PHYS Summer Professor Caillault Homework Solutions. Chapter 5

Transcription

1 PHYS Summe Pofesso Caillault Homewok Solutions Chapte 5

2 7. Pictue the Poblem: The ball is acceleated hoizontally fom est to 98 mi/h ove a distance of 1.7 m. Stategy: Use equation 2-12 to detemine the acceleation of the ball, and then use Newton s Second Law (equation 5-1) to find the net foce on it. Solution: 1. (a) Use equation 2-12 to find the acceleation of the ball: 2. Use equation 5-1 to find the net foce: v 2 x = v 2 0x + 2aΔx a = v 2 2 x v 0x ˆx = 2Δx ( 98 mi/h m/s mi/h) 2 0 ˆx = 564 m/s 2 ˆx m F = ma = ( 0.15 kg) ( 564 m/s 2 ˆx ) = ( 85 N) ˆx 3. (b) If the mass of the ball is inceased, the foce must be inceased to poduce the same acceleation. Insight: While the foce is not lage (85 N is about 19 lb), the acceleation is impessive, about 57 times the acceleation of gavity! We kept an exta significant digit in step 1 in ode to avoid ounding eo. 15. Pictue the Poblem: The paent and child push off each othe while on ice skates. The child is pushed in the positive x diection and the paent is pushed in the negative x diection. Stategy: The situation is simila to example 5-3, whee the foce F paent exeted on the paent by the child is equal and opposite to the foce F child exeted on the child by the paent. Solution: 1. (a) The magnitude of the foce expeienced by the child is the same as the magnitude of the foce expeienced by the paent, but they point in opposite diections. 2. (b) The acceleation of the child is moe than the acceleation of the paent. The child expeiences the same foce as the paent but has less mass, so she expeiences the lage acceleation. 3. (c) Set the magnitudes of the foces equal to find the acceleation of the paent using equation 5-1: F child = F paent m child a child = m paent a paent a paent = m child 19 kg a m child = paent 71 kg 2.6 m/s2 = 0.70 m/s 2 Insight: If the child wee olde and had the same mass as the paent, they would each expeience the same magnitude foce and the same magnitude acceleation, but in opposite diections. 23. Pictue the Poblem: The two teenages pull on the sled in the diections indicated by the figue at ight. Stategy: Wite Newton s Second Law in the x diection (paallel to a ) in ode to find the acceleation of the sled. Solution: F x Wite Newton s Second Law in the x diection: = 2F cos35 57 N = ( m sled + m child ) a x 2F cos35 57 N a x = m sled + m child a x = 2 ( 55 N )cos35 57 N kg = 1.5 m/s 2 Insight: Some of the foce exeted by the teenages is exeted in the y diection and cancels out; only the x

3 components of the foces move the sled. 27. Pictue the Poblem: The uncoupled tain ca coasts up the incline, slowing down unde the influence of gavity, and biefly comes to est befoe acceleating backwads down the incline. Stategy: Afte the last ca beaks fee, it will continue coasting up the incline until the foce of gavity bings it momentaily to est. Let the ˆx diection point uphill and paallel to the incline. The only foce acting on the tain ca is the paallel component of its weight, W x = mg sinθ. Use the known acceleation a x = g sinθ to find the time equied to bing the tain ca to a stop, and then use equation 2-10 to detemine the distance it tavels along the incline while slowing down. Solution: 1. (a) Find the time equied to stop the tain ca: t = v v 0 a x = 0 v 0 g sinθ = 3.15 m/s = 4.7 s ( 9.81 m/s 2 )sin (b) Use Eq to find the distance taveled while stopping: x = 1 ( v v )t = 1 ( m/s) 4.7 s 2 = 7.4 m Insight: In most cases when motion occus along an inclined suface it is much simple to tilt the coodinate axes until they ae paallel and pependicula to the incline, as it was in this case. 33. Pictue the Poblem: The elevato acceleates up and down, changing you appaent weight W a. A fee body diagam of the situation is depicted at ight. Stategy: Thee ae two foces acting on you: the applied foce F = W a of the scale acting upwad and the foce of gavity W acting downwad. The foce W a epesents you appaent weight because it is both the foce the scale exets on you and the foce you exet on the scale. Use Newton s Second Law togethe with the known foce W a acceleation to detemine the acceleation a. Solution: 1. (a) The diection of acceleation is upwad. An upwad acceleation esults in an appaent weight geate than the actual weight. 2. (b) Use Newton s Second Law togethe with the known foces to detemine the acceleation a. = W a W = ma a = W a W m = W a W W g N = 9.81 m/s N = 1.9 m/s 2 3. (c) The only thing we can say about the velocity is that it is changing in the upwad diection. That means the elevato is eithe speeding up if it is taveling upwad, o slowing down if it is taveling downwad. Insight: You feel the effects of appaent weight twice fo each ide in an elevato, once as it acceleates fom est and again when it slows down and comes to est. 35. Pictue the Poblem: The seeds fall fom the tee at constant speed. Stategy: Because the fuit is falling with constant speed, we conclude that the net foce on it is zeo. Use Newton s Second Law to detemine the foce the ai exets on the seeds.

4 Solution: 1. (a) Use Newton s Second Law to find the foce of ai esistance: F = F ai + W = 0 F ai = W = mg = ( kg) 9.81 m/s 2 = N 2. (b) If the constant speed of descent is geate than 1.1 m/s, the acceleation still emains zeo, so that the foce of ai fiction emains the same as in pat (a). Insight: The foce of ai fiction depends upon speed and the shape of the object that is passing though the ai. A seed with a moe steamlined shape will fall at a geate and geate speed until F ai inceases to N and the net foce becomes zeo once again. At that point the seed is said to have achieved its teminal velocity. 39. Pictue the Poblem: The child sits on a chai and the chai sits on the floo. The fee body diagams of the child and the chai ae shown at ight. Stategy: Thee ae two foces acting on the child: the nomal foce N of the chai acting upwad and the foce of gavity W child acting downwad. Thee ae thee foces acting on the chai: the nomal foce N of the floo acting upwad, the weight of the baby acting downwad, and the foce of gavity W chai acting downwad. Wite Newton s Second Law in the vetical diection fo each case and then use the equations to Solution: 1. (a) Wite out = N mg = 0 Newton s Second Law in the vetical N = mg = 9.3 kg diection and solve fo N: = 91 N 2. (b) Wite out Newton s Second Law in the vetical diection fo the chai and solve fo N: 9.81 m/s 2 = N m baby g m chai g = 0 g ( 9.81 m/s ) 2 N = m baby + m chai = kg N = 130 N = 0.13 kn (a) child Insight: The nomal foce is lage in case (b) because the floo must suppot the weight of the child plus the weight of the chai, wheeas the chai must only suppot the weight of the child.

5 43. Pictue the Poblem: The fee body diagam of the lawn mowe is shown at ight. Stategy: Wite Newton s Second Law in the vetical diection to detemine the nomal foce. Solution: 1. (a) Use Newton s Second Law to find N: = N F sinθ mg = ma y = 0 N = F sinθ + mg = ( 209 N)sin kg N = 290 N = 0.29 kn 9.81 m/s 2 2. (b) If the angle between the handle and the hoizontal is inceased, the nomal foce exeted by the lawn will incease because it must still balance the weight plus a lage downwad foce than befoe. Insight: The vetical acceleation of the lawn mowe will always emain zeo because the gound pevents any vetical motion.

6 51. Pictue the Poblem: The gasshoppe is acceleated along a staight line by the foce geneated by its hind legs. Stategy: Use equation 2-12 to find the acceleation of the gasshoppe, and then use Newton s Second Law to find the magnitude of the aveage net foce. Solution: 1. (a) Find the acceleation of the gasshoppe using equation 2-12: 2. (b) Apply Newton s Second Law to find F legs : a = v2 v Δx ( 2.7 m/s) = = 99 m/s m = F legs mg = ma F legs = m a + g = 0.22 N = ( kg) m/s 2 3. (c) As long as the final speed and acceleation distance emain unchanged, the takeoff acceleation will stay the same even though the mass inceases, because the acceleation is independent of mass. 4. (d) If the mass of the gasshoppe inceases, the equied takeoff foce will incease because thee is moe inetia. Insight: The foce geneated by the gasshoppe s legs is moe likely to stay the same as the mass of the gasshoppe inceases, so that its takeoff acceleation will decease and it won t be able to jump as fa.