Welcome back to Physics 211

Transcription

1 Welcome back to Physics 211 Today s agenda: Work Power Physics 211 Fall 2012 Lecture

2 Current assignments HW#9 due this Friday at 5 pm. Short assignment SAGE (Thanks for the feedback!) I am using it to hopefully improve. I hope it is useful to you (feel free to provide feedback about SAGE.) Different lecturer next week: Simon Catterall I will have limited contact (I ll be in Slovenia) EXAMS: handed back on Friday in workshop/ recitation Physics 211 Fall 2012 Lecture

3 Midterm 2 grades 25 Midterm 2 20 number of students Percentage grade Average: 44/64 (69%), Median:46/64 (72%) Physics 211 Fall 2012 Lecture

4 Rough grade estimate for percentages (# points/64): A+ (94-100%) A (88-93) A- (82-87) B+ (76-81) B (70-75) B- (65-70) C+ (58-64) C (52-57) C- (46-51) D (40-45) You might want to see me in office hours F (less than 40%) You should probably come see me during office hours Physics 211 Fall 2012 Lecture

5 Total Energy E = U g + K Gravitational Potential energy: for an object of mass m: U g = mgh h is height above arbitrary reference line Kinetic energy: For an object of mass m moving with speed v: K = (1/2)mv 2 Total Energy is conserved: E i = E f Physics 211 Fall 2012 Lecture

6 Springs -- Elastic potential energy Force F = -kx (Hooke s law) Area of triangle lying under straight line graph of F vs. x = (1/2)(+/-x)(-/+kx) frictionless table F U s = (1/2)kx 2 F = -kx x Physics 211 Fall 2012 Lecture

7 (Horizontal) Spring x = displacement from relaxed state of spring frictionles s table Elastic potential energy stored in spring: U s = (1/2)kx 2 (1/2)kx 2 + (1/2)mv 2 = constant Physics 211 Fall 2012 Lecture

8 9-2.1A 0.5 kg mass is attached to a spring on a horizontal frictionless table. The mass is pulled to stretch the spring 5.0 cm and is released from rest. When the mass crosses the point at which the spring is not stretched, x = 0, its speed is 20 cm/s. If the experiment is repeated with a 10.0 cm initial stretch, what speed will the mass have when it crosses x = 0? cm/s 2. 0 cm/s cm/s cm/s Physics 211 Fall 2012 Lecture

9 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 211 Fall 2012 Lecture

10 What is work? Assume constant force in 1D Consider: v F 2 = v I 2 + 2a Multiply by m/2 à Δx (1/2)mv F 2 - (1/2)mv I 2 = m a Δx But: F = ma à (1/2)mv F 2 - (1/2)mv I 2 = F Δx Physics 211 Fall 2012 Lecture

11 Work-Kinetic Energy theorem (1D) (1/2)mv F 2 - (1/2)mv I 2 = F s Define s = Δx = displacement W = Fs à defines work done on particle = force times displacement K = (1/2)mv 2 à defines kinetic energy =1/2 mass times square of v Physics 211 Fall 2012 Lecture

12 Improved definition of work For forces, write F à F AB Thus W = F s à W AB = F AB Δs A is work done on A by B as A undergoes displacement Δs A Work-kinetic energy theorem: W net,a = Σ B W AB = ΔK Physics 211 Fall 2012 Lecture

13 The Work - Kinetic Energy Theorem W net = ΔK = K f - K i The net work done on an object is equal to the change in kinetic energy of the object. Physics 211 Fall 2012 Lecture

14 9.2-2 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 211 Fall 2012 Lecture

15 9-2.3 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. It takes equal time intervals to stop each ball. 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 211 Fall 2012 Lecture

16 Work and Kinetic Energy in 2D Work done on object 1 by object 2: W (on 1 by 2) = F 1,2 Δ s of 1 Kinetic energy of an object: K = 1 2 mv2 [or: 1 2 m( v v)] Physics 211 Fall 2012 Lecture

17 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) A B = ABcosθ = A x B x + A y B y + A z B z Physics 211 Fall 2012 Lecture

18 9-2.4 Is the scalar ( dot ) product of the two vectors 1. positive 2. negative 3. equal to zero 4. Can t tell. Physics 211 Fall 2012 Lecture

19 9-2.5 Is the scalar ( dot ) product of the two vectors 1. positive 2. negative 3. equal to zero 4. Can t tell. Physics 211 Fall 2012 Lecture

20 9-2.6 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. negative and equal in absolute value to W 1 2. negative and less in absolute value than W 1 3. positive and equal in absolute value to W 1 4. positive and less in absolute value than W 1 Physics 211 Fall 2012 Lecture

21 9-2.7 A person carries a book horizontally at constant speed. The work done on the book by the person s hand is 1. positive 2. negative 3. equal to zero 4. Can t tell. Physics 211 Fall 2012 Lecture

22 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 211 Fall 2012 Lecture

23 Sample problem: The two ropes seen in the figure are used to lower a 255 kg piano 5 m from a second-story window to the ground. How much work is done by each of the three forces? Physics 211 Fall 2012 Lecture

24 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. less than the speed of block B. 2. equal to the speed of block B. 3. greater than the speed of block B. 4. Can t tell. Physics 211 Fall 2012 Lecture

25 Solution Which forces do work on block? Which, if any, are constant? What is F Δs for motion? Physics 211 Fall 2012 Lecture

26 Work done by gravity N Work W = -mg j Δs j mg Δs Therefore, W = -mgδh N does no work! i Physics 211 Fall 2012 Lecture

27 Work done on an object by gravity W (on object by earth) = m g Δh, where Δh = h final h initial is the change in height. Object moves Change in height Δh Work done on object by earth upward positive negative downward negative positive Physics 211 Fall 2012 Lecture

28 Defining gravitational potential energy W = ΔK (on obj. by earth) 0 = ΔK W (on obj. by earth) 0 = ΔK + ΔUg 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 211 Fall 2012 Lecture

29 Curved ramp Δs = W = F Δs = 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 211 Fall 2012 Lecture

30 Demo: 4 tracks Physics 211 Fall 2012 Lecture

31 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 U g only depends on (vertical) position of object U g = U g (h) Elastic forces also conservative elastic potential energy U = (1/2)kx 2... Physics 211 Fall 2012 Lecture

32 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 211 Fall 2012 Lecture

33 Demo: work done by gravity Physics 211 Fall 2012 Lecture

34 Nonconservative forces friction, air resistance, plastic deformation... Potential energies can only be defined for conservative forces Physics 211 Fall 2012 Lecture

35 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 211 Fall 2012 Lecture

36 Conservation of (mechanical) energy If we are dealing with a potential energy corresponding to a conservative force 0 = ΔK+ΔU Or K + U = constant Physics 211 Fall 2012 Lecture

37 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 211 Fall 2012 Lecture

38 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 211 Fall 2012 Lecture

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 µ k = 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 211 Fall 2012 Lecture