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Unformatted text preview: FACULTY OF SCIENCE
FINAL EXAMINATION PHYS 239
Dynamics of Simple Systems Examiner: Professor D. Hanna Thursday, December 9, 2004
Associate Examiner: Prof. K. Regan 9:00 — 12:00
LNSTRUCTlONS: Answer all questions; the weight of each question is printed next to the
question.
Answers are to be given in the examination booklets.
No notes or books are perrrritted.
Approved calculators may be used. Casio fx—3(l0l\’IS Casio fX—QQl Casio fx—270VV Sharp Eli—546 Sharp EL—SlO Sharp Eli520 V Sharp Eli—5331 V rI‘l—SUX IIS
Fromm—English Dictionaries may he used.
Constants arid formulae are found on the last page.
This exam comprises 7 C11.l('!53tl(.mS and 6 pages. Students may keep the exam. PHYS 230, Final Exam December 9, 2004 Page 2 1. (4 points) rl‘wo masses, ml = 1 ﬂag and 7712 = 3 kg, are arranged on a frictionless
surface, as shown in the ﬁgure. ml is attached to a spring with force constant
I: : 16 N /m. and 7112 is next to. but not attached to. m]. The spring is
compressed a distance 11 = —.5 m from its relaxed position. The masses
are released and the spring is free to expand. ( a) Calmilate the ﬁnal velocity of m2 (1)) Calculate how far beyond its relaxed length the spring stretches. 2. (/1 points)
A falling stone takes 0.28 s to travel past a. window 2.2 m tall (see the ﬁgure). From what height. above the top of the window did the stone fall? l 2.2 m PHYS 230. Final Exam Deeen'iber 9, 2004 Page 3 3. (4 points) Rocket A passes Earth at a speed of 0.75 C. At the same time, rocket B
passes Earth, going at speed 0.95 c in the same direction as rocket A. (Both
speeds are as measured by observers on Earth). (a) How fast. is B moving in A’s reference frame when it passes A? (b) If both roekets are 100 m long in their rest frames. how long does rocket B appear to he, as measured by an observer on Earth? 4. (7 points) A bend of mass m slides without friction on a. smooth rod along the 2; axis.
The rod is equidiel_.e'tnt between two spheres of mass 114. The Spheres are
located at :r: :: (J, y : :lza as shown. They attract the bead gravitationally.
(8.) Find the potential energy of the head as a function of 11:. (b) The bead is 1‘(‘:l(':used at :1: = 3a with velocity 110 toward the origin. Find
the speed of the head as it passes the origin. ((2) Find the frequency of small oscillations of the bead about the origin. 1'11 PHYS 230, Final Exam December 9, 2004 Page 4 5. (6 points) Calculate the minimum kinetic energy needed by a proton to create a pair
of charged kaons. The reaction, as seen in the laboratory frame, is: p+p—> gJ+p+K+ +K" where one of the initial protons is moving and one is at rest. The rest mass
of each K particle is 494 Mel/7612 and the proton rest mass is 931 M eV / (:2.
Recall that, at threshold, the ﬁnal state particles are at rest in the centre of mass. 6. (5 points) An object of mass m, travelling with velocity on along the x axis, collides
with another object, also with mass m, which is at rest at the origin. If the
collision is elastic and both objects have the same final speed, (a) what is the value of their ﬁnal speeds? (b) what. angle with respect to the :1: axis does each velocity vector make? Assume all velocites are small (ie do not use special relativity.) V
0 BEFORE . AFTER \‘ ~. PH Y8 230, Final Exam December 9,. 2004 Page 5 7'. (4 points) TWO events have the following C(x'n't'iinates in a frame 3. Event 1: ml = 3:0, t1 = ate/(1, yl = 0, 21 : 0 Event 2: 372 : 2170, t2 = 3330/01 yg 2 0, 22 z: 0 (a) Find the velocity of the frame .5”. which moves in S: in which these events
occur at the same place. (b) Make a i\*‘ii1’1k0WSki diagram showing the relevant axes of the two frames and plot the two points, showing that they have the same :1:’ coordinate. PHYS 230, Final Exam December 9, 2004 Page 6 Useful Formulae and Constants:
g .—: 9.8 m..92
c = 3 x108 m/s :‘1:’ = 7(11", 7 wt) 5 I 10/?” 2‘: ’7?“ 1L _ '11 7’“
1 1—L'1..thc'2 , f ”In!
My 15(1—1mI/C'3] II I f _ , u:
U“; _ 7(171111I/(32)
E ; 7mm? E2 = 19282 + mﬁrﬂl ...
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