Download 2016.17 PH2a FALL TERM QUIZ 1 THERE WILL BE A QUIZ

OCTOBER 07, 2016
2016.17 PH2a
FALL TERM
QUIZ 1
Your solutions are DUE ON TUESDAY, OCTOBER 11, 2016, at the beginning of lecture (11:00 am), in
the locked section boxes outside 201 E. Bridge.
Late quizzes will not be accepted, except in very special circumstances.
The quiz is open book, open lecture, open section notes, open homework sets and solutions.
Calculators may be used. Symbolic manipulators, other than your brain are not allowed.
Please justify your answers and show all work.
TIME LIMIT: 90 MINUTES
IMPORTANT: PLEASE WRITE YOUR NAME, YOUR SECTION NUMBER AND
YOUR T.A.’S NAME ON THE FRONT OF YOUR SOLUTION SHEET.
THERE WILL BE A QUIZ REVIEW ON
SATURDAY OCTOBER 8, 2016
1:00 PM TO 2:00 PM
IN
201 E. BRIDGE
QUIZ 1
PROBLEM 1
An oscillator consists of a mass m hanging from a vertical spring with Hooke’s constant k. The
mass is immersed in a pot of oil. The mass exhibits small oscillations of its vertical position y but
the oil dampens this motion with a drag force that is proportional to the speed of the mass with
FD = −b!y , where b is a positive constant.
a) If b = km and the pendulum is set in motion from rest at some initial finite displacement
y0 , will the mass undergo any oscillations? Explain why or why not. [1 pt]
b) Sketch the displacement and speed of the mass as a function of time for the case given in part
(a). Label your axes and clearly mark the coordinates where the curves achieve their maxima
and minima, and where the curves intersect the time and y-axis. [1 pt]
c) What is the value of b needed to achieve critical damping? [1 pt]
d) At time t=0, the mass is at its equilibrium position but is given a kick such that the initial
velocity is 0.1 m/s. Assuming critical damping and using k=10 N/kg and m=1 kg, calculate y
and y! at time t=10 s. [2 pts]
e) Now suppose that a driving force F = F0 cos ω t is applied to the oscillator. What driving
frequency ω will impart the largest amplitude of oscillation to the oscillator? At this
resonant frequency, calculate the amplitude of oscillation and the phase difference between
the driving force and mass oscillations. Express your answer in terms of the variables m, k, b,
and F0. [2 pts]
PROBLEM 2
L
m
A mass hangs from a rubber band. A rubber band has a force law that obeys:
1/2
1/2
F = a ( y + L ) . When hanging with gravity the total force is Ftot = −a ( y + L ) + mg which at
y = 0 has Ftot = 0 .
a. What is the effective spring constant for small displacements from equilibrium?
b. At what value of y does the force differ from Hooke’s Law by 10%?
c. Is the frequency of oscillation in increased or decreased as the amplitude exceeds the value in
part b?