Problem Set 7: Potential Energy and Conservation of Energy AP Physics C Supplementary Problems
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1 Proble Set 7: Potential Energy and Conservation of Energy AP Pysics C Suppleentary Probles 1. Approxiately 5.5 x 10 6 kg of water drops 50 over Niagara Falls every second. (a) Calculate te aount of potential energy lost every second by te falling water. (b) Calculate te power output of an electric generating plant tat could convert all of te water s potential energy. (c) If te utility copany sold tis energy at an industrial rate of 1 cent/kw., wat would be teir yearly incoe fro tis source?. You drop a pysics textbook, wose ass is.0 kg, to a friend wo is standing on te ground 10 below. (a) Calculate te potential energy of te book just before you release it. (b) Calculate its kinetic energy just before your friend catces it in er outstretced ands, wic are 1.5 above te ground level. (c) How fast is te book oving as it is caugt? 3. A frictionless roller coaster car starts at point A wit speed v o. Calculate te speed of te car (a) at point B, (b) at point C, and (c) at point D. Assue tat te car can be considered a particle and tat it always reains on te track. A v o B / C D 4. A volcanic as flow is oving across orizontal ground wen it encounters a 10 o upslope. It is observed to travel 90 on te upslope before coing to rest. Te volcanic as contains trapped gas, so te force of friction wit te ground is very sall and can be ignored. At wat speed was te as flow oving just before encountering te upslope? 5. Te agnitude of te gravitational force of attraction between a particle of ass 1 and one of ass is given by 1 F ( r) = G, r were G is a constant and r is te distance between te particles. (a) Deterine te potential energy function U(r). Assue tat U ( r) 0 as r. (b) How uc work is required to increase te separation of te particles fro r = r 1 to r = r 1 +d? Probles selected fro Halliday, D., & Resnick, R. (1993). Fundaentals of Pysics (4 t ed.). New York: Jon Wiley & Sons, Inc.
2 Potential Energy and Conservation of Energy 6. A ball wit ass is attaced to te end of a very ligt rod wit lengt. Te oter end of te rod is pivoted so tat te ball can ove in a vertical circle. Te rod is pulled aside to te orizontal and given a downward pus so tat te rod swings down and just reaces te vertically upward position. (a) Calculate te cange in potential energy of te ball. (b) Calculate te initial speed iparted to te ball. 7. Te grap below sows te force in newtons as a function of stretc or copression in centieters for a cork gun containing a spring, also sown. Te spring is copressed by 5.5 c and used to propel a cork of ass 3.8 g fro te gun. (a) Calculate te speed of te cork if it is released as te spring passes troug its relaxed position. (b) Suppose now tat te cork sticks to te spring, causing te spring to extend 1.5 c beyond its unstretced lengt before separation occurs: Calculate te speed of te cork at te tie of release in tis case. 0.4 force (N) stretc (c) An ideal assless spring can be copressed.0 c by a force of 70 N. A block wose ass is 1 kg is released fro rest at te top of an incline, te angle of te incline being 30 o. Te block coes to rest eentarily after is as copressed tis spring by 5.5 c. (a) How far as te block oved down te incline at tis oent? (b) Calculate te speed of te block just as it touces te spring. 30 o
3 Potential Energy and Conservation of Energy 3 9. A.0 kg block is dropped fro a eigt of 40 c onto a spring of force constant k = 1960 N/. Find te axiu distance te spring will be copressed. 40 c 10. A projectile wose ass is 9.4 kg is fired vertically upward. On its upward fligt, an energy of 68 kj is dissipated because of air resistance. How uc iger would it ave gone if te air resistance ad been ade negligible? 11. During a rockslide, a 50 kg rock slides fro rest down a illslope tat is 500 long and 300 ig. Te coefficient of kinetic friction between te rock and te ill surface is 0.5. (a) Calculate te potential energy of te rock just before te slide. (Take y = 0 to be at te botto of te ill.) (b) Calculate te work done on te rock by frictional forces during te slide. (c) Calculate te kinetic energy of te rock as it reaces te botto of te ill. (d) Calculate its speed. 1. A.5 kg block collides wit a orizontal assless spring wose force constant is 30 N/. Te block copresses te spring a axiu distance of 7.5 c fro its rest position. Te coefficient of kinetic friction between te block and te orizontal surface is 0.5. (a) Calculate te work done by te spring in bringing te block to rest. (b) Calculate te work done by te force of friction wile te block was being brougt to rest by te spring. (c) Calculate te speed of te block wen it it te spring. 13. Two blocks are connected by a string. Tey are released fro rest. Sow tat, after tey ave oved a distance, teir coon speed is given by v = ( 1 ) 1 µ + in wic µ is te coefficient of kinetic friction between te upper block and te surface. g 1
4 Potential Energy and Conservation of Energy A certain spring is found not to confor to Hooke s law. Te force (in newtons) it exerts wen stretced a distance x (in eters) is found to ave a agnitude 5.8x x in te direction opposing te stretc. (a) Copute te total work required to stretc te spring fro x = 0.50 to x = (b) Wit one end of te spring fixed, a particle of ass.17 kg is attaced to te oter end of te spring wen it is extended by an aount x = If te particle is ten released fro rest, copute its speed at te instant te spring as returned to te configuration in wic te extension is x = (c) Is te force exerted by te spring conservative or nonconservative? Explain. 15. A sall particle slides along a track wit elevated ends and a flat central part. Te flat part as a lengt =.0. Te curved portions of te track are frictionless. For te flat part, te coefficient of kinetic friction is µ k = 0.0. Te particle is released at point A wic is a eigt = 1.0 above te flat part of te track. Were does te particle finally coe to rest? A 16. Below is a siple pendulu of lengt. Its bob is observed to ave a speed v o wen te cord akes an angle θ wit te vertical. (a) Derive an expression for te speed of te bob wen it is in its lowest position. Calculate te least value tat v o can ave if te cord is to swing up to (b) a orizontal position and (c) a vertical position wit te cord reaining straigt. θ v o
5 Potential Energy and Conservation of Energy A sall block of ass slides along a frictionless loop-te-loop track. (a) Te block is released fro rest at point P. Calculate te net force acting on it at point Q. (b) Calculate te eigt above te botto of te loop fro wic te block sould be released so tat it is on te verge of losing contact wit te track at te top of te loop. P 5R R Q Answers: 1. a).7 x 10 9 J/s b).7 x 10 9 W c) $.4 x a) 196 J b) 167 J c) 1.9 /s 3. a) v o b) v o + g /s G 5. a) 1 G b) 1d r r r + d) 1 ( 1 c) v o + g 6. a) g b) g 7. a).8 /s b).7 /s 8. a) 0.35 b) 1.7 /s c a) 1.5 x 10 6 J b) -5.1 x 10 5 J c) 9.9 x 10 5 J d) 6 /s 1. a) J b) J c) 1.0 /s 13. proof 14. a) 31 J b) 5.3 /s c) conservative because only dependent on position. 15. in te iddle, after ½ oscillations. 16. a) θ b) g cosθ g (1 cos ) + vo c) g ( 3 + cosθ ) a) 8g toward center; g downward b) R
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