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Unformatted text preview: Physics 421: Exam 1 Oct.9, 2003
(7—9PM) Show all work and deﬁne all constants you use. You will do 5 problems in all. 1) Do m2 of the following three problems: i. (10 pts.) A particle’s motion is described in cylindrical coordinates by the
following equations:
R0) = a(1——e””’)
$0) = CU?
z(t) = btz where (o, 1:, a and b are all positive constants. a. Find the velocity and speed (magnitude of the velocity) of the
particle as functions of time. _ ' r ' b. Describe and sketch the motion (particle begins motion at time
1:0). ' 2. (10 pts.) For a critically damped harmonic oscillator with fundamental
angular frequency (90:
a. Show, that for initial conditions given as x(t = 0) z a (a > 0) and
v(r = 0) = v0, it is possible for the. mass to cross through its equilibrium
position x = 0.
b. What are the conditions on v0 for which this can occur? 0. How long does it take to pass through the equilibrium position?
d. Sketch x as a function of tin this case (Where motion passes through
the equilibrium positibn) and describe features of the motion. 3. (10 pts.) Analyze the motion of a damped harmonic oscillator that is driven
by an external driving force FD consisting of a succession of rectangular pulses FD=F0 NT—é—aStSNTiéa
FD = 0 otherwise
where N = 0,i1,i 2,. .., Tis the time from one pulse to the next and a is the rwidth of each pulse. The mass is m, the fundamental angular frequency of
the oscillator is (00 and the coefﬁcient of viscous drag is c. a. Plot FD versus t.
b. Make a Fourier expansion of ED and show explicitly the first four terms of the expansion.
c. Give the steady state equation of motion, x(t), being sure to deﬁne all quantities in terms of quantiﬁes given above.
2) Do all of the following problems: For problems 4 and 5 below assume that at t = 0 the object is at position x0 with
velocity v0. 4. (5 pts.) An objeCt of mass m is subject to the following force: F = Foe“ (F0 and c are constants). Find the position and velocity, x(t) and v(t),
respectively, as functions of time. '5. (5 pts.) An object of mass m is subject to the following force: F = F0 cos(cx)
(F0 and c are constants). Find the velocity as a function of position. 6. (5 pts.) A block of mass m slides on a horizontal surface that has been
lubricated with a heavy oil such that the viscous resistance varies as the 3/2
power of the velocity: F = —cv3’2 (c > 0). If the initial speed of the block is v0 at x = 0 , ﬁnd the distance the block travels (in terms of stated constants). m ~_ﬂw..._;u.w._ﬁ.m inaugnhmg; ﬁwmﬂmwmwww . R1? 1‘ _ are _MWUEEABW‘ﬁ'WﬁIféf“;§:ff
' v 7 ._ ..._ , ,, a. V = axz + [33:2 + n2 (5pts.)
b. v =‘ ce<w?+ﬁy+w (5pts.)
0. V = er" in spherical coordinates (5pts..) (1. In each of above cases, the particle has velocity v0 at position (1,2,3) , what is its velocity at position (0,0,0)? Note that these positions are given in Cartesian coordinates.
(5pts. Each) ' " Efraﬁihﬂff " 39448,: _, r
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 Spring '10
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