If you have a conflict, you should have already requested and received permission from Prof. Shapiro to take the make-up exam.

Transcription

1 Reminder: Exam this Sunday Nov. 9. Chapters , 3.4,.0, 6, 7. Time: 6:0 7:30 PM Look up locations online. Bring calculator and formula sheet. If you have a conflict, you should have already requested and received permission from Prof. Shapiro to take the make-up exam.

2 ELASTIC POTENTIAL ENERGY AND ENERGY CONSERVATION Review of last lecture ork done by varying force. curved path: F ds Hooke s Law for Springs: F x kx kx displacement from equilibrium Power: P t F x t F v Gravitational Potential Energy: U g mgy ork Energy Theorem: other E E here E = K + U (Total Energy) If: 0 then E E 0 other

3 i-clicker Abbie and Bonita decide to race up a hill that is 30 meters high. Abbie takes a path that is 60 meters long while Bonita uses a path that is 00 meters long. It takes Abbie 40 sec since her route is steep, while Bonita runs up her path in 30 sec. They both start from rest at the same height and stop at the top. They have the same weight. The average power generated going up the hill is A.) greater for Abbie than Bonita. B.) greater for Bonita than Abbie. C.) the same for both. D.) Not enough information provided.

4 i-clicker In each case below, a toboggan starts from rest and slides without friction down a hill. The toboggans are all identical, and the vertical starting heights and angles of the hills are given. Rank these situations by the speed of the toboggan at the bottom.. A > B > D > E > F > C. D > B > A > E > C > F 3. A = B = D > E > C = F 4. A = F > B = C > D = E

5 i-clicker Amanda and Bertha are in a car race. Their cars have the same mass. At one point in the race, they both change their speeds by 0 m/s in seconds. Ignore air friction. hich is true about the power generated by the car while speeding up? A. Amanda s is greater than Bertha s B. Amanda s is less than Bertha s C. Amanda s is the same than Bertha s D. It cannot be determined without knowing Force E. It cannot be determined without knowing Distance

6 i-clicker Three identical balls are thrown with equals speeds from the top of a building at angles to the horizontal as shown. Neglect air resistance. hat is the relationship between their speeds when the balls hit the ground? A) v = v = v 3 B) v = v < v 3 C) v < v < v 3 D) v > v > v 3 K U 0

7 ELASTIC POTENTIAL ENERGY F x Spring at equilibrium x = Displacement from equilibrium ork done on spring by force F to create displacement x : BUT kx kx kx 0 ( x As spring is stretched, it applies an equal and opposite force on whatever is causing the stretching. here el is work done by the spring. 0) el el kx kx x x If 0 el hen stretched spring retracts: F s el kx kx 0! x x

8 As with gravity, work done by spring can be characterized as change in potential energy: U el kx el kx kx Uel U el U el Recall Energy Displacement diagram: Note: U(x) must be referenced to x = 0! (different from gravity) (This is because the elastic force depends on position while the force of gravity does not) So, if a spring acts on a body, the work done by the elastic force on the body is: x tot el U U el el K K K

9 Suppose gravity and elastic forces are both present? TOT g el other K K other : ork done on object by non-gravitational and non-elastic forces (e.g., friction, muscle force) other K K ) ( ( K U g U ) ( ) el K U g U el K U ) ( K ) el ( U g ( K K) ( U g U g) ( Uel Uel) E E STRATEGY Define system. For U g, the system includes the object and the earth! Define initial () and final () states. Identify other forces that cannot be described by potential (such as friction, muscle, motor). Positive work done by other forces increases the total mechanical energy of the system. Negative work done by other forces (such as friction) decreases the total mechanical energy.

10 If ONLY elastic force acts on body: other E E 0 E E mv kx mv kx EXAMPLE : It takes a force of 800 N to extend a spring 0.0 m. 0. m 5 kg (a) hat is the PE stored in the spring? F = kx 800 N = k (0.0 m) k = 4000 N/m U el = (/)kx = (/)(4000 N/m)(0.04 m ) = 80 J (b) A 5 kg mass is attached to the spring and then released. hat is the speed of the mass at the instant the spring returns to its equilibrium position? (no friction) E - E = 0: E = K + U el = J E = K + 0 = (/) mv E = E (/)(5 kg) v = 80 J v = (80 J)/(5 kg) = 3 m /s v = 5.6 m/s

11 Example : A.0 kg block is pushed against a spring (k = 500N/m), compressing it 0.0 m. Block is released and slides along a frictionless surface up a 45 incline. How far up does it go? h = l sinθ Use conservation of mechanical energy, potential energies for gravity and spring, no other forces. K U U E E K g el g el U U Final state: K U U g el 0 mgl sin 0 Initial state: mgl sin kx K U U g el 0 0 kx l kx mgsin (500 N/m)(0.0 m) (.0)(9.8 m/s )sin m

12 i-clicker speed is changed U g mgy U el kx

13 Last week s i-clicker A skateboarder is launched by a giant spring initially compressed a distance x as shown at right. His speed on the horizontal portion of the ramp is v. He then conducts a second launch with the spring initially compressed a distance x. For the second launch, what can you say about the speed of the skateboarder on the horizontal portion of the ramp? A. The speed is v. B. The speed is v. C. The speed is 4v. D. None of the above. Last week: System = skateboarder K K Now: System = spring + skateboarder 0 = E 0 = K + U 0 = mv kx

14 Last week s i-clicker A skateboarder is launched by a giant spring initially compressed a distance x as shown at right. He rises to a height H after he leaves the ramp. He then conducts a second launch with the spring initially compressed a distance x. For the second launch, what can you say about the height of the skateboarder on the horizontal portion of the ramp? A. The height is H. B. The height is H. C. The height is 4H. D. None of the above. Last week: System = skateboarder + earth: non grav kx E mgh E Now: System = skateboarder + earth + spring: 0 = E 0 = K + U g + U el 0 = 0 + mgh kx

15 Example: You are designing a delivery ramp for heavy crates (470 N) moving at top with v =.80 m/s. The ramp is 8.00 m long, inclined at and exerts frictional force f k = 550 N. At the bottom, the crate compresses the spring by d = 0.45 m and stops - the static friction force prevents it from rebounding. hat is spring constant k? Identify: Use potential energies for gravity, spring (U g, U el ) and include other (friction). Execute: friction E E K U g U el K U g U el friction

16 Final state: Initial state: K U U g el 0 0 kd K U U g mv mgh 0 el wl sin Also: friction f k l (why minus sign?) Substitute into work-energy theorem: kd mv wl sin f k l 470N (.8 m/s) o (470 N)(8 m)sin( ) 550 N8 m 9.8m/s kd 48 J use d 0.45 m k = 450 N/m

17 CONSERVATIVE AND NON-CONSERVATIVE FORCES A conservative force allows two-way conversion between kinetic and potential energy. Properties of work done by conservative force: ork done can be expressed as difference between initial and final states of a potential energy function. ork done is reversible. ork done on a body is independent of path - it depends only on starting and ending points. ork done through a closed path (same beginning and end point) is zero. cons. force U (gravity, spring, electrostatic forces) If there are only conservative forces, total mechanical energy is conserved: E K U constant E 0 Non-conservative forces include friction, air resistance, muscle force, chemical reactions, If there are non-conservative forces, include their work: non conserv. E

18 LA OF CONSERVATION OF ENERGY Non-conservative forces cause change of internal energy of object(s) (e.g., friction causes heating, and temperature change is related to change of internal energy). K U non cons. 0 e.g., for friction, cons. U non internal K U U internal 0 Define total energy: E total K U U internal So, if you account for all forms of energy, the total energy of a closed system is conserved: E total 0 hat is a closed system? No energy gets in or out. Energy is neither created or destroyed, it only changes form. Although U int formally looks like a potential energy, it is not: Consider that heat cannot easily be converted back to kinetic energy!

19 Force and Potential Energy Conservative force is intimately related to potential energy function. For conservation force in -D, c U For small displacement Check: F x For spring, For gravity: In 3-dimensions: U x U U kx mgy F x, c F x x U du ( x) F x dx F F du dx kx Restoring Force du dy mg (weight is in - y direction) du du du ( i j k) dx dy dz

20 ENERGY DIAGRAMS For a spring, potential energy function: U( x) kx Energy diagram: At equilibrium: du F kx 0 dx General case: For spring, equilibrium is stable, because F attracts displaced object back to x = 0

21 i-clicker A spaceship is located in a region of space where there are conservative forces acting on it. Below is a graph of the potential energy as a function of the spaceship s position. The points A-F represent specific locations along the path. Rank the magnitude of the force on the spaceship at the labeled points.. E > A = B > C = D > F. A = B > C = D > E = F 3. A > B > C = D > E > F 4. F = E > D = C > B = A 5. F = E > B = A > C = D

22 i-clicker E K K mv

23 Reminder: Exam this Sunday Nov. 9. Chapters , 3.4,.0, 6, 7. Time: 6:0 7:30 PM Look up locations online. Bring calculator and formula sheet. If you have a conflict, you should have already requested and received permission from Prof. Shapiro to take the make-up exam.