physics 218, lecture xiii1 physics 218 lecture 13 dr. david toback

Physics 218, Lecture XIII 1

Physics 218Lecture 13Dr. David Toback

Physics 218, Lecture XIII 2

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

•Things that were due Last Monday:–Chap 5&6 turned in on WebCT

•Things that were due for Wednesday’s Recitation:–Problems from Chap 7

•Things due for this coming Monday:–Problems from Chap 7 on WebCT–Chaps 5&6 if you haven’t done them already

Physics 218, Lecture XIII 3

The ScheduleThis week: (2/25) • HW on Chaps 5&6 on WebCT• 3rd and 4th lectures (of six) on Chapters 7, 8 & 9• Chapter 7 in recitationNext week: (3/3) • Chapter 7 due in WebCT• 5th and 6th lectures (of six) on Chapters 7, 8 & 9• Chapter 8 in recitation Following week: (3/10) Spring Break!!!Following Week: (3/17)• Chapter 8 due in WebCT• Reading for Chapters 10 & 11• Lecture on Chapters 10 & 11• Chapter 9 and Exam 2 Review in recitation Following Week: (3/24)• Chapter 9 due in WebCT• Exam 2 on Tuesday• Recitation on Chapters 10 & 11• Reading for Chapters 12 & 13 for Thursday• Lecture 12 & 13 on Thursday

Physics 218, Lecture XIII 4

Last time:– Work and Energy– The Work-Energy relationship

This time and next time:– Potential Energy– Conservation of Mechanical Energy

– Conservation of Energy– Lots of problems

Chapters 7, 8 & 9 Cont

Physics 218, Lecture XIII 5

Physics 218, Lecture XIII 6

Different Style Than the Textbook

I like teaching this material using a different style than the textbook

1.Teach you the concepts2.Give you the important

equations3.Then we’ll do lots of

problems

Physics 218, Lecture XIII 7

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

Physics 218, Lecture XIII 8

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

Physics 218, Lecture XIII 9

Other Potential Energies: Springs

Last week we calculated that it took ½kx2 of work to compress a spring by a distance xHow much potential energy does it now how have?U(x) = ½kx2

Physics 218, Lecture XIII 10

Force and Potential EnergyIf we know the potential energy, U,

we can find the force

This makes sense… For example, the force of gravity points down, but the potential increases as you go up

dxdU

xF

Physics 218, Lecture XIII 11

Force and Potential Energy

Draw some examples…

–Gravity–Spring

Physics 218, Lecture XIII 12

Mechanical Energy

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

•Define E≡K+U

Physics 218, Lecture XIII 13

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

Physics 218, Lecture XIII 14

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

Physics 218, Lecture XIII 15

Problem Solving

For Conservation of Energy problems:

BEFORE and AFTER diagrams

Physics 218, Lecture XIII 16

Conservation of Energy Problems

Before…

Physics 218, Lecture XIII 17

After

Physics 218, Lecture XIII 18

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?

Physics 218, Lecture XIII 19

Potential EnergyA brick held 6 feet in the air has potential energy

•Subtlety: Gravitational potential energy is relative to somewhere!

Example: What is the potential energy of a book 6 feet above a 4 foot high table? 10 feet above the floor?

• U = U2-U1 = Wext = mg (h2-h1)•Write U = mgh•U=mgh + Const

Only change in potential energy is really meaningful

Physics 218, Lecture XIII 20

Z Z

Before After

C

Falling onto a Spring

We want to measure the spring constant of a certain spring. We drop a ball of known mass m from a known height Z above the uncompressed spring. Observe it compresses a distance C.

What is the spring constant?

Physics 218, Lecture XIII 21

Quick Problem

A refrigerator with mass M and speed V0 is sliding on a dirty floor with coefficient of friction .

Is mechanical energy conserved?

Physics 218, Lecture XIII 22

Non-Conservative Forces•We’ve talked about three different types of forces:

1.Gravity: Conserves mechanical energy

2.Normal Force: Conserves mechanical energy (doesn’t do work)

3.Friction: Doesn’t conserve mechanical energy

•Since Friction causes us to lose mechanical energy (doesn’t conserve mechanical energy) it is a Non-Conservative force!

Physics 218, Lecture XIII 23

Law of Conservation of Energy

• Mechanical Energy NOT always conserved

• If you’ve ever watched a roller coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc.

• Energy = Kinetic Energy + Potential Energy + Heat + Others…

–Total Energy is what is conserved!

Physics 218, Lecture XIII 24

Conservative ForcesIf there are only conservative forces in the

problem, then there is conservation of mechanical energy

• Conservative: Can go back and forth along any path and the potential energy and kinetic energy keep turning into one another– Good examples: Gravity and Springs

• Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost.– Good example: Friction (like on Roller

Coasters)

Physics 218, Lecture XIII 25

Law of Conservation of Energy

•Even if there is friction, Energy is conserved

•Friction does work– Can turn the energy into heat– Changes the kinetic energy

•Total Energy = Kinetic Energy + Potential Energy + Heat + Others…

– This is what is conserved•Can use “lost” mechanical energy to estimate things about friction

Physics 218, Lecture XIII 26

Roller Coaster with FrictionA roller coaster of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2).

Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

Physics 218, Lecture XIII 27

Energy SummaryIf there is net work on an object, it

changes the kinetic energy of the object (Gravity forces a ball falling from height h to speed up Work done.)

Wnet = KIf there is a change in the potential

energy, some one had to do some work: (Ball falling from height h speeds up→ work done → loss of potential energy. I raise a ball up, I do work which turns into potential energy for the ball)

UTotal = WPerson =-WGravity

Physics 218, Lecture XIII 28

Energy Summary

If work is done by a non-conservative force it does negative work (slows something down), and we get heat, light, sound etc.

EHeat+Light+Sound.. = -WNC

If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost)

K1+U1 = K2+U2+EHeat…

K1+U1 = K2+U2-WNC

Physics 218, Lecture XIII 29

Friction and SpringsA block of mass m is traveling on a rough surface. It reaches a spring (spring constant k) with speed Vo and compresses it a total distance D. Determine

Physics 218, Lecture XIII 30

l

l

Bungee JumpYou are standing on a

platform high in the air with a bungee cord (spring constant k) strapped to your leg. You have mass m and jump off the platform.

1.How far does the cord stretch, l in the picture?

2.What is the equilibrium point around which you will bounce?

Physics 218, Lecture XIII 31

Coming up… •Lectures:

– Last lectures on Chaps 7, 8 and 9•HW due in WebCT on Monday

– Chapter 7•Reading for Lecture next week

– Chaps 10 & 11: Momentum•Recitation next week

– Chapter 8

Physics 218, Lecture XIII 32

Physics 218, Lecture XIII 33

Roller CoasterYou are in a roller coaster car of mass

M that starts at the top, height Z, with an initial speed V0=0. Assume no friction.

a)What is the speed at the bottom?b)How high will it go again?

c)Would it go as high if there were friction?

Z

Physics 218, Lecture XIII 34

Energy•Potential Energy & Conservation of Energy problems

•The relationship between potential energy and Force

•Energy diagrams and Equilibrium

Physics 218, Lecture XIII 35

Energy Review

If there is net work on an object, it changes the kinetic energy of the object (Gravity forces a ball falling from height h to speed up Work done.)

Wnet = KIf there is a change in the potential energy, some one had to do some work: (Ball falling from height h speeds up→ work done → loss of potential energy. I raise a ball up, I do work which turns into potential energy for the ball)

UTotal = WPerson =-WGravity

Physics 218, Lecture XIII 36

Energy Review

If work is done by a non-conservative force it is negative work (slows something down), and we get heat, light, sound etc.

EHeat+Light+Sound.. = -WNC

If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost)

K1+U1 = K2+U2+EHeat…

K1+U1 = K2+U2-WNC

Physics 218, Lecture XIII 37

Potential Energy Diagrams• For Conservative

forces can draw energy diagrams

• Equilibrium points

– Motion will move “around” the equilibrium

– If placed there with no energy, will just stay (no force) 0F dx

dUx

Physics 218, Lecture XIII 38

Stable vs. Unstable Equilibrium Points

The force is zero at both maxima and minima but…

– If I put a ball with no velocity there would it stay?

– What if it had a little bit of velocity?

Physics 218, Lecture XIII 39

Roller Coaster with FrictionA roller coaster car of mass m starts at rest

at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2).

Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

Physics 218, Lecture XIII 40

Roller Coaster with FrictionA roller coaster car of mass m starts at rest

at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2). An odometer tells us that the total scalar distance traveled is d.

Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

Assuming that the magnitude and angle of the force of friction, F, between the car and the track is constant, find |F|.

Physics 218, Lecture XIII 41

Bungee JumpA jumper of mass m

sits on a platform attached to a bungee cord with spring constant k. The cord has length l (it doesn’t stretch until it has reached this length).

How far does the cord stretch y?

l

Physics 218, Lecture XIII 42

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

Physics 218, Lecture XIII 43

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?

2

0

2

0 x3x

1F F(x)