Name:______________________________
PHYSICS 306 – MID-TERM EXAM sample
Instructions:
 You are allowed no more than 8.5”x11” pages of hand written notes (1-side) for
mid-term Exam.
 Please put away all materials except for the above notes, pens, pencils, erasers and
your calculator.
 Turn off any electronic communication devices that you have with you.
Possibly Useful Formulas
a b  a b
 
det 
 ad  bc
c d c d
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If
ax 2  bx  c  0
When x<<1 then
then x 
 b  b 2  4ac
2a
1  x n  1  nx
x3
sin x  x 
3!
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sin a  b   sin a cos b  cos a sin b
sin a  b   sin a cos b  cos a sin b
cosa  b   cos a cos b  sin a sin b
cosa  b   cos a cos b  sin a sin b
___________________________________________________________________________
Name:______________________________
Answer all questions and show all work for full credit.
Problem 1 (35 points total)
Consider a string with one end exhibiting a time-dependent displacement along the
vertical y direction



yt   0.5m  sin 30 s 1 t
a) (5pts) What are the amplitude and period of the oscillatory motion of the end point?
b) (10pts) Find out velocity v(t) and acceleration a(t) for the end point of the string?
c) (5pts)The three graphs below (i, ii, iii) represent time variation of the displacement
y(t), velocity v(t) and acceleration a(t). Please assign the quantities with the
corresponding graphs.
(i)
t
(ii)
t
t
(iii)
d) (10pts) If we start to count at the time when the end point passes a position of
y=0.25m moving upwards, when will be the shortest time for the end point going
through the equilibrium position y=0? What is its velocity at that time?
e) (5 pts) A transverse wave is releasing from the end point and travelling along the
string with a wavelength of 10m, determines the wave number and wave phase
velocity.
Problem 2 (35 points total)
A physical pendulum of length l=1.25 m oscillates back and forth in the vertical plane.
a) (5pts) first let’s consider on a frictionless case. What is the angular frequency of
the oscillation 0?
b) (10pts) Next a frictional force for the pendulum -rv is introduced. There are three
traces (blue, black and red) in the plot below for three values of r. Which trace is
corresponding to lightly damped case? Which trace is corresponding to critically
damped case? Rank the value r from high to low for three traces.
c) (10 pts) If we know r = 0.1 kg/s, determine the quality factor of the oscillator and
the number of cycles required for the energy to decay to 1/e of its initial value.
d) (5 pts) In addition to the damping force, the oscillator is subjected to a harmonic
force F = F0 sin(t) where F0 = 2 N. Determine the ratio between steady-state
amplitude of oscillation at zero frequency and at the resonance 0?
e) (5pts) finally we consider average power versus p() supplied to the pendulum by
the driving force above. Determine the full width of half maximum of the response
curve.
Problem 3 (30points total)
A system consists of two masses and three massless springs moving on a frictionless
plane and connected to the wall. Stiffness of the spring is k. The positions of masses
from equilibrium are labeled x1 and x2.
x1
x2
a) (15 points) Derive the equations of motion for this system
b) (15 points) Find the frequencies of the normal modes for this system