In-Class Problems 11-13: Uniform Circular Motion Solutions

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Phyic Phyic 8.01T Fall Tem 004 In-Cla Poblem 11-13: Unifom Cicula Motion Solution Section Table and Goup Numbe Name Hand in one olution pe goup. We would like each goup to apply the poblem olving tategy with the fou tage (ee below) to anwe the following two poblem. I. Undetand get a conceptual gap of the poblem II. Devie a Plan - et up a pocedue to obtain the deied olution III. Cay ou you plan olve the poblem! IV. Look Back check you olution and method of olution 1

2 In-Cla-Poblem 11: Whiling Object Two object of equal ma m ae whiling aound a haft with a contant angula velocity ω. The fit object i a ditance d fom the cental axi, and the econd object i a ditance d fom the axi. You may aume the ting ae male and inextenible. You may ignoe the effect of gavity. Find the tenion in the two ting. A you begin thi poblem, conide what type of foce diagam you may need. In paticula what body o ytem of bodie will ue fo you fee body diagam. It neve hut to daw a many fee body diagam a you can. Solution: Fee Body Diagam: foce diagam fo each object. v v Newton Second Law, F = m 1 a, in the adial diection fo the inne object i 1 1 ˆ : T T 1 = mdω. v v Newton Second Law, F = m 1 a, in the adial diection fo the oute object i 1 1 ˆ : T = m(d)ω. We can now olve fo the tenion in the ting between the inne object and the oute object. Fom the foce equation fo oute object, the tenion in the ting between the inne object and the oute object i

3 T = m(d)ω The tenion in the ting between the haft and the inne object can be found by uing the eult fom pat b) in the foce equation fo the inne object yielding, m( d)ω = md T 1 ω. Solving fo the tenion in the ting between the haft and the inne object give T 1 = 3mdω 3

4 In-Cla Poblem 1: Object undegoing Cicula Motion on the Inide of a Cone Conide an object of ma m that move in a cicula obit with contant velocity v 0 along the inide of a cone. Aume the wall of the cone i fictionle. The cone make an angle φ with epect to a vetical axi. a) Find the adiu of the obit of the object in tem of the given infomation. In the Look Back check you olution and method of olution pat of ou methodology anwe the following quetion. b) I the nomal foce a centipetal foce o i a compopnent of the nomal foce the centipetal foce in thi poblem? c) I the adiu of the cicula obit independent of the ma of the object? Explain why o why not. Solution: Fee Body Diagam: 4

5 Chooe cylindical coodinate a hown in the above figue. Chooe the unit vecto ˆ to point in the adial outwad diection. Since we ae now woking in thee dimenion, chooe the unit vecto kˆ to point upwad. (Note that the unit vecto θˆ i now automatically defined to point into the page. Since thee ae no foce in thi diection, we need not woy about it while olving the poblem.) The foce diagam on the object i hown in the figue above. The two foce acting on the object ae the nomal foce of the wall on the object and the gavitational foce. Then Newton Second Law become: v v F = ma become: mv ˆ : N co φ = 0 kˆ : N in φ mg = 0 Thee equation become N co φ = mv 0 N in φ = mg We can divide thee two equation, N in φ mg = N coφ v m 0 yielding tan φ = g v 0 Thi can be olved fo the adiu. = v 0 tan φ. g b) The centipetal foce in thi poblem i the vecto component of the contact foce that point adially inwad, F cent = N co φ. 5

6 c) The adiu i independent of the ma becaue the component of the nomal foce in the vetical diection mut balance the gavitational foce o the nomal foce i popotional to the ma. Theefoe the adially inwad component of the nomal foce i alo popotional to ma. Since thi foce i cauing the object to acceleate inwad, by Newton Second Law, the ma will cancel and the acceleation i independent of the ma. Theefoe the adiu i alo independent of the ma. 6

7 In-Cla Poblem 13: Ca on a Banked tun A ca of ma m i going aound a cicula tun of adiu R with contant peed v, which i banked at an angle φ with epect to the gound. The ca i held up on the bank by tatic fiction between the wheel and the oad with a coefficient of tatic fiction µ. Let g be the magnitude of the acceleation due to gavity. Deive an expeion fo the minimum velocity neceay to keep the ca moving in a cicle without lipping down the embanked tun. Expe you anwe in tem of the given quantitie. Solution Ca on banked tun undegoing cicula motion A you ty to Undetand get a conceptual gap of the poblem conide the following quetion. Thee ae all type of eitive foce acting on the ca along the diection of motion of the ca; fo example: ai fiction and olling eitance. Do you need to know thee in ode to model thi poblem? Explain you eaoning. Chooe cylindical coodinate. Chooe unit vecto ˆ pointing in the adial outwad diection and kˆ pointing upwad (figue 6..5). (Once again, the unit vecto θˆ i implicitly defined.) The foce diagam on the ca i hown in figue 6..5 when the ca i jut about to lide down the embanked tun. Fee body diagam fo ca v When v min < v < 0, the ca would lide down the banked tun if tatic fiction did not v hold it up. i the velocity uch that f = 0, at v= v. 0 tatic 0 7

8 v v Newton Second law, F = ma, become (in the adial and vetical diection) ˆ : N inφ + f tatic coφ = mv kˆ : N coφ + f tatic inφ mg = ma z When v= v min, the jut lipping condition i that the acceleation in the z-diection i zeo, and the tatic fiction ha it maximum value: o the foce equation become a z = 0 and f tatic = µ N Dividing thee equation yield N inφ µ N coφ = mv min + N coφ + µ N inφ = mg inφ + µ coφ v = min coφ + µ inφ g that can then be olved fo the minimum peed, v min, neceay to avoid liding down the embanked tun. v min inφ µ coφ = g coφ µ inφ + 1 8