Conservative Forces Nonconservative Forces

Lesson 5

Physics 168 Luis Anchordoqui

Tuesday, September 26, 17 1 Overview

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 2 Work Done by a Constant Force Overview distance moved times component of force in direction of displacement

W = Fd cos ✓

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 3 Work Done by a Constant Force (Cont’d) Overview In SI system ➤ units of work are joules: 1J = 1 N m · As long as this person does not lift or lower bag of groceries he is doing no work on it

Force he exerts has no component in direction of motion

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 4 Work Done by a Constant Force

WhatOverview about centripetal forces?

Centripetal forces do no work as they are always perpendicular to direction of motion

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 5 Kinetic Energy and Work-Energy Principle If constant net forceOverview acts on particle that moves along x axis Newton’s second law leads to

Fnet,x = max If net force is constant ☛ acceleration is constant 2 2 convince yourself that ☛ vf = vi +2ax x

Solving for ax 1 a = (v2 v2) x 2x f i ⬇ 1 1 F x = mv2 mv2 net,x 2 f 2 i ⬇ 1 1 W = mv2 mv2 net 2 f 2 i

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 6 Kinetic Energy and Work-Energy Principle If we write accelerationOverview in terms of velocity and distance we find that work done is

1 1 W = mv2 mv2 net 2 2 2 1

1 2 We define kinetic energy ☛ KE = mv 2

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 7 Kinetic Energy and Work-Energy Principle Overview Work done is equal to change in kinetic energy

Wnet =KE

If net work

Is positive Is negative kinetic energy increases kinetic energy decreases

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 8 Work Done by a Varying Force For varyingOverview force work can be approximated by dividing distance up into small pieces finding work done during each and adding them up

As pieces become very narrow work done is area under force vs. distance curve

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 9 Work done on a variable force – straight line motion Overview

x2 W =lim Fxi xi W = F dx xi 0 x ! i X Zx1 Replacing for Fx = max x2 x2 d2x x2 dv W = madx = m dx = m dx dt2 dt Zx1 Zx1 Zx1 v2 1 2 2 W = mvdv = m v2 v1 v 2 Z 1 ⇣ ⌘ C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 10 Potential Energy Overview

An object can have potential energy by virtue of its surroundings

A wound-up spring

Familiar examples A stretched elastic band of potential energy

An object at some height above ground

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 11 Gravitational Potential Energy Overview In raising a mass m to a height h work done by external force is

Wext = Fext d cos 0 = mgh = mg(y y ) 2 1

We therefore define gravitational potential energy

PEgrav = mgy

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 12 Gravitational Potential Energy (cont’d) Overview

This potential energy can become kinetic energy if object is dropped

Potential energy is a property of a system as a whole not just of object (because it depends on external forces)

If PEgrav = mgy ☛ where do we measure y from?

It turns out not to matter as long as we are consistent about where we choose y =0 Only changes in potential energy can be measured

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 13 Elastic Potential Energy Overview Potential energy can also be stored in a spring when it is compressed

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 14 Elastic Potential Energy (cont’d) Overview Potential energy can also be stored in a spring when it is compressed

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 15 Potential Energy (cont’d) Overview Potential energy can also be stored in a spring when it is compressed

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 16 Elastic Potential Energy (cont’d) Force required to compressOverview or stretch a spring is

F = kx s spring constant k needs to be measured for each spring

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 17 Elastic Potential Energy (cont’d) Force increases as springOverview is stretched or compressed further Potential energy of compressed or stretched spring measured from its equilibrium position ☛

xf xf 2 2 xf xi Wby spring = Fx ddxx = k xdx = k x x 2 2 Z i Z i ⇣ ⌘

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 18 Scalar product Consider a particle movingOverview along an arbitrary curve

Component F is related to angle (between directions of F ~ and dl ~ ) by k F = F~ cos k Work done by F for displacement dl ~ is dW = F dl = F cos dl k This combination of two vectors and cosine of angle between their directions is called scalar product Scalar product of two general vectors A~ and B ~ is A~ B~ = AB cos · ➤ angle between A ~ and B~

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 19 Properties of Scalar Products If Overview Then

A~ and B ~ are perpendicular ☛A~ B~ =0(because = 90, cos =10 ) · A~ and B~ are parallel ☛A~ B~ = AB (because =0, cos =1) · A~ B~ =0 ☛ A~ =0 or B~ =0 or A~ B~ · ? A~ A~ = A2 ~ · ☛ Because A is parallel to itself

A~ B~ = B~ A~ ☛ Commutative rule of multiplication · ·

(A~ + B~ ) C~ = A~ C~ + B~ C~ ☛ Distributive rule of multiplication · · ·

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 20 Scalar product (cont’d)

Product of A andOverview projection of B ~ on A ~ ➤ and vice versa

Scalar product is distributive over addition (A~ + B~ ) C~ = A~ C~ + B~ C~ · · ·

Rule of differentiating a scalar product is d dA~ dB~ (A~ B~ )= B~ + A~ dt · dt · · dt

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 21 Conservative and Nonconservative Forces Work done by a conservativeOverview force on a particle is independent of path taken as particles moves from one point from another

A force is conservative if work it does on a particle is zero when particle moves around any closed path returning to its initial position A force is said to be non conservative if it does not meet definition of conservative forces

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 22 Conservative and Nonconservative Forces If friction is present work done dependsOverview not only on starting and ending points but also on path taken

Friction ☛ nonconservative force

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 23 To calculate work done byOverview a force F around a closed curve (path) C we evaluate F~ d~l · IC Circle on integral means that integration is evaluated for one complete trip around C Calculate integral around closed path shown in figure if

F~ = Axˆı

Force is described by Hooke’s law Integral is zero as force for a spring is conservative

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 24 Conservative and Nonconservative Forces (cont’d) Overview We distinguish between: work done by conservative forces and work done by nonconservative forces

Work done by nonconservative forces is equal to

total change in kinetic and potential energies

WNC =KE +PE

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 25 Mechanical Energy and Its Conservation Overview If there are no nonconservative forces sum of changes in kinetic energy and in potential energy is zero

Kinetic and potential energy changes are equal but opposite in sign

Define total mechanical energy:

E = KE + PE

and its conservation: E2 = E1 = Constant

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 26 Problem Solving Using Conservation of Mechanical Energy

In leftOverview image total mechanical energy is: 1 E = KE + PE = mv2 + mgy 2

Energy buckets (right) show how energy moves from all potential to all kinetic

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 27 Problem Solving Using Conservation of Mechanical Energy

If there Overviewis no friction speed of a roller coaster will depend only on its height compared to its starting height

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 28 Work done on a skier You and your friend areOverview at a ski resort with two ski runs a beginner's run and an expert's run Both runs begin at top of ski lift and end at finish line at bottom of same lift Let h be vertical descent for both runs Beginner's run is longer and less steep than expert's run You and your friend, who is a much better skier than you, are testing some experimental frictionless skis To make things interesting you offer a wager that if she takes expert's run and you take beginner's run her speed at finish line will not be greater than your speed at finish line Forgetting that you study physics she accepts bet Conditions are that you both start from rest at top of lift and both of you coast for entire trip Who wins bet? (Assume air drag is negligible)

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 29 Work done on a skier (cont’d) Overview

1 1 W = mv2 mv2 total 2 f 2 i

Wtotal = WFn + WFg F d~` W =0 n ? ) Fn WFg = mgy KE = mgy v = 2gh ) f h Final speed depends only on which is same for both runsp Both of you will have same final speed YOU WIN!!

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 30 Bungee Jumping A 62 kg bungee jumperOverview jumps from a bridge He is tied to a bungee cord whose un-stretched length is L1 = 12 m and falls a total of L1 + L2 = 31 m Calculate ⓐ spring stiffness constant k of bungee cord, assuming Hooke's law applies ⓑ Calculate maximum acceleration he experiences ⓒ Calculate velocity just before cord is starting to stretch ⓒd Calculate the position of maximum velocity

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 31 ⓐ Conservation of energy Overview

x1 =0

x2 = L2

1 1 1 1 mv2 + mgy + kx2 = mv2 + mgy + kx2 2 0 0 2 0 2 2 2 2 2

1 2 2mgy0 mgy0 = kx2 k = 2 = 104.4N/m 2 ) x2

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 32 ⓑ Maximum acceleration occurs when bungee cord has maximum stretch Overview F = F mg = kx mg = ma net cord 2 kx a = 2 g = 22 m/s2 =2.2g m

v @ kx mg =0 x = mg/k =5.8m max )

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 33 Energy Conservation with Dissipative Processes Overview

If there is a nonconservative force such as friction where do kinetic and potential energies go?

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 34 Others Forms of Energy Overview

Some other forms of energy

Electric energy nuclear energy thermal energy chemical energy

Work is done when energy is transferred from one object to another

Accounting for all forms of energy we find that total energy neither increases nor decreases

Energy as a whole is conserved

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 35 Conservative and Nonconservative Forces Overview Potential energy can only be defined for conservative forces

Conservative forces Nonconservative forces

Gravitional Friction

Elastic Air resistance

Tension in cord

Motor or rocket propulsion

Push or pull by a person

C.Luis B.-Champagne Anchordoqui 2 Tuesday, September 26, 17 36