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Work, Energy and Power

Chapter 6 READ!1TodayDefinition of WorkWhat is work?How does it relate to force?2All objects or systems have an energy associated with itE = Emotion +Egrav+ Eelastic + Ethermal+ Echem + Enuc + Enuc Energy can be transferred from the environment to an object or system in two waysWORK mechanical transfer of energy (apply force over a distance)HEAT nonmechanical transfer of energy (due to temperature difference)3Work When a force acts on an object over a distance, it is said that work was done upon the object. Work tells us how much a force transfers energy to a system.

dot productComponent of force parallel to displacementAngle between F and DdWork a bridge between force (a vector) and energy (a scalar)4Dot ProductThe product of the magnitudes of 2 vectors and the angle between them. The result is a SCALAR.

ABqResult is a scalar that measures how much of one vector lies along the direction of the other.Bcosq5Work

dot productComponent of force parallel to the displacementSI Unit of work (and Energy) is Joule: J = (N)(m)Work is a SCALAR6

Work QuestionIs this bellhop doing work on the suitcases as he walks forward at constant speed?DxFGFapp7

If a man lifts a 50 kg barbell 2 m, how much work does he do on the barbell? hFGFapp

8Example: Work done on a crate. A 50 kg crate is pulled 40 m along a horizontal floor by a constant force exerted by a, FP = 100 N, which acts at a 370 angle. The floor is rough and exerts a friction force fK = 50 N. Determine the work done by each force on the crate and the net work done on the crate.Dx = 40m

FGFNFP=100fK3709Example: Work done on a crate. Dx = 40m

FGFNFPfK370

10Force and direction of motion both matter in defining work!There is no work done by a force if it causes no displacement. Forces can do positive, negative, or zero work. When an box is pushed on a flat floor, for exampleThe normal force and gravity do no work, since they are perpendicular to the direction of motion. The person pushing the box does positive work, since she is pushing in the direction of motion. Friction does negative work, since it points opposite the direction of motion.11Example: Work done on a crate. Dx = 40m

FGFNFPfK370

Appears as kinetic E. Dx12

Question: Work on a backpack. a) How much work does a hiker do on a 150kg backpack to carry it up a hill of height h = 100m. Determine b) the work done by gravity on the backpack, c) the net work done on the backpack. Assume the motion is smooth and at constant velocity.DdFGFHh=100q13

Question: Work on a backpack. dFGFHh=100q90-q90+q

Gravity does work only in vertical directionWg always mgh14Does Earth do work on the Moon? The Moon revolves around the Earth in a circular orbit, kept there by the gravitational force exerted by the Earth. Does gravity do a) positive work, b) negative work, or c) no work at all?FG

v

15Energy has the ability to do work

16Work Energy TheoremWork on a system by a force over a distance transfers energy to the system!W=DEIf an applied force does positive work on a system, it increases the energy. If an applied force does negative work, it decreases the energy.W=DE = DK +DUg+ DUS + DEthermal+ DEchem+DEnuc+The two forms of mechanical energy are called kinetic energy and potential energy.17Kinetic Energy, K

The NET work done on an object changes its kinetic energy.for FnetK will change if there is a Fnet (acceleration)Work-Energy Theorem18Kinetic Energyenergy of translational motion

SI unit of kinetic energy: JouleK is a scalarK of a group of objects is the algebraic sum of the Ks of the individual objectsIf positive net work done on an object, its K increases

If negative net work done on object, its K decreases19Kinetic Energy, K Energy due to motion

Unit: Joule

Sample: A 10.0 g bullet has a speed of 1.2 km/s.a)What is the kinetic energy of the bullet?b)What is the bullets kinetic energy if the speed is halved?c)What is the bullets kinetic energy if the speed is doubled?20Question: KE and work done on a baseball. A 145 g baseball is thrown with a speed of 25 m/s. a) What is its kinetic energy? b) How much work was done on the baseball to make it reach this speed if it started from rest?

21Question: Work on a car, to increase its KE. How much work is required to accelerate a 1000 kg car from 20 m/s to 30 m/s?

How much work is required to increase the cars speed another 10 m/s, from 30 m/s to 40 m/s?

22Question: A 15 g acorn falls from a tree and lands on the ground 10 m below with a speed of 11.0 m/s.What would the speed of the acorn have been if there had been no air resistance? Did air resistance do positive, negative or zero work on the acorn? Why?How much work wasdone by air resistance?What was the average force of air resistance?

FGFa10m232 m/s1 kg3 m/s1 kg-2 m/s1 kg2 m/s2 kgABCDRank from greatest to least the kinetic energies of the sliding pucksB > D > A = C24PowerPower is the rate of which work is done. P = W/Dt =(Fdcosq)/t= Fvcosq= DE/tW: work in JoulesDt: elapsed time in secondsWhen we run upstairs, t is small so P is big.When we walk upstairs, t is large so P is small.

25Unit of Power

SI unit for power is the Watt 1 W = 1 J/sW: work in Joules Dt: elapsed time in secondsNamed after the Scottish engineer James Watt (1776-1819) who perfected the steam engine. 26How We Buy Energy

The kilowatt-hour is a commonly used unit by the electrical power company.Power companies charge you by the kilowatt-hour (kWh), but this not power, it is really energy consumed. 1 kW = 1000 W 1 h = 3600 s 1 kWh = 1000J/s 3600s = 3.6 x 106J

27Sample problem: A record was set for stair climbing when a man ran up the 1600 steps of the Empire State Building in 10 min and 59 sec. If the height gain of each step was 0.20 m, and the mans mass was 70.0 kg, what was his average power output during the climb?

FGFapp

28Sample problem: Calculate the power output of a 1.0 g fly as it walks straightup a window pane at 2.5 cm/s.

FGFfly

29Sample problem: Cary pushes a 15 kg lawn mower across a lawn at a constant speed by pushing with a force of 115 N along the direction of the handle which makes a 22.50 angle with the horizontal. a) If Cary develops 64.6 W for 90.0 s, what distance is the lawn mower pushed?b) What is the work done by friction?What is the coefficient of friction between the lawn mower and the lawn? If the initial speed of the lawn mower is 1 m/s and Cary then increases her pushing force so that the lawn mower speeds up to 2 m/s after 20 m, what is Carys new pushing force?54.7 m-5814 J0.56316 N30Work Done by a Varying Force31

Work Done by a Constant Force

W = AREA under F-x curveThe force shown is a constant force. W = FDx can be used to calculate the work done by this force when it moves an object from xi to xf. The area under the curve from xi to xf can also be used to calculate the work32Work Done by a Varying ForceThe force shown is a variable force. W = FDx CANNOT be used to calculate the work done by this force. The area under the curve from xi to xf can STILL be used to calculate the work

Work = AREA under F-x curve33

Sample Problem How much work is done by the force shown when it acts on an object and pushes it from x = 2.0 m to x = 4.0 m?+ 0.6J34

Sample Problem How much work is done by the force shown when itacts on an object and pushes it from x = 0.25 m tox = 0.75 m?35More Work Done by a Varying Force

SPRINGS36Springs When a spring is stretched or compressed from its equilibrium position, it does negative work since the spring pulls in the direction opposite the direction of motion. Force of a spring varies with DxFspFpull0xDx37Springs When a spring is stretched or compressed from its equilibrium position, it does negative work since the spring pulls in the direction opposite the direction of motion.Hookes Law: The force exerted by a spring is directly proportional to the distance the spring is stretched from the equilibrium position.

Fs = k x F : Force (N)k : spring constant (N/m) Describes the stiffness of the springx : displacement from equilibrium (m)FspFpull38Springs When a spring is stretched or compressed from its equilibrium position, it does negative work since the spring pulls in the direction opposite the direction of motion.FspFpullFsp = -kx

39Sample Problem: It takes 180 J of work to compress a certain spring 0.10 m.What is the spring constant of the spring?To compress the spring an additional 0.10 m, does it take 180J, more than 180 J, or less than 180J. Verify your answer with a calculation.

40.140DO NOW

You have two springs that are identical except that spring 1 is stiffer than spring 2 (k1>k2). On which spring is more work done

a) if they are stretched using the same force?

b) if they are stretched the same distance?Spring2, k2 smaller/looser so spring stretches farther with same forceSpring1, k1 larger/stiffer so need larger F to stretch same distance41Potential EnergyEnergy associated with forces that depend on the position and/or configuration of an object.Eg. Wound up clock spring has PE because as it unwinds it can do work moving the clock hands.Eg. Gravitational PE. Heavy brick held high in air has PE because of its position. When released it has ability to do work.42

FGFHGravitational Potential Energy, Ugy2y1h=1m

To lift an object of mass m vertically a height h (constant velocity):m=1kg

Hand increases the energy of ball by 10JGravity decreases the energy of ball by 10JBut where did the energy that the gravitational force took from the ball go?43

FGFHGravitational Potential Energy, GPEh2h1h=1mm=1kg

The 10 J of work done lifting the book was stored by the gravitational force and then converted to kinetic energy. The gravitational field stores POTENTIAL ENERGY mgh. Gravity is a conservative force.But where did the energy that the gravitational force took from the ball go?44FGh2h1DhLooses UgFallRiseGains UgWhen gravity (or any conservative force) does negative work, system gains U.When gravity (or any conservative force) does positive work, system looses U.

If only conservative forces act on isolated system, total mechanical E is conserved and No work is done by external forces that would transfer E in or outBallNOT IsolatedWG

BallEarthIsolated systemE conservedWDRAGBallEarthNOT IsolatedE lost45Gravitational Potential Energy, UgEnergy associated with an objects position in the gravitational field.

SI unit of PE: JoulePE is a scalarThe higher an object is above the ground, the more gravitational potential energy it hasOnly changes in gravitational PE are relevant and depend ONLY on change in vertical height and NOT on the path taken.46

47How does the gravitational PE depend on the path taken to get to h?dq

FGFHDh

FN

Path-INDEPENDENTDh48Energy and Conservative ForcesForces such as gravity that store potential energy and for which the work done does not depend on the path taken, but only on initial and final positions, are called conservative forces.FGh

In ALL cases49What about Friction?

Hand increases the energy of book by 10JFriction decreases the energy of ball by 10JBut where did the energy that the frictional force took from the book go?

Dx = 1m

FGFNFPush=10fK

50What about Friction?

But where did the energy that the frictional force took from the book go?

Dx = 1m

FGFNFPushfKIt is NOT stored as potential energy. It is converted to heat energy (nonmechanical) and dissipated. Friction is a nonconservative force.51

xf

Elastic Potential Energy, USFspring=kxFPushDxFspring= - kx

Hand increases the balls energySpring decreases the balls energy

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Elastic Potential EnergyFspringFPushDx

But where did the energy that the spring force took from the ball go?The work done compressing the spring was stored by the spring and converted to kinetic energy. The spring stores Elastic POTENTIAL ENERGY 1/2kx2. Elastic force is conservative.

53Potential Energy

If a conservative force does positive work on an object, potential energy is lost (K gained)If a conservative force does negative work on an object, potential energy is gained (K lost)In general, the change in PE associated with a particular conservative force is equal to the negative of the work done by that force if object moved from one point to another.

54Conservative ForcesNonconservative Forces

Gravitational (Ug=mgh)FrictionElastic (US=1/2kx2)Air resistanceElectricTension in a cordMotor or rocket propulsionApplied push or pull Path dependentdo not store energy that is available to convert to K Path independentstore energy that is available to convert to K55 Work is path independent.Work can be calculated from the starting and ending points only.The actual path is ignored in calculations.Work along a closed path is zero.If the starting and ending points are the same, no work is done by the force.Work changes potential energy.ExamplesGravitySpring ForceConservation of mechanical energy holds!Conservative Forces56 Work is path dependent.Knowing the starting and ending points is not sufficient to calculate the work.Work along a closed path is NOT zero.Work changes mechanical energy.ExamplesFrictionDrag (air resistance)Conservation of mechanical energy does not hold!Non-conservative Forces57

Energy and Conservative ForcesQ: Assume a conservative force moves an object along the various paths. Which two works are equal?Q: Which two works when added together, give a sum of zero?A: W2 = W3 (path independence)A: W1 + W2 = 0 or W1 + W3 = 0 (work along a closed path is zero)58Work-Energy Theorem

General form:Nonconservative forces change mechanical energy. If nonconservative work is negative, as it often is, the mechanical energy of the system will drop.

59Conservation of Mechanical Energy

If only conservative forces act, the total mech energy of an isolated system neither increases or decreases.

If only conservative forces act on a system:

60Conservation of Mechanical Energy

In any isolated system, the total energy remains constant: DE = 0Energy can neither be created nor destroyed, but can only be transformed from one type of energy to another.An isolated system is one in which no work is done on the system so that no energy is transferred into or out of the system. If gravity and a spring interact with an object to store energy or transform energy, then earth and spring must be part of the system61

Conservation of Mechanical EnergyOnly gravity acts on the rock and FG is conservative.

at any pointhttp://www.batesville.k12.in.us/physics/phynet/Mechanics/Energy/EnergyIntro.html 62

Falling Rock: If the original height of the stone is 3.0 m, calculate the stones speed when it has fallen to 1.0 m above the ground.

63

9.0 mvi=2.0 m/sThe greased pig pushes off the slide at a speed of 2.0m/s, 9.0 m above the ground. How fast will the pig be traveling at the bottom if the slide is frictionless?

c) At the bottom of which slide is the pig moving fastest? Does the shape of the slide matter? 123b) If a pig with twice the mass went down the slide, how would the speed and kinetic energy of the heavier pig be related to that of the smaller pig. 1=2=364

9.0 mvi=2.0 m/sThe greased pig pushes off the slide at a speed of 2.0m/s, 9.0 m above the ground. How fast will the pig be traveling at the bottom if the slide is frictionless?

c) At the bottom of which slide is the pig moving fastest? Does the shape of the slide matter? 123b) If a pig with twice the mass went down the slide, how would the speed and kinetic energy of the heavier pig be related to that of the smaller pig. d) At the bottom of the slide on the flat, the coefficient of friction between the pig and the ground is 0.6. How far will the pig slide before coming to rest.

FgFNfk////1=2=365

Roller coaster: If a 200 kg roller coaster car is pulled up to point 1, starts from rest and coasts down the track, What will be its energy at point 1 and point 4?What will its speed be at point 4? Draw energy bar diagrams at point 1 and 4. Assume no friction between the car and track and no air drag.

A)B)66

Roller coaster: d) What would the total and kinetic energy be at point 4 if there was friction between point 3 and 4? The average frictional force between the car and the track is 400 N and the distance between point 3 and 4 is 25m.

E1 = E2 = E3 = 68,600JEnergy loss due to frictionC)67Roller coaster:

E1 = E2 = E3 = 68,600JEnergy loss of 10,000J due to friction

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69

slopeflat70

ExampleA dart of mass 0.100 kg is loaded in a toy dart gun. The spring in the gun has a spring constant of 250 N/m. The spring is compressed 6.0 cm and then released. What speed does the dart leave the gun with? vf = 3.0 m/s

71

Fspr=-kxFPushDxFsprSample Problem A vertical spring (ignore its mass) with spring constant 900 N/m, is attached to a table and compressed 0.150 m with a ball. When released, what is the launch speed of the 0.30 kg ball? How high does the ball go?Dx

12Fg72

Fspr=-kxFPushDxSample Problem A vertical spring (ignore its mass) with spring constant 900 N/m, is attached to a table and compressed 0.150 m with a ball. When released, what is the launch speed of the 0.30 kg ball? How high does the ball go?Dx13

73A roller coaster car begins at rest at height h above the ground and completes a loop along its path. In order for the car to remain on the track throughout the loop, what is the minimum value for h in terms of the radius of the loop, R? What is the acceleration of the car at the top of the loop. Assume no friction

At A, for the car not to fall off, FCFg and ac=gFg

74

What is the ACCELERATION and NORMAL force of the car at the at the Side and Bottom of the loop? Assume no friction

FgFgFN

BottomSideFN75Pendulums and Energy ConservationEnergy goes back and forth between K and Ug.

At highest point, all energy is Ug.

As it drops, Ug goes to K.

At the bottom , energy is all K. simulation of energy transformations 76vB = 2.62 m/s1.5m400xhABExample: Pendulum. What is the speed of the pendulum bob at point B if it is released from rest at point A?77

How fast do you have to go here to get here?(h = 456ft = 139m)(0.0006214 mi/m)Kingda Ka

78m1m2.Two masses, m1 and m2, hang from the ends of a rope that passes over a small frictionless pulley. The system is released from rest. Mass 1 is 2.00 kg and mass 2 is 8.00 kg, and the two masses are released from rest. After the two masses have each moved 2.0 m, what are their velocities?h=0h=2h=0h=2

Vf = 4.85m/s79Springs and Energy ConservationEnergy goes back and forth between K and US.

When fully stretched or compressed, all energy is US.

When passing through equilibrium, energy is all K.

At other points in cycle, energy is a mix of US and K. 80X=0Spring Energyall USall Kall US

For any two points 1 and 2:

For max and min displacement from equilibrium:simulation of energy transformations 81http://www.science-animations.com/support-files/energy.swf Great animations of conservation of energy82Spring example: A 1.60 kg block slides with a speed of 0.950 m/s on a frictionless, horizontal surface until it encounters a spring with a force constant of 902 N/m. The block comes to rest after compressing the spring 4.00 cm. Find the spring potential energy, US, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for the following compressions: 0 cm, 2.00 cm, 4.00 cm.v = 0.95 m/s0x83Spring example: A 1.60 kg block slides with a speed of 0.950 m/s v = 0.95 m/s0x84Work Done by Nonconservative Forces

Nonconservative forces change mechanical energy. They add or remove energy to the system. If nonconservative work is negative, as it is for friction, the mechanical energy of the system will drop and the amount of energy lost is equivalent to the work done by the non conservative force.85Sample problem: Catching a wave, a 72-kg surfer starts with a speed of 1.3 m/s, drops through a height of 1.75 m, and ends with a speed of 8.2 m/s. How much non-conservative work was done on the surfer?

The surfer GAINS energy due to the work added by the water86Sample problem: A 1.75-kg rock is released from rest at the surface of a pond 1.00 m deep. As the rock falls, a constant upward force of 4.10 N is exerted on it by water resistance. Calculate the nonconservative work, WNC, done by the water resistance on the rock, the gravitational PE of the system, GPE, the kinetic energy of the rock, KE, and the total mechanical energy of the system, E, for the following depths below the waters surface: d = 0 m, d = 0.500 m, d = 1.00 m. Let GPE be zero at the bottom of the pond.h=0d=0d=0.5d=1

Fg (conservative)Fwater (Nonconservative)E0 E0.5 E1 As the rock sinks, it looses energy since the water resistance force is doing negative work. The amount of energy lost is equal to the work done by the force of the water.87Sample problem: A 1.75-kg rock is released from rest at the surface of a pond 1.00 m deep. As the rock falls, a constant upward force of 4.10 N is exerted on it by water resistance. Calculate the nonconservative work, Wnc, done by the water resistance on the rock, the gravitational potential energy of the system, U, the kinetic energy of the rock, K, and the total mechanical energy of the system, E, for the following depths below the waters surface: d = 0.00 m, d = 0.500 m, d = 1.00 m. Let potential energy be zero at the bottom of the pond.882. The bar graph shows the energy of the Skater, where could she be on the track?

ABCDE89D4. If the ball is at point 4, which chart could represent the balls energy?

4KEPE

A.B.C.D.321905. If a heavier ball is at point 4, how would the pie chart change?

2134No changesThe pie would be largerThe PE part would be largerThe KE part would be larger

KEPE916. As the ball rolls from point 4, the KE bar gets taller. Which way is the ball rolling?

2134

At 4 Next stepUp Down not enough info92