|MadSci Network: Physics|
The different configurations could be ("constrain" is the end where the spring is attached to a wall or anything with infinite mass. The direction of g is indicated in all the cases):
Throughout the present expln. the springs are considered massless. In case you want to discuss the case of springs with mass, send your question to Madsci.
Also, before we come to the individual cases, I will start with the concept common for all cases (but the last one), though I am sure you are completely aware of it: simple harmonic motion. Skip this part if you understand SHM pretty well.
The motion of mass with a massless spring attached to it is usually simple
harmonic motion (SHM). It is characterized by a sinusoidal motion in time
a single resonant frequency:
X = A sin(w t)
X = displacement from mean position
A = amplitude
t = time
The acceleration is found by diffrentiating the above equation wrt time:
Accn = -A w^2 sin(w t) = -w^2 X
Note that whenever the acceleration (and hence the restoring
force) is directly proportional to the
negative displacement from the neutral postion (-X), the motion would be a
What this means is that the acceleration increases as displacement from the
neutral positon increases. The negative sign creeps in since the
in a direction opposite to the displacement
For the case of a simple spring with a mass hanging at its end this
Accn = F/m = -k X/m = -w^2 X
where k is the Hooke's constant for the spring.
Thus, w = sqrt (k/m), where "sqrt" stands for square root.
Thus the period of oscillation is given by:
T = (2 pi)/w = 2 pi sqrt(m/k)
Now consider the individual cases:
Case 1: Two horizontal springs and mass at end and horizontal displacement on mass
| |^^^^^^0^^^^^^^OMO -> |The spring is given a displacement in the horizontal plane
Case 2: Two horizontal springs and mass at the middle and horizontal displ. on mass
| | |^^^^^OMO^^^^^^| | |In this case the displacement on both the springs is the same thus reducing the situation to:
Case 3: Two vertical springs in series and mass at end (vertical displacement)
____ < > < > < 0 > < > < > < MMBasically the same situation as described in with a small variation: since gravity comes to play here, the neutral position is not the unextended position of the spring. There is an intial displacement given by:
Case 4: Two vertical springs in series and mass in the middle end (vertical displacement)
____ < > < > < MM > < > < > < ---Again this is a case similar to case 2 with the initial displacement (at the neutral postion) given by:
Two springs in series and mass in the middle (VERTICAL displacement).
Intial: | | |^^^^^^^^^M^^^^^^| | | Displaced state: | | |# #| | # # | | # # | | # # | | # # | | # # | | M |The intial lengths of the two springs are l1 and l2 and their spring constants are k1 and k2 respectively. If their displacement at equilibrium position is x01 and x02 respectively, the equations before the horizontal perturbation are given is given by:
If you want to solve nice problems on SHM I recommend I.E.Irodov - I
enjoying the large set of problems given there. For the case of springs
mass, which is a rather interesting case, do ask me via madsci.
Try the links in the MadSci Library for more information on Physics.