Spring 2002 Test 2 Solutions. Problem 1 (Short Answer: 15 points)

Transcription

1 Spin 2002 Test 2 Solutions Poble 1 (Shot Answe: 15 points) A sled on ice oves in the was descibed in questions 1-7 below. Fiction is so sall that it can be inoed. A peson weain spiked shoes standin on the ice can appl a foce to the sled and push it alon the ice. Choose the one foce (A thouh G) which would keep the sled ovin as descibed in each stateent below. You a use a choice oe than once o not at all but choose onl one answe fo each blank. If ou think that none is coect, answe choice J. The two ke points to this poble ae that (1) the applied foce is the net foce and ust point in the diection of the acceleation, and (2) since the otions all have eithe constant o zeo acceleation, this eans onl esponses B, D, & F appl since onl these esponses descibe constant foces.

2 Poble 2 (Estiation Poble: 15 points) You have joined a voluntee fie depatent. The ae lookin to bu a new escue net because the old one boke. (A escue net is the cicula tapoline-like net fiean hold to catch people fallin fo buildins.) The cost inceases daaticall with the stenth of the net and ou fie depatent has a ve liited budet. You need to bu the cheapest escue net that eets ou depatent s needs. Assuin that the foce eeted b the escue net is constant and that the tallest buildin in ou eion is 5 stoies hih, what is the aiu foce the escue net would need to withstand? Reebe the idea is stop people befoe the hit the ound. Fo a student solution: D. Saul s coents: Althouh issin soe details that ake it had to follow and one istake, this is a ood solution. The analsis of the otion is divided into two pats. Motion 1 is fo the peson fallin fo the buildin and Motion 2 is when the peson is bein stopped b the net. Let s efe to the point whee the peson has left the buildin as point 0. The point iht befoe the peson hits the net is point 1 and the point whee the peson coes to a stop is point 2. We also need to define a coodinate sste, so let s define down as + (this is consistent with the above solution). Fo otion 1: v 0 0 /s and a 1, > v 1 v f /s ( note that if is, so is the displaceent. Thus we ae not takin the squae oot of a neative nube.)

3 Fo otion 2: Initial v v /s and final v v 2 0 /s > a /s/s So fa, so ood but now coes the istake, usin a 2 to find the foce. Usin ewton s 2 nd law, the net foce is 47 kewtons as shown above. Howeve, the question asked to find the aveae foce of the net on the peson which net foce (peson) Weiht foce of the eath on the peson, So F (net > peson) net F (> peson) W(eath > peson) (200 k) * (- 9.8 /s/s) 49 k

4 Poble 3 (Essa 10 points) You a use diaas and equations but no calculations in ou esponse fo this poble. A cat can ove to the iht o left alon a hoizontal tack (the positive pat of the ais) as shown in the fiue below. Assue that fiction is sall enouh that it can be inoed. A sonic ane is used (as shown) to easue the position, velocit, and acceleation of the cat. The tack is not necessail flat o hoizontal. In addition, the tack a be tipped o the cat a be pulled o pushed. Fo the fist un, the sonic ane displas a aph of the velocit that looks like the aph shown at the iht. Velocit Descibe the otion of the cat in wods Daw aphs showin what the sonic ane would displa fo the cat's position and the cat's acceleation. USE WHAT YOU VE LEARED FROM tie CLASS SO FAR to eplain in wods how ou cae up with ou answes (ak point A as the point whee v 0 /s.) Fo a student solution: The cat stats with an initial positive velocit which eans it is ovin awa fo the sonic ane. The ate at which it is tavelin oes down until it eaches 0 at point A. Then it continues down then havin a neative velocit so ovin towads the ane. In class we ve leaned that velocit, acceleation, and position ae all intetwined. Fo the velocit aph iven ou can find the position aph b the inteal o aea unde the aph of the velocit. We see at point A it [position] and because the aea afte A in v vs. t is neative (below t-ais) that the values fo vs. t decease. We also leaned in class that the a vs. t aph is the deivative (o aph of the slope) of the velocit vs. tie aph. As we can see fo the v vs. t aph, the slope is alwas the sae and alwas neative, so the a vs. t values coespond ivin a constant neative value. D. Saul s coent: Althouh chopp, this essa eceived 10 out of 10 points and is coect in each detail. Howeve, it could be ipoved b notin in the fist paaaph that the slope is constant and theefoe the acceleation is constant. In addition, one could note that since velocit is the slope of the vs. t aph, the slope of the vs. t aph should decease in slope thouhout.

5 Poble 4 (15 points) A cow is sittin on a telephone wie nea a field, waitin patientl fo the fae to leave so he can eat the fae s con. The cow is standin on a sall piece of wie ade staiht b the foce of his feet. a. Identif all the foces actin on the cow and on the piece of wie unde the cow s feet. Give an elationships between these foces that ou can and eplain whee the coe fo. Fo ewton's second law, since neithe the cow no the wie is acceleatin (the ae both at est) we know that the net foce on both objects is zeo and theefoe the su of the - coponents and the -coponents of the foces eeted on the two objects is also zeo. Thus, W wie cow eath wie The su of the vetical foces actin on the wie 0 The su of the hoizontal foces actin on the wie 0, 0 TLeft cable wie + TRiht cable wie TRHS wie cosθ RHS TLHS wie cosθ T RHS wie cos θ RHS T LHS wie cos θ LHS Fo ewton's Thid law, we know that cow wie wie cow Theefoe, W cow wie eath cow cow LHS

6 b. The anles that the left piece of wie and the iht piece of wie ake with the flat piece on the botto ae the sae and equal to θ. If the cow has a ass of 0.5 k, and θ 150º, find the tension in the wie. Coents: This is a pett ood solution statin fo the coponent vesions of ewton's 2 nd Law and wokin towad the sbol solution befoe pluin in nubes and calculatin the final answe. Howeve, thee ae soe ino eos. In the -coponent equation, T 1 cos θ T 2 cos θ should have been set 0 since a a 0. In the -coponent equation shown above, it appeas that the is sain that T 1 cos θ T 2 cos θ is defined to be T 1 cos θ. Thee is no weiht foce eeted b the cow on the wie (the avitational foce eeted b the cow on the wie is ve sall and can be nelected). Howeve, thee is a contact foce eeted b the cow on the wie which b ewton's thid law is equal in anitude to the contact foce that the wie eets on the cow which in tun is equal in anitude to the weiht foce the eath eets on the cow since the cow is not acceleatin and theefoe the net foce on the cow is zeo. (Thee is a sall weiht foce eeted on the wie b the eath but we will assue that it is sall and can be nelected.) Also, if we take the coponent equations diectl fo the vecto fo of ewton's 2 nd Fwie + T2 + cow wie law a, then + T2 cosθ T2 cosθ a 0 > T 1 T 2 (since the anles ae the sae) and + T2 + F cow wie sinθ + T2 sinθ a 0 > 2T 1 sin θ T 1 T 2 / 2 sin θ

7 Poble 5 (20 points) a. eed fee bod diaas to see this. Since v constant, F net 0 fo both the ope and the cate (ewton s fist law of otion). Let s assue that the anitude of the tension in the ope is constant so that Tbo ope Tope bo Two ke ope floo bo Since the net foce 0, the hoizontal foces actin on the bo add up to f floo bo zeo. Since thee ae onl two of the, these two foces ae equal in anitude and opposite in diection. Thus, the foce that the woke Tope bo eets is equal in anitude and opposite in diection to the fiction foce W eeted b the floo on the bo. The anitude of this fiction foce eath bo µ floo bo. Howeve, since the noal foce is one of onl two vetical F 0 foces actin on the bo with the othe foce bein the Weiht foce eeted b the Eath on the bo, the two vetical foces ae of equal anitude. Theefoe Two ke ope f floo bo µ floo bo µ WEath bo µ and The foce eeted b the woke on the ope points to the iht. b. Yes, how had the woke pulls does depend on whethe o not he little bothe is on top. As we showed above, how had the woke pulls depends on the stenth of the noal foce eeted b the floo on the bo. As we saw in the two-book poble in class, placin the little bothe on top of the bo inceases the noal foce the floo eets on the bo. Lookin at the foces actin on the bo, the net foce on the bo is zeo since it is stationa and not ovin. So the noal foce of the floo on the bo ust be equal in anitude to the weiht foce of the Eath on bo plus the noal foce of the little bothe on the bo. Since the weiht foce on the bo has not chaned, the noal foce eeted b the floo on the bo is lae than if the bothe wee eoved. (See the fee bod diaas in pat c). This inceases the fictional foce eeted b the floo on the bo, which eans the woke ust pull hade to keep the bo ovin with constant velocit. c. F W bo bothe Eath bothe 0 F f 0 Twk ope floo bo W 300 floo bo T eath bo bothe bo ope bo Since F 0, then F F 0 Fo little bothe, 0 F bo bothe + W b b W E b ( b ) Fo ewton s 3 d law of otion b b b b b Eath bothe Fo the bo 0 F f b + b b + W E b v f b b b W E b f b ( b ) ( b) ( b + b T f µ ( + ) wk ope fl b fl b µ b b ) T (0.4) [(30 k) + (50 k)] (9.8 /s 2 ) wk ope b