μ = Physics 1401 Homework Solutions - Walker, Chapter 6

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1 Homewo Solution - Wale, Chapte 6 Conceptual Quetion CQ15. Sitting on the tun of the ca help becaue it inceae the nomal foce that the gound eet on each patch of tie that i in contact with the gound. Thi inceae the maimum alue of the foce of tatic fiction, ince ( ) ma f i popotional to the nomal foce: ( f) ma Poblem and Conceptual Eecie 4. Pictue: a = 1.6 m/ = μ..0 Conide FBD fo the child: f = μ co.0 a = a = 1.6 m/ in.0 w= ewton econd law applied to the child gie: F = ma F = 0 Conide the diection. Filling in the foce in the diection, I get: in.0 μ = m ( 1.6 m/ ) Soling fo μ, I get: in.0 μ = m ( 1.6 m/ ). But I need to now what i. I get it fom the diection: F = 0 1

2 Homewo Solution - Wale, Chapte 6 Put thi into the epeion fo μ : co.0 = 0 = co.0. ( ) in.0 ( ) in.0 m 1.6 m/ m 1.6 m/ μ = = co.0 co.0 co.0 μ = tan.0 μ = ( ) 1.6 m/ 9.81 m/ co.0 8. Hooe law a that in ode to compe the ping a ditance and hold it thee, the peon hand will hae to eet a foce of magnitude. B ewton thid law, the ping will puh bac on the peon hand with a foce of equal magnitude,. But and hee i the e point, a fa a the bloc i concened the ping will alo puh on the bloc with a foce of magnitude. (That i, when it i compeed a ditance, the ping puhe on the object attached to eithe of it end with a foce of magnitude.) So the fee-bod diagam fo the bloc will loo lie: f F = w= (a.) The poblem a fo the minimum compeion needed to eep the bloc fom falling. Suppoe we tat b compeing the ping quite a lot (moe than i necea to eep the bloc fom falling), and then gaduall let the ping epand o that it i not compeed quite a much. A we let the ping epand towad it elaed

3 Homewo Solution - Wale, Chapte 6 length, the foce it eet on the bloc will deceae. At ome point, the bloc will be on the ege of falling. At thi point, we hae the minimum compeion necea to eep the bloc fom falling an le compeion, and it will fall. What i the condition fo the bloc being on the ege of falling? The condition i that f = μ. Appling ewton econd law to the bloc in the and diection, I find: Let conide the diection fit: F = 0 F = 0 = 0 Which gie: =. But what i? Well, conide the diection: μ =. μ Plugging thi into the epeion fo gie: = 0 = (1) μ So the minimum compeion needed i: = ( 0.7 g)( 9.81 m/ ) ( 0.46)( 10 /m) = m = 4.8 cm (b.) Ye. i diectl popotional to the ma, o if the ma wee doubled, the minimum compeion would be doubled. (See Equation 1 aboe.) To put it anothe wa, if the ma wee doubled, f would hae to go up b a facto of two (to eep the bloc in equilibium in the etical diection). But that mean would hae to go up b a facto of two. But that mean that would hae to go up b a facto of two. So would hae to go up b a facto of two. 44. Thi poblem i almot tiial, if ou thin about the entie tem compoed of m 1, m, m, and the two ting (and thi i how the intend fo ou to thin about the poblem). A fa a thi tem i concened, the net etenal foce i jut the gaitational foce that the eath eet on m... that i, the weight of m. (The tenion in the ting ae intenal to thi tem.) ow, ewton econd law a: F = ma,

4 Homewo Solution - Wale, Chapte 6 that i, the net foce on an object (o a tem of object) equal the ma of the object (o tem) time the acceleation of the object (o tem). In addition, the net foce that appea in ewton econd law i the net etenal foce on the object (o tem). So if we hae a tem of object, onl the etenal foce lead to acceleation of the whole tem. (Thi point wa nee woth maing when we wee taling about foce on a ingle object becaue in that cae the tem conited of jut one object, o ee foce on that object wa an etenal foce.) So fo the thee mae in thi poblem, it a though ou had the following ituation: the tem m m 1 m F etenal = m g So what the acceleation of thi tem? Well, ewton econd law applied to the tem a (fo the diection): ( m1 + m m ) a tem m g = +. So: a tem = ( m + m + m ) 1 m g = 6 ( 9.81 m/ ), o a = 4.91m/. tem Ea, ight? A a chec, we might wo thi poblem in a moe familia wa. Daw FBD fo each indiidual ma. Let the tenion in the left ting be called and the tenion in the ight ting be called T. Then the FBD loo lie: Fo m 1 : a ewton econd law a: T1 = m1a (1) m 1 g m 1 4

5 Homewo Solution - Wale, Chapte 6 Fo m : ewton econd law a: = T m a () a m T g m And fo m : T ewton econd law a: m = g T m a () m a g m ow do a little algeba to ole fo a: Subtitute (1) into (). Then () become: T m1a = ma, o Then ubtitute thi into () to get: = ( m m )a. + ( m + m ) a m a m g 1 = a =, ( m + m + m ) which i eactl the ame eult we got b the fit method! 1, o: 5

6 Homewo Solution - Wale, Chapte (a.) At the top of the Fei wheel, ou FBD loo lie: a cp = So ewton econd law a: a: = m, which gie the appaent weight (the nomal foce) = m. At the bottom of the Fei wheel, the FBD loo the ame, a fa a the diection of the foce on ou ae concened, but the centipetal acceleation i now upwad (the cente of the Fei wheel i now aboe ou), o ewton econd law gie: = m, o = + m So ou appaent weight at the bottom i geate than what it i at the top. Thi i wh ou feel heaie at the bottom and lighte at the top. (b.) Thi i jut plugging in numbe, ecept fo one thing: What i? Well, if the Fei wheel complete one eolution ee 8, then the ditance it goe in 8 i the cicumfeence of the Fei wheel, π. So the peed of the Fei wheel (auming thi peed i contant, o we can jut ue the definition of aeage peed) i: So the appaent weight at the top i: At the bottom, the appaent weight i: ( 7. m) total ditance π = = = m/. total time 8 ( )( ) ( ) ( ) m/ = m = 55 g 9.81 m/ 55 g = m = + m =

7 Homewo Solution - Wale, Chapte 6 6. Refeing to Poblem 6, I find that it bea a tiing imilait to the ituation we jut dicued: It jut lie going oe the top of the Fei wheel. So we ll hae (b the ame agument I jut made fo Poblem 61): = m, and we d lie to now what need to be in ode fo the people in the ca to feel weightle. (I want to put emphai on the wod feel hee... the people in the ca ae nee eall weightle, of coue... the alwa hae weight equal to thei ma time g.) Well, the e to thi poblem i to ealize that the people feel weightle when the no longe feel the eat puhing upwad on them... in othe wod, when become zeo. You can thin of it thi wa, if ou lie: If ou wee to go oe the bump epeatedl, each time inceaing ou peed, the tem m in the aboe epeion fo would be lage each time. At ome point, ou d go oe the bump with a peed that would mae the tem m equal to. A ou go oe the top of the bump at thi peed, the nomal foce goe to zeo, accoding to the epeion fo aboe. In thi condition, the people in the ca no longe feel the eat puhing upwad on them. The people actuall leae the uface of the eat... the go into fee-fall fo a hot time, until the fall bac down to the eat. The peed equied fo thi to happen i then gien b: m =, o: = g = ( 9.81)( 5) = 19 m/ 81. A diagam of the ituation i hown below. Let the tenion in the thee ope be called, T, and T, a indicated in the diagam. T A T 45 B Fee-bod diagam fo bloc A: 7

8 Homewo Solution - Wale, Chapte 6 f A w A = m g A (a.) We want to find f. Well, appling ewton econd law to Bloc A in the diection, I find: F = 0 f = 0 f =. So we need to find. (otice that we can t a ege of liding.) ow, how to get? Well, thee i anothe object that 1 f = μ hee becaue we ae not told that Bloc A i on the T act on: the not whee the thee ting come togethe. Thi not i in equilibium, o the um of all the foce on it mut be zeo. Conide the FBD fo the not: T 45 T co 45 T in 45 T Appling ewton econd law to the not in the diection, I find: T = co T 8

9 Homewo Solution - Wale, Chapte 6 But what i T? Well, conide the foce on the not in the diection. ewton econd law a: F = 0 T in 45 = T T T =. in 45 But what T? Well, thee anothe object (beide the not) that T act on, and it Bloc B. Hee the FBD fo Bloc B: T B Bloc B i alo in equilibium, o = 0 So: F. Thi implie that T mb g ( )( ) T.86. in45 T = = =. in 45 And: T T ( ) = co 45 =. co 45 = And finall, f = T1 =.86 =.9, eeping thee ignificant figue. w B = m g B = =. g 9.81 m/ =.86. (b.) If the ma of Bloc A i doubled, f ta the ame. Changing the ma of Bloc A doen t do anthing to, and f mut alwa equal if Bloc A i to be in equilibium. 9

Worked Examples. v max =?

Worked Examples. v max =? Exaple iction + Unifo Cicula Motion Cicula Hill A ca i diing oe a ei-cicula hill of adiu. What i the fatet the ca can die oe the top of the hill without it tie lifting off of the gound? ax? (1) Copehend