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1 Name: Depatment: Student ID #: Notice ˆ + ( 1) points pe coect (incoect) answe. ˆ No penalty fo an unansweed question. ˆ Fill the blank ( ) with ( ) if the statement is coect (incoect). ˆ : coections to an incoect answe. ˆ : solution. ˆ Textbook: Walke, Halliday, Resnick, Pinciples of Physics, Tenth Edition, John Wiley & Sons (014). 8-1 Potential Enegy 1. The change U in gavitational potential enegy is defined as being equal to the negative of the wok done on an object by the gavitational foce. (a) ( ) The absolute value of the potential enegy is not unique but the change is uniquely defined. (b) ( ) The change U in the gavitational potential enegy is U = W g. (c) ( ) The gavitational potential enegy U in the unifom gavitational field with the gavitational acceleation g = gĵ is U = U 0 = U 0 F g dx ( ) mgĵ y = U 0 + mg dy y 0 = U 0 + mg(y y 0 ), (î dx + ĵ dy + ˆk ) dz whee y is the height paallel to g fom the oigin of a fame of efeence and U 0 is the value of the potential enegy at the efeence point of height y 0. A conventional choice is U 0 (0, 0, 0) = 0. Then, U(x, y, z) = mgy.. The change U in sping potential enegy is defined as being equal to the negative of the wok done on an object by the sping foce. (a) ( ) The absolute value of the potential enegy is not unique but the change is uniquely defined. (b) ( ) The change U in the sping potential enegy is U = W sping. (c) ( ) The sping potential enegy U is U = U 0 = U 0 = U 0 + k F sping dx ( kxî) (î dx) xdx = U k(x x 0), whee k is the sping constant and x is the displacement fom the elaxed point. The value of U 0 can be set abitaily. A conventional choice is = 0. Then, U(x) = 1 kx. 3. Consevative and Nonconsevative Foces: When the system configuation changes, the foce does wok (call it W 1 ) on the paticle-like object, tansfeing enegy between the kinetic enegy K of the object and some othe type of enegy of the system. When the configuation change is evesed, the foce eveses the enegy tansfe, doing wok W in the pocess. (a) ( ) In a situation in which W 1 = W is always tue, the othe type of enegy is a potential enegy and the foce is said to be a consevative foce. Depatment of Physics, Koea Univesity Page 1 of 5

2 (b) ( ) A foce that is not consevative is called a nonconsevative foce. The kinetic fictional foce and dag foce ae nonconsevative. (c) ( ) Duing the sliding of a block on a fictional floo, a kinetic fictional foce fom the floo slows the block by tansfeing enegy fom its kinetic enegy K to a type of enegy called themal enegy E themal, which has to do with the andom motions of atoms and molecules. (d) ( ) Themal enegy cannot be tansfeed back to the kinetic enegy of the block by the kinetic fictional foce. (e) ( ) When only consevative foces act on a paticle-like object, we can geatly simplify othewise difficult poblems involving motion of the object by intoducing the consevation of mechanical enegy, the sum of the kinetic enegy and the potential enegy. 4. Path Independence of Consevative Foces (a) ( ) The net wok done by a consevative foce on a paticle moving aound any closed path is zeo. (b) ( ) The wok done by a consevative foce on a paticle moving between two points does not depend on the path taken by the paticle. (c) ( ) The gavitational potential enegy associated with a paticle Eath system depends only on the vetical position y (o height) of the paticle elative to the efeence position y = 0, not on the hoizontal position. 5. ( ) The unit of a potential enegy can be any one of the following: W s, J, kg m /s. 6. ( ) The dimensions of potential enegy can be expessed as [U] = [F ][L]. whee [F ], and [L] ae the dimensions of foce, and length. 7. Conside a nonconsevative foce. (a) ( ) A nonconsevative foce does not violate Newton s thee laws of motion. (b) ( ) A nonconsevative foce does wok. (c) ( ) A nonconsevative foce espect the enegy consevation. 8. Two paticles inteact by consevative foces. In addition, an extenal foce acts on each paticle. They complete ound tips, ending at the points whee they stated. (a) ( ) The total kinetic enegy of the two-paticle system may not be conseved. (b) ( ) The total potential enegy of the two-paticle system is conseved. (c) ( ) Because of the extenal foce, the total linea momentum of the two-paticle system may not be conseved. 9. ( ) A foce on a paticle is consevative, if its wok depends on the endpoints of the motion, not the path between. 8- Consevation of Mechanical Enegy 1. The mechanical enegy E mechanical of a system is the sum of its potential enegy U and the kinetic enegy K of the objects within it: E mechanical = K + U. When a consevative foce does wok W on an object within the system, that foce tansfes enegy between kinetic enegy K of the object and potential enegy U of the system.. ( ) (a) ( ) The change K in kinetic enegy is K = K K 1 = W. (b) ( ) The change U in potential enegy is U = U U 1 = W. (c) ( ) The mechanical enegy is conseved: K 1 + U 1 = K + U. Depatment of Physics, Koea Univesity Page of 5

3 The long pendulum shown is dawn aside until the ball has isen h. It is then given an initial speed of v0. The speed of the ball at its lowest position is q vbottom = v0 + gh. Fo a block of mass m to slide without fiction up the ise of height h shown, it must have a minimum initial kinetic enegy of mgh. 3. A small object slides along the fictionless loop-the-loop with a diamete of D. At the top of the loop, the object emained in contact with the loop. The speed of the object at the top is gd v=. 5. A block of mass m attached to an ideal sping with a sping constant of k oscillates on a hoizontal fictionless suface. At t = 0 the sping is longe than its equilibium length by d, the speed of the block is v0. (a) ( ) The total mechanical enegy of the system is (a) ( ) The kinetic enegy of the object at the bottom is 5 Kbottom = mgd. 4 (b) ( ) The weight of the object at the bottom is Wbottom = 6mg. (c) ( ) The nomal foce of the tack at the top is ( ) 1 1 Emechanical = mv0 + kd. (b) ( ) The geatest speed of the block is vmax = v0 + k d. m (c) ( ) The maximum distance D of the block fom the elaxed point is m D = d + v0. k 6. A block of mass m is initially moving to the ight on a hoizontal fictionless suface at a speed v. It then compesses a sping of sping constant k. Conside the instant t = t1 when the kinetic enegy of the block is equal to the potential enegy of the sping. (a) ( ) The speed of the block is v v(t1 ) =. Depatment of Physics, Koea Univesity Page 3 of 5

4 (b) ( ) The sping is compessed by a distance of m x(t 1 ) = v k. 8-3 Reading a Potential Enegy Cuve 1. A consevative foce F can be computed analytically fom the potential enegy U(x) as F (x) = d dx U(x). (a) ( ) The sping potential enegy is U(x) = 1 kx. Then the sping foce can be computed as F (x) = d ( ) 1 dx kx = kx. (b) ( ) The speed v 0 of the paticle at the oigin is v 0 = v + d m (a + bd ). (c) ( ) The foce on the paticle at x is F (x) = ax 4bx ( ) The potential enegy of a body of mass m is given by U(x) = mgx + 1 kx. The coesponding foce is F (x) = mg kx. 4. A block is eleased fom est at point P and slides along the fictionless tack shown. (b) ( ) The gavitational potential enegy is U(y) = mgy. Then the gavitational foce can be computed as F (y) = d dy (mgy) = mg.. A paticle moves along the x axis unde the influence of a consevative foce. The potential enegy is given by U(x) = ax + bx 4, whee a and b ae constants and x is the coodinate of the paticle. The speed of the paticle at x = d was v. (a) ( ) The units of the constants can be chosen as [a] = J/m, [b] = J/m 4. (a) ( ) At point Q, its speed is v Q = g(h 1 h ). (b) ( ) If v P is not zeo, then the speed at Q is v Q = vp + g(h 1 h ). 5. Suppose that U(x) is given on a gaph. whee E mechanical is the mechanical enegy of the system. (a) ( ) At any value of x, the foce F (x) is the negative of the slope of the cuve at [x, U(x)]. (b) ( ) The kinetic enegy K of the paticle is given by K(x) = E mechanical U(x). (c) ( ) A tuning point is a point x at which the paticle eveses its motion. At this point K(x) = 0. Depatment of Physics, Koea Univesity Page 4 of 5

5 6. The paticle is in equilibium at points whee the slope of the U(x) cuve is zeo: F (x) = 0. (a) ( ) A mable placed on a hoizontal tabletop is in that state is in neutal equilibium. (b) ( ) A mable balanced on top of a bowling ball is in unstable equilibium. (c) ( ) A mable placed at the bottom of a hemispheical bowl is in stable equilibium. 8-4 Wok Done on a System by an Extenal Foce 1. (a) ( ) When a kinetic fictional foce acts within the system, then the themal enegy E themal of the system inceases. This enegy is associated with the andom motion of atoms and molecules in the system. The wok done by the extenal foce on the system is then W extenal = E mechanical + E themal. (b) ( ) The wok done by the nonconsevative foce is the change of the mechanical enegy: W extenal + W fiction = E mechanical. (c) ( ) The change E themal is elated to the magnitude f k of the fictional foce and the magnitude d of the displacement caused by the extenal foce by E themal = f k d.. ( ) The themal enegy of a system consisting of a thown ball, the Eath, and the ai is most closely associated with motions of individual paticles within the ball and the ai. 3. An elevato is ising at a constant speed. Conside the following statements: (a) ( ) The acceleation of the elevato is zeo. (b) ( ) The upwad cable foce is constant. (c) ( ) The kinetic enegy of the elevato is constant. (d) ( ) The gavitational potential enegy of the Eath-elevato system is inceasing with a constant time ate. 8-5 Consevation of Enegy 1. Consevation of Enegy: (a) ( ) The total enegy E of a system is the sum of its mechanical enegy and its intenal enegies, including themal enegy. (b) ( ) It can change only by amounts of enegy that ae tansfeed to o fom the system. (c) ( ) If wok W is done on the system, then W = E = mechanical + themal + intenal. (d) ( ) If the system is isolated (W = 0), this gives mechanical + themal + intenal = 0. (e) ( ) At two diffeent instants 1 and, the mechanical enegies ae elated as E mechanical,1 = E mechanical, themal intenal. Depatment of Physics, Koea Univesity Page 5 of 5

Potential Energy and Conservation of Energy

Potential Energy and Conservation of Energy Potential Enegy and Consevation of Enegy Consevative Foces Definition: Consevative Foce If the wok done by a foce in moving an object fom an initial point to a final point is independent of the path (A