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Introductory mechanics
Chapter 13  Periodic Motion
Due at 11:59pm on Tuesday, December 9, 2008
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Springs in Parallel
In this problem you will study two cases of springs connected in parallel that will enable you to draw a general conclusion.
Two springs in parallel
Consider two massless springs connected in parallel. Springs 1 and 2 have spring constants
and
and are connected via a
thin, vertical rod. A constant force of magnitude
is being exerted on the rod. The rod remains perpendicular to the direction
of the applied force, so that the springs are extended by the same amount.
This system of two springs is equivalent to a single spring, of spring constant
.
Part A
Find the effective spring constant
of the twospring system.
Hint A.1
Freebody diagram
Hint not displayed
Part A.2
Extension of two springs
Part not displayed
Part A.3
Magnitude of
Part not displayed
Give your answer for the effective spring constant in terms of
and
.
ANSWER:
=
Three springs in parallel
Now consider three springs connected in parallel as shown.
The spring constants of springs 1, 2, and 3 are
,
, and
.
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MasteringPhysics
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The springs are connected by a vertical rod, and a force of
magnitude
is being exerted to the right.
Part B
Find the effective spring constant
of the threespring system.
Give your answer in terms of
,
, and
.
ANSWER:
=
You have now seen that the general formula for the overall spring constant of a set of springs in parallel is the sum of the
spring constants of the individual springs. You will see this again when you study electric circuits and learn about the
total capacitance of capacitors in parallel.
Springs in Series
In this problem you will study two cases of springs connected in series that will enable you to draw a general conclusion.
Two springs in series
Consider two massless springs connected in series. Spring 1 has a spring constant
, and spring 2 has a spring constant
. A
constant force of magnitude
is being applied to the right. When the two springs are connected in this way, they form a
system equivalent to a single spring of spring constant
.
Part A
What is the effective spring constant
of the twospring system?
Hint A.1
Freebody diagram
Hint not displayed
Hint A.2
Freebody diagram for spring 2
Hint not displayed
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Part A.3
Determine the extension of spring 2
Part not displayed
Part A.4
Determine the extension of spring 1
Part not displayed
Part A.5
Determine the total extension of the two springs
Part not displayed
Part A.6
Replace
and
Part not displayed
Part A.7
Determine the extension of the equivalent system
Part not displayed
Hint A.8
Solving for
Hint not displayed
Express the effective spring constant in terms of
and
.
ANSWER:
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This note was uploaded on 04/07/2010 for the course PHYS 3163 taught by Professor Pickett during the Spring '10 term at CSU Long Beach.
 Spring '10
 PICKETT
 Physics, mechanics

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