Announcements. If you think there was an error in the scoring, fill out a regrade form and had back to ME (not TAs)

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1 Exam 1 scores posted on Canvas: Ø Announcements If you think there was an error in the scoring, fill out a regrade form and had back to ME (not TAs) Ø Must return regrade forms before next Wednesday, October 7 Quiz #4 on Wednesday: Ø Covers units #6 and #7 (Friction, Work-KE theory) Mechanics Lecture 8, Slide 1

2 Exam 1 stats Number of Students Histogram (by percent) Mean 64.74% Median 65.77% Standard Devia3on 18.33% More Score (%) Mechanics Lecture 8, Slide 2

3 Classical Mechanics Lecture 8: Conservative Forces and Potential Energy Today s Concepts: 1) Potential Energy 2) Total Mechanical Energy Mechanics Lecture 8, Slide 3

4 ΔK W NET Summary Lecture 7 Work Kine=c Energy theorem ΔU W E K + U Lecture 8 For springs & gravity (conservative forces) Total Mechanical Energy E = Kinetic + Potential ΔE =W NC Work done by any force other than gravity and springs will change E Mechanics Lecture 8, Slide 4

5 Relax! There is nothing new here! It s just re-writing the work-ke theorem: Everything except gravity and springs gravity ΔK =W tot = W gravity + Wsprings + W ΔU Δ ΔK + ΔU + ΔU = W springs ΔK + ΔU = NC W NC gravity U springs NC ΔE =W NC ΔE = 0 If non-conservative forces aren't doing work Mechanics Lecture 8, Slide 5

6 Finding the potential energy change: Use formulas to find the magnitude Check the sign by understanding the problem Mechanics Lecture 8, Slide 6

7 CheckPoint: Three Balls Three balls of equal mass are fired simultaneously with equal speeds from the same height h above the ground. Ball 1 is fired straight up, ball 2 is fired straight down, and ball 3 is fired horizontally. Rank in order from largest to smallest their speeds v 1, v 2, and v 3 just before each ball hits the ground. A) v 1 > v 2 > v 3 B) v 3 > v 2 > v 1 C) v 2 > v 3 > v 1 D) v 1 = v 2 = v 3 h Mechanics Lecture 8, Slide 7

8 CheckPoint Results: Three Balls A) v 1 > v 2 > v 3 2 B) v 3 > v 2 > v 1 C) v 2 > v 3 > v 1 D) v 1 = v 2 = v 3 h 1 3 Sample answers for D: ΔE = ΔK + ΔU = 0 All of the balls started from the same position and will ultimately reach the same destination, the ground, and the force of gravity is conservative which means it is independent of the path taken. because each of them starts off with the same amount of kenetic energy and they are all acted on by gravity over the same distance(the distance between the origanal position and the ground) they are all traveling at the same speed when they hit the ground. Mechanics Lecture 8, Slide 8

9 ACT: Checkpoint followup Three balls of equal mass are fired simultaneously with equal speeds from the same height h above the ground. Ball 1 is fired straight up, ball 2 is fired straight down, and ball 3 is fired horizontally. Which of the following quan==es are NOT the same for the three balls as they move from height h to the floor: 2 h 1 3 A C B D A) The change in their kinetic energies B) The change in their potential energies C) The time taken to hit the ground Mechanics Lecture 8, Slide 9

10 Exercise: Block and Spring A block of mass m is launched toward a spring with initial speed v. The spring is attached at one end to a wall and has a spring constant k. What is the maximum compression of the spring? m v ΔL m Mechanics Lecture 8, Slide 10

11 CheckPoint: Box and Spring A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x 1 from its relaxed position while momentarily coming to rest. If the initial speed of the box were doubled, how far x 2 would the spring compress? x 2 = 2x 2x x = x = 4x 2 1 A) B) C) x About 56% of you answered correctly. Mechanics Lecture 8, Slide 11

12 Revote: Box and Spring x A C B D K = 1 2 mv2 U = 1 2 kx2 K i = U f 1 2 mv2 = 1 2 kx2 x 2 = 2x 1 2 2x1 x = 1 A) B) C) x 2 = 4x A) Because the potential energy given for a spring is (1/2)mD^2 B) The work done by the spring varies with the square of the compression. The kinetic energy varies with the square of the velocity. So these balance and the compression of the spring should vary linearly with the speed. C) Because kinetic energy is equal to the square of velocity, a doubled velocity would equal 4 times more energy. Mechanics Lecture 8, Slide 12

Revote: Box and Spring x K i = U f A) B) C) 1 2 mv2 = 1 2 kx2 x 2 = 2x 1 2 2x1 x = 1 A) Because the potential energy given for a spring is (1/2)mD^2 x 2 = 4x B) The work done by the spring varies

13 Revote: Box and Spring x K i = U f A) B) C) 1 2 mv2 = 1 2 kx2 x 2 = 2x 1 2 2x1 x = 1 A) Because the potential energy given for a spring is (1/2)mD^2 x 2 = 4x B) The work done by the spring varies with the square of the compression. The kinetic energy varies with the square of the velocity. So these balance and the compression of the spring should vary linearly with the speed. C) Because kinetic energy is equal to the square of velocity, a doubled velocity would equal 4 times more energy. Mechanics Lecture 8, Slide 13

14 ACT: Block on Spring, part I A block attached to a spring is oscillating between point x (fully compressed) and point y (fully stretched). The spring is unstretched at point O. At point O, which of the following quantities is at its maximum value? A C B D x o y A) The block s kinetic energy B) The spring s potential energy C) Both A) and B) Where is the block moving fastest? Mechanics Lecture 8, Slide 14

15 ACT: Block on Spring, part II A block attached to a spring is oscillating between point x (fully compressed) and point y (fully stretched). The spring is unstretched at point O. A C B D At point x, which of the following quantities is at its maximum value? x o y A) The block s kinetic energy B) The spring s potential energy C) Both A and B Where is the block moving slowest? Since E tot = K + U = const Mechanics Lecture 8, Slide 15

16 ACT: Block on Spring, part III A block attached to a spring is oscillating between point x (fully compressed) and point y (fully stretched). The spring is unstretched at point O. At which point is the acceleration of the block zero? A C B D A) At x x o y B) At O ~a = ~ F /m Where is F = 0? C) At y Mechanics Lecture 8, Slide 16

17 Vertical Springs M kx 2 x Mechanics Lecture 8, Slide 17

18 Exercise: Block on Ramp A block of mass m is launched up a frictionless ramp with an initial speed v. How high does it go? m v h Mechanics Lecture 8, Slide 18

19 ACT: Block on Ramp A block of mass m is launched up a frictionless ramp with an initial speed v and reaches a maximum vertical height h. A second block having twice the mass (2m) is launched up the same ramp with the same initial speed (v). What is the maximum vertical height reached by the second block? A C B D A) h 2 B) h C) 2h v m h D) 4h U = K Analysis: U f 0= (0 K i ) mgh = 1 mv 2 2 h = 1 v 2g 2 independent of mass Mechanics Lecture 8, Slide 19

20 ACT: Block on Ramp, II A block with initial speed v slides up a frictionless ramp to a height h. How high does the block go if it starts with initial speed 2.45v? A C B D A) 2.45h B) 2.45h C) 6.0h D) 4.9h m v? Mechanics Lecture 8, Slide 20

21 Example problem A block of mass 3 kg is moved up an incline that makes an angle of 37 with the horizontal under the action of a constant horizontal force of 40 N. The coefficient of kinetic friction between the block and the incline is 0.1. The block is initially at rest. What is the kinetic energy of the block after it has been displaced 2 m along the incline? Mechanics Lecture 8, Slide 21

22 ACT: Earth s gravity An object of mass m is released from a height h above the earth s surface (not necessarily close to its surface). If it starts from rest, what is its kinetic energy when it lands on earth? A C B D R E h Mechanics Lecture 8, Slide 22

23 ACT: Earth s gravity An object of mass m is released from a height h above the earth s surface (not necessarily close to its surface). If it starts from rest, what is its kinetic energy when it lands on earth? 1 GmM E R E 1 1 R E + h GmM E R E + h mgh mg h R E A) B) C) D) A C B D ( ) R E h Mechanics Lecture 8, Slide 23

24 CheckPoint: Earth s Gravity In Case 1 we release an object from a height above the earth s surface equal to 1 earth radius, and we measure its kinetic energy just before it hits the earth to be K 1. In Case 2 we release an object from a height above the earth s surface equal to 2 earth radii, and we measure its kinetic energy just before it hits the earth to be K 2. Compare K 1 and K 2. A) K 2 = 2K 1 B) K 2 = 4K 1 C) K 2 = (4/3)K 1 D) K 2 = (3/2)K 1 Case 1 Case 2 Mechanics Lecture 8, Slide 24

25 What is the work done by gravity in Case 1? A) ACT: Earth s Gravity W case1 = GM e m 1 1 2R e R e = 1 2 GM e m R e A C B D B) W case1 = GM e m 1 1 R e 2R e = 1 2 GM e m R e C) W case1 = GM e m 1 1 R e 2R e = GM m e R e Case 1 Case 2 R E Mechanics Lecture 8, Slide 25

26 What is the work done by gravity in Case 2? A) ACT: Earth s Gravity W case2 = GM e m 1 1 3R e R e = 2 3 GM e m R e A C B D B) W case2 = GM e m 1 1 R e 3R e = 2 3 GM e m R e Case 1 Case 2 R E Mechanics Lecture 8, Slide 26

27 W case1 = 1 2 GM e m R e W case2 = 2 3 GM e m R e A C B D ΔK 2 ΔK 1 What is the ratio? K 2 K 1 = W 2 W 1 = 2/3 1/2 = 4 3 A) 2 B) 4 C) 4/3 D) 3/2 Case 1 Case 2 Mechanics Lecture 8, Slide 27

28 CheckPoint Results: Earth s Gravity In Case 1 we release an object from a height above the surface of the earth equal to 1 earth radius, and we measure its kinetic energy just before it hits the earth to be K 1. In Case 2 we release an object from a height above the surface of the earth equal to 2 earth radii, and we measure its kinetic energy just before it hits the earth to be K 2. Compare K 1 and K 2. A) K 2 = 2K 1 B) K 2 = 4K 1 C) K 2 = (4/3)K 1 D) K 2 = (3/2)K 1 Case 1 Case 2 Mechanics Lecture 8, Slide 28

Classical Mechanics Lecture 8

Classical Mechanics Lecture 8 Today's Concepts: a) Poten2al Energy b) Mechanical Energy Mechanics Lecture 8, Slide 1 Stuff you asked about: Gravity is the law. violators will be brought down. How were