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Unformatted text preview: Physics GB January 28, 2008
Katsushi Arisaka Last Name: First Name: Student ID No. Enrolled Lecture: 1 {llaml l' 2 {Boon} Room: HEINES 391' KNSY PV 12203 Exam Time: 11am! Noon Important Remarks: ' Sorry to tell you but during the exam, close books, close notes. no calculator is allowed; please
just rely on your own brain.  Please use any open space on the exam for yotu‘ solution. Please write down how you derived
concisely {winch helps you to get a partial credit even if the ﬁnal answer is wrong.) " If you ﬁnish early, please be seated quietly until the exam time is over. Points: m—m
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Question : In this exam, we explore a simple massspring system only. A mass (with mass m) is connected to a
spring {with spring constant k) as shown below. There is no friction on the ﬂoor. At t = 0, the mass
stays at the initial position x = 0, where there is no force from the spring. Then the mass is suddenly
hit by a hammer (from the right to the left) and acquires the initial velocity v = we (to the left
direction). Please note that 'right‘ is the positive 1: direction and ‘leﬁ‘ is the negative .1: direction. Hit Walter/m 1. First. let’s analyze this system using Newton‘s law, step by step. 1:) Write down Hooke’ 3 law, when the mesa is located at position x. b) By combining it with Newton' 5 law, derive the second—order differential equation (in time) for
the position x. c) Which one do you think' is the correct solution of 1:? (Assume A' [5 a positive value.)
1) .r= —A sin(e1r)
2) x— ~ —A sin(mt‘)
3) x = A costar)
4) x = —A costar)
d) Derive the velocity v by differentiating the above expression of x by r.
e) Derive the acceleration a by differentiating the above expression of v by r. _
i) Show that the your answer in c) satisﬁes the secondorder differential equation {given in h) :5 above), if the angular frequency meatiaﬁee 1o  JE .
m
Express the amplitude A by the initial velocity 1)., and a1 (Recall at 1= 0, V'" — we.) :9 m 1,) Fzr'rML , ﬂaw—+922 ﬂrﬁx
C) l)13“/4MCWJ
0K). ’lf’ﬂ Hummer) 2. Now we are ready to express the complete solution graphically. Plot the following as a function of
time.  :1} Graph of x as a function of r. (Clearly mark the amplitude A and the period T.) _
1:) Graph of velocity v as a function of r. (Clearly mark the initial ‘Iurelooityr v9 and the period T.)
0} Graph of acceleration a as a function of t. (Clearly mark the period T.) d) What is the relation between angular frequency a; period T and frequency f in general? 3. In general. it is rare that we can actually solve the secondorder diﬂ'erential equation (Le.
Newton‘s law). In such a case, the energy conservation law gives another powerful tool to
analyze a system. So let’s try this approach next. a) What are the general forms of the potential energy U and the kinetic energy K in the case of a
simple massspring system? (You do not have to derive it.) 1:) Just after the mass is hit by a hammer at r = 0, what is the potential energy U and the kinetic
energy K"? So what is the total energy E? (Express it by v0.) c] At x = A, the mass stops. What is the potential energy U and the kinetic energy If ? Then what
is the total energy E? (Express it by A.) d} According to the energy conservation law, the total energy given by b} and that given by c)
above must be equal. Using this fact, express the ampIitude A of by the initial velocity v” and or. [l’ou willﬁno‘ out that it is the same answer as l—g}.] a 0) oz: ml, [GEL/W2] L) M. kiwi, sew: imivﬁ =0 Bails/“r" 2
1__ﬂjl__ 1
A bliu
__ d
W
‘ 32";
“ﬂLW 4. Next, let’s consider the more complex system as shown below. The mass is supported by a spring
on the leﬁ side (with Spring Constant= it). II' :1. I! 1 i in H sicall ' 4: t iﬂ _L' ' ". As a result, when the mass goes to the right side (I z» o}, it will be released into curved slope {with a
curvature L}. Like the previous problem, at t = (l, the mass is suddenly hit by a hammer (from the right) and
acquires the initial velocity v = va (i.e. to the leﬁ direction). When I < 0, the situation is exactly
the same as before, where the mass has angular ﬁ'equency in and amplitude A. A; ,5 2" ID, the mass behaves like a pendulum {with a string length L} because of the gravity mg doanards. Let' s assume that the angular frequency becomes of and amplitude heeomcsA’.
(Please assume that A‘ is much shorter than L.) _
 at
a) At 1: I" 0, there is a force F which brings the mass back to the left (due to gravity). Express F in
terms ofm, 3,1; and x.
b} At 1: 2> 0, what is the angular frequency of '? (ExpreSs it by L and g).
c) What is the height of the mass (it in the picture), when it stops at x = A’ '2’ (Express it by vs and
s). h
i '\. «’1. t9 l. ‘. i
i
i
t
s ‘l
l
i
i
i
I Eprhgct i j _ («mag ff, r‘a 5. Let‘s assume af = 2m. Now we are ready to express the complete motion ni‘the mass graphically. a) Plot the graph (If): as a function of I. (At least plot for the time period when the mass goes left
and right for a couple of times.) b) What is the relation between L, Jr, and m, if m‘ satisﬁes w‘ = 2m ? (Express L in terms of k, m
and g}. I: 5. Let’s assume a: = Zw'. Now We are ready to express the complete motion of the mass graphically.
a) PIot the graph ofx as a function of I. (At least plot for the time period when the mass goes left and right for a couple of times.)
b) What is the relation betWeen L, k, and m, if m satisﬁes to = 200' 1’ (Express 1. in terms of k, m and s)
o) I ...
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 Spring '08
 Arisaka

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