Work, Energy and Power Chapter 6 – READ!

How does it relate to force? Today Definition of Work What is work? How does it relate to force?

All objects or systems have an energy associated with it E = Emotion +Egrav+ Eelastic + Ethermal+ Echem + Enuc + Enuc … Energy can be transferred from the environment to an object or system in two ways WORK – mechanical transfer of energy (apply force over a distance) HEAT – nonmechanical transfer of energy (due to temperature difference)

Work – 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 product Angle between F and Dd Component of force parallel to displacement Work a bridge between force (a vector) and energy (a scalar)

Dot Product The product of the magnitudes of 2 vectors and the angle between them. The result is a SCALAR. A B q Result is a scalar that measures how much of one vector lies along the direction of the other. Bcosq

Work dot product SI Unit of work (and Energy) is Joule: J = (N)(m) Component of force parallel to the displacement SI Unit of work (and Energy) is Joule: J = (N)(m) Work is a SCALAR

Work Question Fapp Dx FG Is this bellhop doing work on the suitcases as he walks forward at constant speed? FG Fapp Dx

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

Example: Work done on a crate Example: 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. FG FN FP=100 fK 370 Dx = 40m

Example: Work done on a crate. FG FN FP fK 370 Dx = 40m

Force 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 example… The 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.

FN FP Dx = 40m fK FG Example: Work done on a crate. 370 Appears as kinetic E

Question: Work on a backpack 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. FG FH Dd h=100 q

FH d h=100 FG q Question: Work on a backpack. 90-q 90+q Gravity does work only in vertical direction Wg always mgh

Does Earth do work on the Moon Does 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? v FG

Energy – has the ability to do work

Work Energy Theorem W=DE Work on a system by a force over a distance transfers energy to the system! W=DE If 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.

The NET work done on an object changes its kinetic energy. Kinetic Energy, K Work-Energy Theorem K will change if there is a Fnet (acceleration) for Fnet The NET work done on an object changes its kinetic energy.

energy of translational motion SI unit of kinetic energy: Joule If positive net work done on an object, its K increases If negative net work done on object, its K decreases SI unit of kinetic energy: Joule K is a scalar K of a group of objects is the algebraic sum of the Ks of the individual objects

Kinetic 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 bullet’s kinetic energy if the speed is halved? c)What is the bullet’s kinetic energy if the speed is doubled?

Question: KE and work done on a baseball Question: 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?

Question: Work on a car, to increase its KE Question: 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 car’s speed another 10 m/s, from 30 m/s to 40 m/s?

Question: 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 was done by air resistance? What was the average force of air resistance? Fa FG 10m

Rank from greatest to least the kinetic energies of the sliding pucks 2 m/s 2 kg 2 m/s 1 kg 3 m/s 1 kg -2 m/s 1 kg A B C D B > D > A = C

Power Power is the rate of which work is done. P = W/Dt =(Fdcosq)/t= Fvcosq= DE/t W: work in Joules Dt: elapsed time in seconds When we run upstairs, t is small so P is big. When we walk upstairs, t is large so P is small.

Unit of Power SI unit for power is the Watt 1 W = 1 J/s W: work in Joules Dt: elapsed time in seconds Named after the Scottish engineer James Watt (1776-1819) who perfected the steam engine.

How 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

Sample 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 man’s mass was 70.0 kg, what was his average power output during the climb? FG Fapp

Sample problem: Calculate the power output of a 1 Sample problem: Calculate the power output of a 1.0 g fly as it walks straight up a window pane at 2.5 cm/s. FG Ffly

Sample 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 Cary’s new pushing force? 54.7 m -5814 J 0.56 316 N 30

Work Done by a Varying Force

Work Done by a Constant Force The 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 work W = AREA under F-x curve

Work Done by a Varying Force The 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 curve

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? + 0.6J

How much work is done by the force shown when it Sample Problem How much work is done by the force shown when it acts on an object and pushes it from x = 0.25 m to x = 0.75 m?

More Work Done by a Varying Force SPRINGS 36

Springs 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 Dx Fsp Fpull Dx x 37

Springs 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. Hooke’s 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 spring x : displacement from equilibrium (m) Fsp Fpull

Springs 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. Fsp Fpull Fsp = -kx

Sample Problem: It takes 180 J of work to compress a certain spring 0 Sample 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. 4 0.1

DO 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 force Spring1, k1 larger/stiffer so need larger F to stretch same distance

Potential Energy Energy 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.

Gravitational Potential Energy, Ug To lift an object of mass m vertically a height h (constant velocity): Hand increases the energy of ball by 10J y2 h=1m FG FH Gravity decreases the energy of ball by 10J But where did the energy that the gravitational force took from the ball go? y1 m=1kg

Gravitational Potential Energy, GPE But where did the energy that the gravitational force took from the ball go? FH FG h2 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. h=1m h1 m=1kg

Rise Fall h2 FG Dh h1 Gains Ug Looses Ug WG WDRAG When gravity (or any conservative force) does negative work, system gains U. When gravity (or any conservative force) does positive work, system looses U. NOT Isolated WG Ball 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 out Ball Earth Isolated system E conserved WDRAG Ball Earth NOT Isolated E lost

Gravitational Potential Energy, Ug Energy associated with an objects position in the gravitational field. SI unit of PE: Joule PE is a scalar The higher an object is above the ground, the more gravitational potential energy it has Only changes in gravitational PE are relevant and depend ONLY on change in vertical height and NOT on the path taken.

How does the gravitational PE depend on the path taken to get to h? FN Dh Dh FH q FG Path-INDEPENDENT

Energy and Conservative Forces Forces 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. h FG In ALL cases

What about Friction? FPush=10 fK Dx = 1m FG FN FPush=10 fK Dx = 1m Hand increases the energy of book by 10J Friction decreases the energy of ball by 10J But where did the energy that the frictional force took from the book go?

What about Friction? FPush fK Dx = 1m FG FN FPush fK Dx = 1m But where did the energy that the frictional force took from the book go? It is NOT stored as potential energy. It is converted to heat energy (nonmechanical) and dissipated. Friction is a nonconservative force.

Elastic Potential Energy, US xf Dx FPush Fspring=kx Fspring= - kx Spring decreases the balls energy Hand increases the balls energy

Elastic Potential Energy But where did the energy that the spring force took from the ball go? Dx FPush Fspring 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.

Potential 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.

Nonconservative Forces Gravitational (Ug=mgh) Friction Elastic (US=1/2kx2) Air resistance Electric Tension in a cord Motor or rocket propulsion Applied push or pull Path independent store energy that is available to convert to K Path dependent do not store energy that is available to convert to K

Conservative Forces 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. Examples Gravity Spring Force Conservation of mechanical energy holds!

Non-conservative Forces 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. Examples Friction Drag (air resistance) Conservation of mechanical energy does not hold!

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

Work-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.

Conservation of Mechanical Energy If only conservative forces act on a system: If only conservative forces act, the total mech energy of an isolated system neither increases or decreases.

Conservation of Mechanical Energy In any isolated system, the total energy remains constant: DE = 0 Energy 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 system

Conservation of Mechanical Energy Only gravity acts on the rock and FG is conservative. at any point http://www.batesville.k12.in.us/physics/phynet/Mechanics/Energy/EnergyIntro.html

Falling Rock: If the original height of the stone is 3 Falling Rock: If the original height of the stone is 3.0 m, calculate the stone’s speed when it has fallen to 1.0 m above the ground.

The greased pig pushes off the slide at a speed of 2. 0m/s, 9 The 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? b) 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. c) At the bottom of which slide is the pig moving fastest? Does the shape of the slide matter? 1=2=3 vi=2.0 m/s 1 2 3 9.0 m

The greased pig pushes off the slide at a speed of 2. 0m/s, 9 The 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? b) 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. c) At the bottom of which slide is the pig moving fastest? Does the shape of the slide matter? 1=2=3 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. vi=2.0 m/s 1 2 3 9.0 m FN // fk // Fg

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)

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 = 68,600J Energy loss due to friction C)

Roller coaster: E1 = E2 = E3 = 68,600J Energy loss of 10,000J due to friction

slope flat

Example A 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

Sample 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? Fspr Fspr=-kx FPush Dx Fg Dx 1 2

Sample 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? Fspr=-kx FPush Dx Dx 1 3

A 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 Fg At A, for the car not to fall off, FC≥Fg and ac=g

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

Pendulums and Energy Conservation Energy 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

Example: Pendulum. What is the speed of the pendulum bob at point B if it is released from rest at point A? vB = 2.62 m/s 400 1.5m x A h B

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

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? m1 m2 . h=2 h=2 h=0 h=0 Vf = 4.85m/s

Springs and Energy Conservation Energy 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.

Spring Energy For any two points 1 and 2: all US all K X=0 all US all K For max and min displacement from equilibrium: all US simulation of energy transformations

http://www.science-animations.com/support-files/energy.swf Great animations of conservation of energy

Spring example: A 1. 60 kg block slides with a speed of 0 Spring 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. x v = 0.95 m/s

Spring example: A 1.60 kg block slides with a speed of 0.950 m/s … v = 0.95 m/s

Work 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.

Sample 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 water

Fwater (Nonconservative) Sample 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 water’s surface: d = 0 m, d = 0.500 m, d = 1.00 m. Let GPE be zero at the bottom of the pond. h=0 d=0 d=0.5 d=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.

Sample 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 water’s surface: d = 0.00 m, d = 0.500 m, d = 1.00 m. Let potential energy be zero at the bottom of the pond.

2. The bar graph shows the energy of the Skater, where could she be on the track?

4. If the ball is at point 4, which chart could represent the ball’s energy? KE PE A. B. C. D. 4 3 1 2

5. If a heavier ball is at point 4, how would the pie chart change? KE No changes The pie would be larger The PE part would be larger The KE part would be larger PE 2 1 3 4

6. As the ball rolls from point 4, the KE bar gets taller 6. As the ball rolls from point 4, the KE bar gets taller. Which way is the ball rolling? At 4 Next step 2 1 3 4 Up Down not enough info