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Unformatted text preview: 1B Physics for Scientists and Engineers: Oscillations, Waves, Electricity
and Magnetism Midterm #1. Wednesday 26th January. Instructor: Steve Cowley Please do not write in this region —
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— Please write your answers in the space provided below. Partial credit is given for answers that are on the right
track so show your working. You may use rough paper but please put the working on these sheets. Question 1. Oscillations of a Baby on a Spring. 20 points One way to keep your baby happy(sometimes) is to put it in a baby bouncer. This can be modelled as suspending
the baby by an ideal spring and letting them perform simple harmonic motion — We will neglect the effects of the
baby touching the ground. 5 (a) The baby mass is 10kg and we take (for simplicity) the gravitational acceleration to be 9 = 10ms'2. When
we put the baby in the harness the spring stretches by 1m to the equilibrium position. Calculate the spring constant. 4 points ﬁPh‘arkA kA: m3
FM 1 “a 9. (b) To get the baby oscillating we push it down 0.1m from the equilibrium position and let it go from rest.
Assuming no damping write down a formula for the oscillation period. Estimate the period to the nearest second. You might want to know that JE/vr ~ 1.0065..... 4 points Wk loo Nw' \r— T 11' E] (c) Write down the solution for the baby’s displacement (x(t)) as a function of: time. 4 points X 1"): *‘OJm x cosﬂﬂs‘ﬁ 15) (d) The baby is subject to a. weak damping force —0.01v where v is the baby’sveioeity. How long does it take
for the amplitude of the baby’s oscillation to decrease by a factor of 1/ e? 4 pointsj “.3. _
x({:) :: Ace 1"“ +1 COSCwb) J Q : 0,01: ____Q_ T _l w n _ _ Jim.
Aka): e 1 :e :7 EMT’1 '92,— q (e), Harder. The baby is actually a twin (they both weigh 10kg) and the bouncer is a tandem bouncer (I do
not know if they exist) where you can put two babies in the harness. One baby is quietly resting in the equilibrium
position when we slip the other into the harness without moving the harness up or{ down. So now both are loaded
and we let them go (from rest). Calculate the subsequent motion (including amplitude and frequency) assuming
again no damping. (Note, the spring constant is still the same as calculated in (11);) 4 points NEW egunllorluwi l3 A=m bdow old Eeqm‘mon‘um "Le, SPru‘v/ul Lg gLreizdneol (LA: ‘LM‘ :) Amp/Glitch, :2 A = 'M a k I \r K : IOU NWW‘ !
wmw‘ﬂ” 1m 4 motes ES Question 2. Guitar Strings. 20 points The E string on a guitar weighs 4g = 40 x 10—4kg and it is 1m long. The actual length of the bit of string we
pluck ( i.e. the distance between the ﬁxed end points) is taken here for simplicity to be 0.5m. (a) Calculate the tension in the string for it to vibrate at a frequency of 1005‘1 in the fundamental mode
(11 = 1). 4 points a Mass. _— x r” — IL \II , , 2L 1
/“ Jonath llo '° kw” I ‘F ' it. /A “'7 T “(71f)”
ﬂ=l _\ ’1 ' __ fr L1,: O.gm T: [#31008 M (b) Suppose we excite the second harmonic (n = 2), what is it's period? Where can you touch the string
without disturbing this oscillation? 4 points I 1.. AV" I
ll (1 b A 2 ] 7c V v if,
A l
l  — :OnOOSS (c) The guitarist puts her ﬁnger on the string pushing it against a ” fret” which shortens it to 0.25111 — calculate
the fundamental frequency. 4 points L—sL./7_:=b zeﬁ‘l ref7% Mabm 9.. (d) Does the frequency of oscillation change when she plucks the string harder? 4 points No, Sus’c blu. amplitude, (Tenuon "S conshv‘l— +° "Fru “PPWM‘M/ml‘l‘ohy (e) Write down the fundamental solution for the original (length 0.5m) string? Make the maximum displacement
of the string in the oscillation 0.01m. 4 points . smnaxwgwwe: E
EUHG)‘: O‘O'M ~ Sin(mm"x)ws(moonr't)
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 Winter '08
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 Simple Harmonic Motion, weak damping force, Wk loo Nw

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