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### Two Masses Attached to a Spring

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Date: 12/27/2000 at 11:14:20
From: Gary Cunningham
Subject: Behavior of springs and masses in a gravitational field

A friend proposed the following, but I can't seem to get the math to
prove him wrong even though my intuition tells me he is wrong.

Two masses are attached to either end of an elastic spring and the
whole system is in a gravitational field. One mass is held in a fixed
position and the second mass is allowed to hang free below and stretch
the spring. Now, let go of the upper mass.

He proposes that the spring force will pull the two masses toward each
other in such a way that the lower mass will be pulled upward and thus
remain stationary in space until all the potential elastic energy of
the spring is exhausted (which will be when the masses are as close
together as the spring allows), at which time the lower mass will
begin to accelerate downward. (In contrast, the upper mass will
accelerate downward at a rate dependent on the force of gravity plus
the spring force.) Until the springs compress to a zero-force state,
the lower mass will not move.

This makes no sense to me. The elastic force in the spring is EXACTLY
EQUAL to the weight of the lower object. BUT, this spring force will
act on BOTH masses to pull EACH TOWARD THE OTHER when the upper mass
is released. This means the spring force (which was exactly equal only
to the weight of the lower mass), will act on EACH mass attached at
the ends. Therefore, since the spring force was exactly equal to the
weight of the lower mass, and since that force is now divided between
two masses, it cannot accelerate the lower mass upward to exactly
counterbalance the gravitational force. Thus the lower mass must begin
to accelerate downward the moment the upper mass is released (although
more slowly than it would without the spring system).

In addition, the spring force is dependent on the degree of
deformation of the spring, which declines as the spring returns to its
neutral state. The spring will exert less force on the masses as it
becomes less deformed. Since the force is lessened as the spring
compresses, the force on the lower mass is progressively lessened. So
even if the force were not divided between two masses to pull them
together (if there were zero mass at the upper end of the stretched
spring) the lower mass would still begin to fall instantly since the
spring force is declining from its maximum.

That's it. Thanks for your help.
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Date: 12/29/2000 at 12:59:35
From: Doctor Rick
Subject: Re: Behavior of springs and masses in a gravitational field

Hi Gary, thanks for writing to Ask Dr. Math.

The motion of a system of two masses attached by a spring can be
broken into two parts. The system as a whole is acted on only by
gravity, so the center of mass of the system accelerates downward with
an acceleration of g. The motion of the masses relative to the center
of mass is independent: gravity is not involved, only the spring
force.

Suppose you had this system on the Space Station. Gravity isn't
operative, so the masses at rest are separated by the rest length of
the spring. Stretch the spring until the separation is the same as you
had in the earthbound scenario, and release them. Assuming that the
spring is constrained to move along a straight line, the two masses
will oscillate, first moving together, then apart, and so on.

If the spring is stretched a distance x beyond its rest length, and
its spring constant is k, then the force that it exerts on each mass
is kx, acting toward the center of mass. When you release the masses,
they begin to accelerate toward each other with accelerations kx/m_1
(for the first mass, m_1) and kx/m_2 (for the second mass, m_2).

What is that initial stretch length, x? Going back to earth, the
hanging mass (m_2) experiences two forces: a gravitational force gm_2,
and a spring force kx acting in the opposite direction. When the mass
is at rest, these opposite forces must be equal:

gm_2 = kx

x = gm_2/k

So that's our starting length in the space-station experiment, and the
acceleration on mass 2 is:

kx/m_2 = k(gm_2/k)/m_2

= g

That's interesting! When we drop the system in the earth-bound
experiment, the center of mass accelerates with an acceleration of g,
and the hanging mass accelerates upward (toward the center of mass)
with an acceleration of g. The net acceleration of the mass is 0. In
other words, at the moment you release the first mass, mass 2 is not
accelerated.

Your friend was thus correct, if we consider only the moment at which
the system is released. The source of your confusion is that the force
is NOT divided between the masses: it acts solely on the hanging mass,
while an equal and opposite force (Newton's third law) acts on the
initially supported mass. Thus the initially supported mass is
accelerated downward at twice the acceleration of gravity, while the
hanging mass is pulled upward just as hard as gravity pulls it down,
and it experiences no acceleration.

On the other hand, you are right in your second argument: the
situation of zero acceleration will not persist. Your friend's
statement that the mass will remain stationary until the spring
reaches its rest state is incorrect. As the spring pulls the masses
together, the force it exerts decreases, because it is proportional to
the stretch length. Thus, a moment after the system is released, the
upward force is no longer sufficient to suspend the mass motionless.
By the time the rest-state of the spring is reached, both masses are
already falling at a full 1 g acceleration.

What's interesting about the motion of the hanging mass is that, at
the moment of release, its motion is described, not by a parabola
(quadratic function of time), but by a cubic or third-power function
of time. That's not as dramatic as being motionless for a period of
time, but it is not the way we are accustomed to seeing things fall.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Physics
High School Physics/Chemistry

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