Physics 2210 Fall smartphysics 08 Conservative Force and Potential Energy 09 Work and Potential Energy, Part II 09/30/2015

Transcription

1 Physics 10 Fall 015 smartphysics 08 Conservative Force and Potential Energy 09 Work and Potential Energy, Part II 09/30/015

2 Conservative forces Definition: Forces whose work done an object is (always) path-independent are called conservative forces The work done by a conservative force from point 1 to any other point, and back to 1 along any closed loop is ALWAYS ZERO. i.e. for any conservative force F C F C dr 0 Integrating over a closed loop Identically zero: True for any loop Subscript C indicates a conservative force

3 Example of non-conservative forces 1. Forces (of magnitude F A ) exerted by a hand in moving an object at constant speed on a rough surface through a closed loop. The kinetic force of friction (of magnitude f s ) on that same object The applied force F A is always in the direction of motion: W A > 0 over closed loop The friction force f k is always opposite the direction of motion: W ff < 0 over closed loop

4 Unit 07 NOTE: x is not spring length

5 Unit 08 This last point is confusing don t use it and don t think about it. I prefer you remember the definition this way: U r = U r 0 W r 0 r = W r 0 r Example: gravity (CHOOSING U = 0 at y = 0) U y = U 0 W 0 y = 0 mm y 0 = mmm

6 Unit 08

7 Gravity is a conservative force r i N f k N r i f k r f N f k m mg y<0 here r f N f k mg mg The work done by the force of gravity (near surface of Earth), a uniformly constant force, is given by (assuming UP to be the +y direction) Vertical component of force of gravity F g W g = F g r W g = mm y mg Potential Energy of gravity U g y U g 0 = W g = mm y U g y = mmm Vertical component of displacement r CHOOSE U g = 0 at y = 0 and UP to be +y

8 Poll 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 is fired straight down, and ball 3 is fired horizontally. Rank the speeds of the balls, v 1, v, and v 3, just before each ball hits the ground. A. v > v 3 > v 1 B. v 3 > v > v 1 C. v 1 > v > v 3 D. v 1 = v = v 3

9 Example (1/3) (a) The change in your gravitational potential energy on taking an elevator from the ground floor to the top of the Empire State Building. The building is 10 stories high (assuming a 3 min ride to the top of the building). (Assuming your mass is 71 kg and the height of one story to be 3.5 m.) Give your answer in kj (b) Find the average force exerted by the elevator on you during the trip in newtons (N) (c) Find the average power delivered by that force in kilowatts (kw) (%i1) DPEg: m*g*h; (%o1) g m H (%i) DPEg, m=71, g=9.81, H=10*3.5; (%o) (%i3) /* change to kj by multiplying 1 = 1kJ/1000J */ %/1000; (%o3) Answer (a) 49 kj

10 Aside: Power Standard Horse Power: 1 hp = W

11 Example (3/3) (b) Find the average force exerted by the elevator on you during the trip. (c) Find the average power delivered by that force in kilowatts (kw) (%i4) /* assuing constant speed then the force exerted by the elevator on you is equal to m*g upward to cancel your weight */ F: m*g; (%o4) (%i5) F, m=71, g=9.81; g m (%o5) Answer (b) 697 N (%i6) /* Work done by elevator = F*H */ W: F*H; (%o6) (%i7) /* power: P=W/t */ P: W/t; g m H g m H (%o7) (%i8) /* t=3 min=180s */ P, m=71, g=9.81, H=10*3.5, t=180; (%o8) (%i9) /* convert to kw by mupltiplying 1=1kW/1000 W */ %/1000; (%o9) Answer (c) 1.38 kw t

12 Other Conservative Forces All central forces are conservative A central force is: one that is directed always towards or away from a center (we usually assign this to be the origin) whose magnitude does not depend on the orientation Whose magnitude depends only on the distance to the center Examples Force exerted by a spring on a body tied to it at one end (often the other end is tied to the Earth) Force of gravity on a small object by another (often much larger) object. Electrical (actually: electrostatic) forces between charged objects (PHYS 0)

13 r i Top View O In the limit N the blue path becomes the black path Frictionless, horizontal surface Spring of force constant k relaxed length L 0 W s = The green path gives the same work as the blue path r f m r f The red path also gives the same work as the green and blue paths Path from r i to r f Approximate path by N that alternate n = 1,,3, N, (n 1)th: radial (n)th: arc F s r for arcs: work done along the even segments vanish F r n r n = 0 F s r for radial steps: Only work done along the odd segments contribute F r n 1 r n 1 = F r n 1 r n 1 F(r)dd lim N F r n 1 r n 1 r i n=1 N F r = k(r L 0 )

14 r f W s F s r dd r i Work done by a Spring r f = k r L 0 dd Change of Variable x = r L 0, x = dr r i r f L 0 W s k xdx = 1 kx r i L 0 r i L 0 W s = 1 k r f L 0 ri L 0 r f = k r L 0 dd r i r f L 0 Or of we let x represent the deformation (which we already did), then W s = 1 k L f Li, or W s = 1 k x f x i Potential Energy of a Spring U S r U S L 0 = W s L0 r = 1 k r L 0 L 0 L 0 = + 1 k r L 0 0 = 1 k r L 0 Taking the relaxed spring to have ZERO potential energy (usually best choice) NOTE: U S = 1 k L, or U S = 1 kx x is not spring length NOTE: x is not spring length

15 Poll 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 would the spring compress? A. x = x 1 B. x = x 1 C. x = 4 x 1 NOTE: x is not spring length

16 Example 08-0 (1/3) A block of mass m is pushed up against a spring, compressing it a distance x, and the block is then released. The spring projects the block along a frictionless horizontal surface, giving the block a speed v. The same spring is then used to project a second block of mass 4m, giving it a speed of 5v. What distance x was the spring compressed in the second case? Answer in terms of a numerical factor times x, the compression of the first block. (%i1) /* total energy */ Energy: 0.5*mass*speed^ + 0.5*k*compression^; (%o1) 0.5 mass speed compression k (%i) E1i: Energy, mass=m, speed=0, compression=x; (%o) 0.5 k x (%i3) E1f: Energy, mass=m, speed=v, compression=0; (%o3) 0.5 m v (%i4) eqn1: E1i = E1f; (%o4) 0.5 k x = 0.5 m v (%i5) Ei: Energy, mass=4*m, speed=0, compression=x; (%o5) 0.5 k x... continued

17 Example 08-0 (/3) Spring on mass m, compression x, results in speed v. same spring on mass 4m: speed of 5v. What distance was the spring compressed in the second case? i.e. x / x =? (%i6) Ef: Energy, mass=4*m, speed=5*v, compression=0; (%o6) 50.0 m v (%i7) eqn: Ei = Ef; (%o7) 0.5 k x = 50.0 m v (%i8) /* strategy: solve for k in each case */ soln1: solve(eqn1, k); m v (%o8) [k = ----] x (%i9) k1: rhs(soln1[1]), numer; m v (%o9) ---- x... continued

18 Example 08-0 (3/3) Spring on mass m, compression x, results in speed v. same spring on mass 4m: speed of 5v. What distance was the spring compressed in the second case? i.e. x / x =? (%i10) soln: solve(eqn, k); 100 m v (%o10) [k = ] x (%i11) k: rhs(soln[1]), numer; 100 m v (%o11) x (%i1) soln3: solve(k1=k, x); (%o1) [x = - 10 x, x = 10 x] (%i13) /* take positive root */ x: rhs(soln3[]); (%o13) 10 x Answer: 10

19 Material covered in homework for Unit 8 stops here

20 Universal Gravitation (in HW for unit 9) r M F G m We treat the gravitational force exerted by a very large spherical mass M on a small mass m as if M is stationary with its center at the origin. Then the force on m always points towards the origin, and with a radial component of F r = GGG r The minus sign means it points inward r f W G = F r r dd r i r f = r i GGG r dd W G = GGG r f = GGG r dd r i 1 r f 1 r i GGm = 1 r 1 r i r f It is conventional to choose U G = 0 at r = U G = 0 GGG 1 r 1 = GGG r U G = GGG r

21 Poll (checkpoint for unit 8) Consider two identical objects released from rest high above the surface of the earth (neglect air resistance for this question). Case 1: we release an object from a height above the surface of the earth equal to 1 earth radius, its kinetic energy just before it hits the earth is K 1. Case we release an object from a height above the surface of the earth equal to earth radii, its kinetic energy just before it hits the earth to be K. Compare the kinetic energy of the two just before they hit the surface of the earth. A. K = K 1 B. K = 4 K 1 C. K = (4/3) K 1 D. K = (3/) K 1

22 Unit 09 In your instructor terms: E = j work done by non conservative force j

23 Unit 09