optional, or to be fought for.

“A force is conservative if the work done by a force on an object moving from one point to another point depends only on the initial and final positions and is independent of the particular path taken”

• (We’ll see why we use this definition later)


Closed LoopsAnother definition: A force is conservative

if the net work done by the force on an object moving around any closed path is zero

This definition and the previous one give the same answer. Why?


Is Friction a Conservative Force?


002

21

002010011

00

10

0

01

010

0

21011

1

10

)()(

constant a is Where

xtvatX

xtv) a(x)(v) a(

dttvat dt vatX

vatV

v) a(

v) a(

a dt at a dt V

ttt

t

t

Integral Examples we know…


Work done to stretch a Spring

2

2X

0

X

0

2

X

0

P

X

X

kX2

1W

kX2

1xk

2

1dxkx

dxFdlF W

Person Wby Work X x to0 xfrom spring aStretch

f

i


Robot ArmA robot arm has a funny Force equation in 1-dimension

where F0 and X0 are constants.What is the work done to move a block from position X1 to position X2?

20

2

0 x

3x1F F(x)


Stretch a Spring

A person pulls on a spring and stretches it a from the equilibrium point for a total distance D. At this distance the force required to keep the spring stretched is F.

How much work is done on the spring in terms of the given variables?

Can we use F.d?

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Physics 218, Lecture XI1 Physics 218 Lecture 11 Dr. David Toback