Welcome back to Physics 215 Todays agenda Work

Welcome back to Physics 215 Today’s agenda: • Work • Non-conservative forces • Power Physics 215 – Fall 2014 Lecture 09 -1 1

Current homework assignment • HW 7: – Knight Textbook Ch. 9: 54, 72 – Ch. 10: 48, 68, 76 – Ch. 11: 50, 64 – Due Wednesday, Oct. 22 nd in recitation Physics 215 – Fall 2014 Lecture 09 -1 2

Work, Energy • Newton’s Laws are vector equations • Sometimes easier to relate speed of a particle to how far it moves under a force – a single equation can be used – need to introduce concept of work Physics 215 – Fall 2014 Lecture 09 -1 3

What is work? • Assume constant force in 1 D • Consider: v. F 2 = v. I 2 + 2 a s • Multiply by m/2 (1/2)mv. F 2 - (1/2)mv. I 2 = m a s • But: F = ma (1/2)mv. F 2 - (1/2)mv. I 2 = F s Physics 215 – Fall 2014 Lecture 09 -1 4

Work-Kinetic Energy theorem (1) (1/2)mv. F 2 - (1/2)mv. I 2 = F s Points: • W = Fs defines work done on particle = force times displacement • K = (1/2)mv 2 defines kinetic energy =1/2 mass times square of v Physics 215 – Fall 2014 Lecture 09 -1 5

Work-Kinetic Energy demo • Cart, force probe, and motion detector • Plot v 2 vs. x – gradient 2 F/m • constant F (measure) -- pulling with string • Weigh cart and masses in advance Physics 215 – Fall 2014 Lecture 09 -1 6

Conclusions from experiment Although the motion of the two carts looks very different (i. e. , different amounts of time, accelerations, and final speeds), there is a quantity that is the same for both at the end of the motion. It is (1/2) mv 2 and is called the (final) kinetic energy of the carts. Moreover, this quantity happens to have the same value as F Ds, which is given the name work. Physics 215 – Fall 2014 Lecture 09 -1 7

Improved definition of work • For forces, write F FAB • Thus W = F s WAB = FAB Ds. A is work done on A by B as A undergoes displacement Ds. A • Work-kinetic energy theorem: Wnet, A = SBWAB = DK Physics 215 – Fall 2014 Lecture 09 -1 8

Non-constant force … Work done in small interval Dx DW = F Dx F Total W done from A to B S F Dx = Area under curve! F(x) A Dx Physics 215 – Fall 2014 B x Lecture 09 -1 9

The Work - Kinetic Energy Theorem Wnet = DK = Kf - Ki The net work done on an object is equal to the change in kinetic energy of the object. Physics 215 – Fall 2014 Lecture 09 -1 10

Suppose a tennis ball and a bowling ball are rolling toward you. The tennis ball is moving much faster, but both have the same momentum (mv), and you exert the same force to stop each. Which of the following statements is correct? 1. It takes equal distances to stop each ball. 2. It takes equal time intervals to stop each ball. 3. Both of the above. 4. Neither of the above. Physics 215 – Fall 2014 Lecture 09 -1 11

Suppose a tennis ball and a bowling ball are rolling toward you. Both have the same momentum (mv), and you exert the same force to stop each. The distance taken for the bowling ball to stop is 1. less than. 2. equal to 3. greater than the distance taken for the tennis ball to stop. Physics 215 – Fall 2014 Lecture 09 -1 12

Two carts of different mass are accelerated from rest on a low-friction track by the same force for the same time interval. Cart B has greater mass than cart A (m. B > m. A). The final speed of cart A is greater than that of cart B (v. A > v. B). After the force has stopped acting on the carts, the kinetic energy of cart B is 1. 2. 3. 4. less than the kinetic energy of cart A (KB < KA). equal to the kinetic energy of cart A (KB = KA). greater than the kinetic energy of cart A (KB > KA). “Can’t tell. ” Physics 215 – Fall 2014 Lecture 09 -1 13

Revised definitions for Work and Kinetic Energy Work done on object 1 by object 2: Kinetic energy of an object: Physics 215 – Fall 2014 Lecture 09 -1 14

Scalar (or “dot”) product of vectors The scalar product is a way to combine two vectors to obtain a number (or scalar). It is indicated by a dot ( • ) between the two vectors. (Note: component of A in direction n is just A • n) Physics 215 – Fall 2014 Lecture 09 -1 15

Is the scalar (“dot”) product of the two vectors 1. 2. 3. 4. positive negative equal to zero “Can’t tell. ” Physics 215 – Fall 2014 Lecture 09 -1 16

Is the scalar (“dot”) product of the two vectors 1. 2. 3. 4. positive negative equal to zero “Can’t tell. ” Physics 215 – Fall 2014 Lecture 09 -1 17

Two identical blocks slide down two frictionless ramps. Both blocks start from the same height, but block A is on a steeper incline than block B. The speed of block A at the bottom of its ramp is 1. 2. 3. 4. less than the speed of block B. equal to the speed of block B. greater than the speed of block B. “Can’t tell. ” Physics 215 – Fall 2014 Lecture 09 -1 18

Solution • Which forces do work on block? • Which, if any, are constant? • What is F • s for motion? Physics 215 – Fall 2014 Lecture 09 -1 19

Work done by gravity Work W = -mg j • s N s mg j Therefore, W = -mg. Dh N does no work! i Physics 215 – Fall 2014 Lecture 09 -1 20

Work done on an object by gravity W (on object by earth) = – m g Dh, where Dh = hfinal – hinitial is the change in height. Physics 215 – Fall 2014 Lecture 09 -1 21

Defining gravitational potential energy The change in gravitational potential energy of the objectearth system is just another name for the negative value of the work done on an object by the earth. Physics 215 – Fall 2014 Lecture 09 -1 22

Curved ramp s = W = F • s = Work done by gravity between 2 fixed pts does not depend on path taken! Work done by gravitational force in moving some object along any path is independent of the path depending only on the change in vertical height Physics 215 – Fall 2014 Lecture 09 -1 23

Hot wheels demo One hot wheels car, car A, rolls down the incline and travels straight ahead, while the other car, B, goes through a loop at the bottom of the incline. When the cars reach the end of their respective tracks, the relative speeds will be: 1. v. A > v. B 2. v. A < v. B 3. v. A = v. B 4. Can’t tell Physics 215 – Fall 2014 Lecture 09 -1 24

A person carries a book horizontally at constant speed. The work done on the book by the person’s hand is 1. 2. 3. 4. positive negative equal to zero “Can’t tell. ” Physics 215 – Fall 2014 Lecture 09 -1 25

A person lifts a book at constant speed. Since the force exerted on the book by the person’s hand is in the same direction as the displacement of the book, the work (W 1) done on the book by the person’s hand is positive. The work done on the book by the earth is: 1. 2. 3. 4. negative and equal in absolute value to W 1 negative and less in absolute value than W 1 positive and equal in absolute value to W 1 positive and less in absolute value than W 1 Physics 215 – Fall 2014 Lecture 09 -1 26

A person lifts a book at constant speed. The work (W 1) done on the book by the person’s hand is positive. Work done on the book by the earth: Net work done on the book: Change in kinetic energy of the book: Physics 215 – Fall 2014 Lecture 09 -1 27

Work-Kinetic Energy theorem (final statement) W 1 net = Fnet • Ds 1 = Kf - Ki Physics 215 – Fall 2014 Lecture 09 -1 28

A person lifts two books (each of mass m) at constant speed. The work done on the upper book by the lower book is positive. Its magnitude is W = m g Ds. Is there work done on the lower book by the upper book? 1. 2. 3. 4. Yes, positive work. Yes, negative work. No, zero work. No, since the lower book does work on the upper book, this is not a meaningful question. Physics 215 – Fall 2014 Lecture 09 -1 29

A person lifts two books at constant speed. The work done on the upper book by the lower book is positive. Work done on lower book by upper book: Net work done on the lower book: Change in kinetic energy of the lower book: Physics 215 – Fall 2014 Lecture 09 -1 30

Conservative forces • If the work done by some force (e. g. gravity) does not depend on path the force is called conservative. • Then gravitational potential energy Ug only depends on (vertical) position of object Ug = Ug(h) • Elastic forces also conservative – elastic potential energy U = (1/2)kx 2. . . Physics 215 – Fall 2014 Lecture 09 -1 31

Nonconservative forces • friction, air resistance, . . . • Potential energies can only be defined for conservative forces Physics 215 – Fall 2014 Lecture 09 -1 32

Nonconservative forces • Can do work, but cannot be represented by a potential energy function • Total mechanical energy can now change due to work done by nonconservative force • For example, frictional force leads to decrease of total mechanical energy -- energy converted to heat, or internal energy • Total energy = total mech. energy + internal energy -- is conserved Physics 215 – Fall 2014 Lecture 09 -1 33

Conservation of (mechanical) energy • If we are dealing with a potential energy corresponding to a conservative force 0 = DK+DU • Or Physics 215 – Fall 2014 K + U = constant Lecture 09 -1 34

Conservation of total energy The total energy of an object or system is said to be conserved if the sum of all energies (including those outside of mechanics that have not yet been discussed) never changes. This is believed always to be true. Physics 215 – Fall 2014 Lecture 09 -1 35

Demo Vertical spring and mass with damping • Damping from air resistance • Amplitude of oscillations decays • Oscillations still about x = -xeq Physics 215 – Fall 2014 Lecture 09 -1 36

Many forces • For a particle which is subject to several (conservative) forces F 1, F 2 … E = (1/2)mv 2 + U 1 + U 2 +… is constant • Principle called Conservation of total mechanical energy Physics 215 – Fall 2014 Lecture 09 -1 37

Summary • Total (mechanical) energy of an isolated system is constant in time. • Must be no non-conservative forces • Must sum over all conservative forces Physics 215 – Fall 2014 Lecture 09 -1 38

A compressed spring fires a ping pong ball vertically upward. If the spring is compressed by 1 cm initially the ball reaches a height of 2 m above the spring. What height would the ball reach if the spring were compressed by just 0. 5 cm? (neglect air resistance) 1. 2 m 2. 1 m 3. 0. 5 m 4. we do not have sufficient information to calculate the new height Physics 215 – Fall 2014 Lecture 09 -1 39

A 5. 00 -kg package slides 1. 50 m down a long ramp that is inclined at 12. 0 below the horizontal. The coefficient of kinetic friction between the package and the ramp is mk = 0. 310. Calculate: (a) the work done on the package by friction; (b) the work done on the package by gravity; (c) the work done on the package by the normal force; (d) the total work done on the package. (e) If the package has a speed of 2. 20 m/s at the top of the ramp, what is its speed after sliding 1. 50 m down the ramp? Physics 215 – Fall 2014 Lecture 09 -1 40

Summary • Work is defined as dot product of force with displacement vector • Each individual force on an object gives rise to work done • The kinetic energy only changes if net work is done on the object, which requires a net force Physics 215 – Fall 2014 Lecture 09 -1 41

Power = Rate at which work is done Units of power: Physics 215 – Fall 2014 Lecture 09 -1 42

A sports car accelerates from zero to 30 mph in 1. 5 s. How long does it take to accelerate from zero to 60 mph, assuming the power (=DW / Dt) of the engine to be constant? (Neglect losses due to friction and air drag. ) 1. 2. 3. 4. 2. 25 s 3. 0 s 4. 5 s 6. 0 s Physics 215 – Fall 2014 Lecture 09 -1 43

Power in terms of force and velocity Physics 215 – Fall 2014 Lecture 09 -1 44

A locomotive accelerates a train from rest to a final speed of 40 mph by delivering constant power. If we assume that there are no losses due to air drag or friction, the acceleration of the train (while it is speeding up) is 1. 2. 3. decreasing constant increasing Physics 215 – Fall 2014 Lecture 09 -1 45

A ball is whirled around a horizontal circle at constant speed. If air drag forces can be neglected, the power expended by the hand is: 1. 2. 3. 4. positive negative zero “Can’t tell. ” Physics 215 – Fall 2014 Lecture 09 -1 46

Reading assignment • Extended objects, center of mass • Begin Chapter 12 in textbook Physics 215 – Fall 2014 Lecture 09 -1 47