Wednesday 11 April 2012
Class meeting
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Textbook sections
Ponderables
Mini-labs (deliv.)
Lab
Demonstrations
Mini-lectures
Quiz
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PHYS 116 SCALE-UP
36
Oscillations, springs
15-1 to 15-5
Shaking Table
Spring-mass oscillations (discussion)
Mini-lab: Spring-mass oscillations
o Equipment: for each group: Stand with vertical scale, set of springs of different
constants, hanging masses, photo-gates, smart timer, stopwatches.
Ponderable: Shaking Table
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Wednesday 11 April 2012
PHYS 116 SCALE-UP
SPRING-MASS OSCILLATIONS
Prelab: A spring with force constant k=100 N/m has two masses hanging from it, as
drawn (M = 1 kg).
a. Without the masses, the free end of the spring is 10 cm below its support
point. What is the equilibrium distance of the upper mass below the
support point when both masses are present?
b. If only one mass is present, what is the equilibrium distance of the mass
below the support point?
c. Suppose that both masses are present and the system is initially
motionless. If the string connecting the two masses is suddenly cut and the
lower mass falls to the floor, find the period of the remaining oscillator.
d. Find the amplitude of the motion.
e. Find the maximum velocity of the mass M during the oscillation.
The springs you will use for these experiments are somewhat delicate; do not overstretch them with
excessive weight (large amplitude), or otherwise contort them in unusual ways. The goals are to a)
determine the dependence of period T on oscillating mass, b) determine the dependence of period T on
the spring constant, and c) check for an amplitude dependence. For each test, you should keep other
quantities constant. Either stopwatches or photo-gates will be available to make the period
measurements.
Explorations
•
Determine the force constant k of your springs by hanging various masses on them and measuring
the displacement. This is best done by plotting F vs. x.
•
Using one spring and one mass, set it oscillating and measure the period T. Change the amplitude
and determine how T depends on the amplitude. Do this using just one spring and one mass.
•
Using one spring, change the mass and determine how T depends on m. For plotting purposes, use
at least 4 different masses, preferably 5. How might you best plot your data to achieve a linear
graph?
•
Using one mass, change the force constant and determine how T depends on k. For plotting
purposes, use at least 3 different springs, preferably 4 (or 5 if they are available). How might you
best plot your data to achieve a linear graph?
The deliverable for this mini-lab is a discussion of your data (what would be the discussion section in a
full lab report). You should strongly consider approaching the problem as you did in the empirical analysis
(ring pendulum) lab. That is, consider a model along the lines of T  Ak n m p and determine the values of
A, n and p. You might also consider expressing the problem in terms of frequency instead of period.
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Wednesday 11 April 2012
PHYS 116 SCALE-UP
Applications
1. Consider the arrangements of springs and masses show below [you may imagine a frictionless
surface supporting the mass in figure (b)]. Assume k1=10 N/m, k2=20 N/m, and m=0.2 kg.
What are the effective spring constants keff and periods of oscillation for the spring-mass systems shown
below in figures (a) and (b)? The effective spring constant is the spring constant of a single spring that
would give the same frequency of oscillation.
2. Refer to the graph (below) of acceleration a(t) versus time t of a particle undergoing a simple
harmonic motion for the following questions.
a) Which of the labeled points correspond to the particle at its maximum positive displacement?
b) At point 8, is the velocity of the particle positive, negative, or zero?
c) At point 3, the particle is at:
a. Its maximum negative displacement -xm
b. Its maximum positive displacement +xm
c. between -xm and 0
d. between 0 and +xm
d) At what point(s) is the velocity of the particle at its maximum?
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Wednesday 11 April 2012
PHYS 116 SCALE-UP
SHAKING TABLE
A “shaking table” is a mechanical device (in the form of a table) that can vibrate objects at controlled
and adjustable frequencies and amplitudes. Shaking tables are used for a variety of purposes, including
earthquake research and mechanical integrity testing. Typical shaking tables have a maximum
displacement on the order of 10 cm, and some are capable of vibrating in more than one degree of
freedom. You are tasked with evaluating shaking tables for several different purposes:
Military specifications often call for electronic devices to be able to withstand accelerations of up to 10g
(10g = 98.1 m/s2). A device is placed on a shaking table and made to oscillate at an amplitude of 1.5 cm.
What should you adjust the frequency to in order to test for compliance with the 10g military
specification?
amax = -w 2 xmax ®
9810cm / sec
= 2p f
1.5cm
®
f = 12.9Hz
A shaking table can be used to measure the coefficient of static friction between materials. Specify the
parameters of a shaking table if coefficients of static friction between 0.1 and 0.9 are to be measured:
for an oscillation amplitude of 2 cm, what range of frequencies must the table be able to support?
Fmax = ms mg = mamax = -mw 2 xmax ® ms g = (2p f )2 xmax ®
f (0.1) =
1
2p
(0.1)980cm / sec
= 1.11Hz
2cm
f (0.1) =
1
2p
f=
1
2p
ms g
xmax
(0.9)980cm / sec
= 3.34 Hz
2cm
A shaking table is to be used to test the design of a simple seismometer. The seismometer is essentially
a mass of 5 grams suspended in space between 2 identical springs (in its simplest form, the mass would
be a pen scrawling its signature on a chart recorder). The seismometer is set up on a shaking table which
rapidly displaces 1 cm and stops, setting the seismometer’s mass in motion. What spring constant is
required to set the seismometer oscillating at a frequency of 5 Hz?
w2 =
2k
mw 2 .005kg
Newtons
® k=
=
(2p × 5Hz)2 = 2.47
m
2
2
meter
(Although the vertical equilibrium position of the mass is not equidistant between springs because of
gravity, the frequency of oscillation is the same as a spring of constant 2k).
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