Prelim1 - Spring 2010 Solutions

Prelim1 - Spring 2010 Solutions - Physics 1116 Prelim#1...

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Unformatted text preview: Physics 1116 : Prelim #1 7:30 - 9:00 PM, March 4, 2010 Name : a Don’t forget to write down your name 0 Breathe deeply and relax. o Calculators are permitted, but laptops, communication devices, and quiet sobbing are not. 0 There are 6 problems (+1 bonus) on the exam and 12 pages, not including the equation sheets. Please check that you have all of them. Feel free to remove the equation sheets from the back of the exam. 0 Please show your reasoning clearly on all problems. Partial credit will not be awarded if we do not understand what you are trying to do. Feel free to use the back of pages for your working (but let us know that you are doing so). 0 Don’t panic if you have difficulty with a problem, and don’t get hung up on any one particular problem. This test is designed to be sufficiently challenging that you do not have to be perfect in order to get a good score. DO NOT GO TO THE NEXT PAGE UNTIL INSTRUCTED TO DO SO 1 E Short Questions Question 1 8 pts Consider an object of mass m = 1 kg moving only along the m-axis. A force is applied to this mass as a function of time, shown in the graph below. On the graph at the bottom of the page, plot the POSITION of this object as a function of time. Assume that the object starts at time t = 0 at .7; = 0 and with velocity v = 0. Distance (m) Time(s) _ . U ‘|_ ’L 4” “E: 07% = 0.2: m c" 601(3)=0X23”%(1)1 Szhla inn/5 D ‘3 I.) [\D Question 2 : True or False (no partial credit) 3 pts each Momentum is conserved unless there is friction. If an object moves along a circular trajectory, the acceleration vector of the object always points directly towards the center of the circle. True The normal force that your chair is exerting on you is equal in magnitude to your weight due to Newton’s Third Law. ‘ The velocity of an object is always tangent to its trajectory. True False Question 3 10 pts An oddly shaped dumbell stands vertically at rest on a frictionless surface of ice. m1 rests on the ice, and is attached by a rigid, massless bar of length L to a second mass m2. m1 is initially positioned at a: = 0 and the surface of the ice is at y = 0. m2 m1 —>+x x = 0 (a) Calculate the center of mass of the dumbell «CM '2 0 cm — = (H )L. W. 4’ ML M. -+ M2, fl L—a- ($CMa’yoM) i0 9 (mu+mz> You give m1 3. very tiny tap along the +11: direction. This tap produces negligible momentum but does cause the dumbell to topple over. Assuming that m1 remains in contact with the ice at all times, at what position 1'2 will m2 strike the ice? key fo,‘,‘7l: {N142 ,‘S 7Q-o74ba/eA—Y) .S‘o 2¢fi~= 72ft; line 2953“; ~ O, 72236, ~ 0/ 50 WW oéeo rw/‘ chanjfl. ma i/EX“ m1 as a Guam L—(li—fl. 1c»... can. : O m,+m-:. .. - — -—:—-—-—— so com l5 l7; 5“ Slog—Q X20 m mz s’lvil‘M @ | L t+mt Lfl ml+m2 :82 = END OF SHORT QUESTIONS Question 4 20 pts Two masses, m1 and 'Il'l-g are connected by a inassless string over a. frictionless pulley of negligible mass. The pulley is accelerated upu-‘arcls with a known constant acceleration A due to an external force. we will solve this problem in the. inertial :L' 7 y frame shown below. A [51) Draw ant abel the free body diagram for the two UMSSCS, labeling all forces. 11 T: 4043!?!“ T 1:31 LX‘ m" _ W1: weigh} 4m. +0 jnw'fi W, may m 49 amt, (1)) Write down the equations of motion for both of the IIIHSSCS m I:-. +T~ Wl,8_ ignore. 7C I M131 = +T~r mag (c) Using the appropriate constraints, determine how the acceleration of m1, jjl, depends on the acceleration of mg, Q2, and A. w""‘ (fir‘fll>+ (fip‘ljb)= L. (L: [Ema-h" Dependence of {91 on A and g2 ; | -‘= 07A '— 9.. ((1) Using your answer for (c), solve the equation of motion for the first mass, m1. Determine ij‘ in terms of m1, 7712, A, and g. ’Wlfiit = T“ mtg. _. MLLQA -3‘] : T“ M2. 'M,ij|+ mag. ~2mzA = "mg + meg, (m.+m73ij. = (wands + szA ‘3' z (ML-m031—QMLA Lm‘é- W11) (Wit—ml) S + 0Q'YV\1'I..A (WM-i- Mt) Qif: ,_. || Question 5 25 pts Consider a circular disc of mass M sitting in a frictionless V—shaped groove as pictured below. lt is subject to gravitational force in the —y direction. (it) Draw a free body diagram of the disc, identifying all forces. E" Di Jail: N: Vari‘fulrl m ” WM (b) Determine the contact force Fix of the vertical wall on the disc, as well as the contact force FD of the diagonal wall on the disc. 311 ll K Co (“>- $1 ll 3 A 9-). + v "‘1 (c) Now, let the V—shaped groove be accelerated along the line y = a: in the positive a: and y direction. The acceleration of the V-shaped gr00ve along this line is A m/s2. Again, determine the contact force FV of the vertical wall on the disc, as well as the contact force FD of the diagonal wall on the disc. A Nowa= 2... M56 = _ MA ._ R 8/3?“ n: —-—> ‘11“ fi<MA+ v)=é—;[2MA+EM3] A A - E=~M(%_+3)° + £2373 FD: M ism) [-2 +3] (d) For what values of the acceleration of the V—shaped groove will the disc lose contact with the groove? I? OQJ‘SC.’ aocalua‘bs lA ix 694.r£¢‘l'\'6n’ MMS‘l' L—Q (1 "Force 3C0“ {4% wall LlL-th-N‘K if 3n:on amalgam/TIA allows +13. stow/*6. ac’uz. Mega/1&3 +5“ will Alisa [use e—CaM amikrate, m -% a 1m ‘l'hm anal £86, (4);“ “OWN—Hut grow/UL Question 6 25 pts Consider two identical masses of mass m sitting on a frictionless table, connected by three identical springs of spring constant I; and rest length l as shown below. The system is initially at rest and all springs have length l. The two masses are now pushed toward each other, each by a distance d. I 1 I (1+! 5 ' (1+1 9 I 7‘). 7% (a) Draw and label the free body diagrams for the two masses, m ,4, m3, properly labeling all forces. T5“ as Flu-Si” F71 s “A ' P” F2. "—3 F' MI" 3"? W M: (b) At time t = 0, the two masses are released from rest. Write down the equations of motion for mA and m3. (“~156ch 7, oefheo'h'é‘r‘ WIA'. ‘WI %A = —-l<(°(A‘/q>+ kCXB- xiv-X) '0 mg: m. x; = —k(9‘e’-ZZX)-l<3LX3-7CA—X) (0) Determine the frequency of oscillation of the masses. To solve the algebra, it may help to think of the problem in terms of a new variable which is the separation between mA and m3, and to solve the problem in terms of that variable. w}; a _2k(7cA) 4, We, We = ~1l<0<e>+kxA+sm N Mia-5&3) = - 3km 3ka - gm. “(1628430 = ~3k (Xv XA +1) M Wg‘JXA-‘Pl = Z wé = ~3k£ L; So solulw'maf quake“ or? mo‘lw‘cm (Pr 2. k a: Amwsz calm w= i;- slaoe, nelqakwe. mo'b‘m ‘4 XLXA hm LO) m ’Xfi& X3 alto Am w:m_'_____— 10 (d) Now consider the situation where the system starts from equilibrium, but now both masses are pushed by a distance d along the +3: direction. Determine the frequency of oscillation of these two masses in this situation. M = “A (761 +X3-*‘ 31) 5‘45“ aw van‘aéé, an+nfl3£ 7 .. 57 k k "37”"7’9 fi=‘(fi);7—=9 tam/g 4/50, can a?! 1'7! flu} way .' 7‘14, .5754)? w,'// a/am7x @74‘7 0% 1) J-‘o f/Wef MW 74m m 144/, 0/ m5. Tithe, Inhmg axe (My/{z acwm 17 i .70“)? .5494 $9; 11 Bonus Problem 3 pts The radius of the hydrogen atom can be expressed in terms of the physical constants in terms of 50, h, me, and e. The vacuum permittivity 60 has units [C2 kg"1 s2 m‘3], where C stands for a Coulomb, the unit of electric charge. h is Planck’s constant which has units [kg m2 8*]. me is the mass of the electron. e is the charge of the electron which has units The radius of the hydrogen atom can be expressed as : mm = [germs [mjpf [m] z [67. Asa, m.3] mtg—I] is]?I 2A= B [m]; (leA =2. 0 [k3]: 0:" [doe 'D=—'2. warm («a-é ms 1(th— al'f‘ / 12 ...
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This note was uploaded on 11/03/2010 for the course PHYS 1116 taught by Professor Elser, v during the Spring '05 term at Cornell.

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Prelim1 - Spring 2010 Solutions - Physics 1116 Prelim#1...

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