Potential Energy and Conservation of Energy 8.01t Oct 18, 2004Conservative Forces Definition: Conservative Work Whenever the work done by a force in moving an object from an initial point to a final point is independent of the path, then the force is called a conservative forceConservative Forces • motion of a object under the influence of a gravitational force near the surface of the earth • work done by gravity only depends on the change in the vertical W = F ∆y = − mg y ∆g g,y position,Change in Potential Energy Definition: Change in Potential Energy The change in potential energy of a bodyassociated with a conservative force is the negative of the work done by theconservative force in moving the body along any path connecting the initialposition to the final position. . B GG ∆U =− W =−∫F ⋅drsystem c c AConservation of Mechanical Energy for Conservative Forces • Change in Potential Energy is the work done by the conservative force K∆ system =Wc =−∆ Usystem • Change in Mechanical Energy is zero E∆ mechanical ≡∆ Ksystem +∆ U = 0systemPRS Question: Energy Curve Consider the following sketch of potential energy for a particle as a function of position. There are no dissipative forces or internal sources of energy. If a particle travels through the entire region of space shown in the diagram, at which point is the particle's velocity a maximum? 1. a 2. b 3. c 4. d 5. ePRS : Energy Curve - Emin 10 6 0 -8 -4 Consider the following sketch of potential energy for a particle as a function of position. There are no dissipative forces or internal sources of energy. What is the minimum total mechanical energy that the particle can have if you know that it has traveled over the entire region of X shown? 1. -8 2. 6 3. 10 4. It depends on direction of travel 5. Can’t say - Potential Energy uncertain by a constantForce and Potential Energy Difference • Potential energy difference: one dimensional motion B Ux (() =Ux ) −∫F dx 0 x A Force is then the derivative of the potential energy dUF =−x dxPotential Energy and Force: Graphs x F x F x F x F 1 2 3 4 Hint: Fx = -kx Which Graph Represents Force of Spring?Change in Potential Energy: Constant Gravity G GForce: F = mg = Fgrav y ˆj =− mg ˆjgrav , y ∆ , Work: Wgravity =Fgravity y ∆=− mg y Potential Energy: ∆Usystem =−Wgravity = mg y mgy − mgy∆ = f 0 Choice of Zero Point: Whatever “ground” isconvenientPRS Question A basketball dropped (from rest) from a height of 1 meter strikes the earth and returns to a height of 1 meter. For which choice of system is energy conserved at all times? 1. the basketball by itself 2. the earth by itself 3. the basketball plus the earth 4. all of the abovePRS Question A 1kg ball falls from a height of 1 meter. The work done on the earth-ball system is: 1. 0 J 2. 10 J 3. - 10 J 4. 20 J 5. -20 JPRS Question An object is dropped to the earth from a height of 10m. Which of the following sketches best represent the kinetic energy of the object as it approaches the earth (neglect friction). 1. a 2. b 3. c 4. d 5. ePRS Question You lift a ball at constant velocity from a height hi to a greater height hf. Considering the balland the earth together as the system, which of the following statements is true? 1. The potential energy of the system increases. 2. The kinetic energy of the system decreases. 3. The earth does negative work on the system. 4. You do negative work on the system. 5. The source energy of the ball increases. 6. Two of the above. 7. None of the above.Change in Potential Energy: Inverse Square Gravity •Force: G =−Gm mˆ 1 2F rm,m 1 2 2r • Work done: rf GGrf Gm m2 ⎞ Gm m2 rf ⎛ 1 1 ⎞1⋅dW =∫Fr=∫⎜⎛− 1 ⎟dr = = Gm m2 ⎜⎜ −⎟1 ⎟r ⎠ r r0r0 ⎝rf ⎠r0• Potential Energy r0 ⎝ 2 Change: ⎛ 1 1 ⎞ 1 − ⎟⎟U∆ gravity ≡− Wgravity =− Gm m2 ⎜⎜ rf⎝ r0 ⎠ • Zero Point: Ugravity (r = ∞) ≡ 00 • Potential Energy 1Function Ugravity (r) = −Gm m2 rChange in Potential Energy: Spring Force G ˆ •Force: F =Fˆi =−kx ix = 2 • Work done: W = xxf (−kx dx =− 1k xf 2 − x0 )∫) ( 2= • Potential Energy xx0 ( 2 Change: ∆Uspring ≡− Wspring = 1k xf 2 − x0 )2 • Zero Point: Uspring (=0)≡0 x • Potential Energy Function Uspring ()= 1kx2 x 2PRS Question • 1. In part (a) of the figure,an air track cart attached to a spring rests on thetrack at the position xequilibrium and the spring isrelaxed. In (b), the cart ispulled to the position xstartand released. It then oscillates about xequilibrium. Which graph correctlyrepresents the potentialenergy of the spring as afunction of the position ofthe cart?Non-Conservative Forces • work done on the object by the force depends on the path taken by the object • Example: friction Wfriction =−µ N∆ < 0xkNon-Conservative Forces Definition: Non-conservative Work Whenever the work done by a force in moving an object from an initial point to a final point depends on the path, then the force is called a non-conservative force.Change in Energy for Conservative Forces and Non-conservative Forces GG G • Total force Ftotal =F total +F total c nc • Total work done is change in kinetic energy B G B G Gtotal total total total Wtotal =∫F ⋅dr G =∫(Fc + F )⋅dr G =−∆ U +W =∆ Knc nc A A • Energy Change: ∆+∆ U total =WK ncPRS Question A person lifts a 1kg cup of water 1 meter at constant velocity. Defining the system as the cup of water plus the earth, the work done on the system is: 1. 10 J 2. - 10 J 3. 0 J 4. none of the abovePRS Question You lift a ball at constant velocity from a height hi to a greater height hf. Considering the ball to be the system, which of the followingstatements is true? 1. The potential energy of the system increases. 2. The kinetic energy of the system decreases. 3. The earth does negative work on the system. 4. You do negative work on the system. 5. The source energy of the ball increases. 6. Two of the above. 7. None of the above.PRS Question A block of inertia m is attached to a relaxed spring on an inclined plane. The block is allowed to slide down the incline, and comes to rest. The coefficient of kinetic friction of the block on the incline is µk. For which definition of the system is the change in total energy (after the block is released) zero? 1. block 2.
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