Spring System Energy
Name: Shawn D.
My physics teacher found a discrepancy in our book.
Say you are given a vertical spring with a mass hanging from it. If
the weight is displaced downward, and let go, the book says that the
potential energy of gravity can be ignored when finding the weight's
maximum velocity. It factors in the change in the spring's potential
energy, but disregards gravity. Please help us restore our faith in the
book we invested so much work on.
The book is correct, but misleading. The condition of MAXIMUM velocity is
the key phrase and the resolution of the apparent paradox. Without going
through the derivation, the maximum velocity occurs when the spring passes
through its equilibrium position. For that instant of maximum velocity the
spring is in its equilibrium so the force of the spring Fs = -k(X - Xo) = 0
just balances the gravitational force
Fg = mgh. All of the energy is in the form of kinetic energy. I do agree
that the book should not have just left such a remark unexplained.
Your book is absolutely right, even though it seems to violate common sense
when you first see it. I think I might be able to convince you with this
We all know that Newton's Laws rules the roost here, so F = ma. Consider
the F. If you consider y to be measured upward from the position of the
the spring is not extended (the mass is not yet attached), then
F = -ky - mg.
Here k is the spring constant and mg is the weight of the mass. Now do a
change of variable to, say, z = y + d, where z is measured from the mass
position when it is hanging on the spring and is at rest. then
F = 0 = -kd - mg and d = -mg/k so
y = z - d.
Notice that then F = -ky - mg = -kz + kmg/k - mg = -kz.
So if you measure the displacement of the mass from its equilibrium
position, the effect of gravity vanishes! What really happens, of course,
is that the
extra spring tension just exactly balances the weight of the mass.
Best, Dick Plano, Professor of Physics emeritus, Rutgers University
The effect of gravity is take into account by your choice of the zero point
for the spring. If you choose the zero point to be the point of
equilibrium, the spring force you are using contains the effect of
gravitational force. The spring potential energy contains the effect of
gravitational potential energy. You have essentially fused them into one
force equation. If instead of equilibrium, you choose the zero point to be
where the spring is when it has no weight hanging from it, then you must
include gravitational potential energy.
Dr. Ken Mellendorf
Illinois Central College
I think the idea that the book intends to get across is that you can
choose arbitrarily the height to which you will assign a gravitational
potential energy of zero. You can do this because you're only interested
in changes in gravitational potential energy.
However, to calculate the maximum kinetic energy that will result from a
given spring extension, you cannot ignore the change in gravitational
potential energy associated with that spring extension. If the spring
force at the given extension is large compared to the weight of the mass,
then you can neglect gravitational potential energy and get approximately
the right answer, but the exact answer has to include it.
While it may not seem intuitive at first, the book is correct, the maximum
velocity is not affected by gravity. This isn't to say that gravity has
no affect on the system, only that its effect on the system does not
affect the velocity.
Recall that the force due to gravity is m * g, or mass times the
acceleration due to gravity (9.8 m/(s*s)).
Recall also that force from the spring is -k * x, or the spring constant
times the displacement.
In its initial state, this spring system has no acceleration, so the total
forces must be zero:
-m*g - k * x = 0
So we can solve this for x:
m * g / k = -x
We will call this initial displacement Xg, or the displacement due to
gravity, and its sign will be negative since it is a downward displacement.
Now, when you pull the spring downward, you will increase the force acting
upward on the spring. We will call this displacement Xd, and its sign
will be negative, since it is a downward displacement.
So when you have displaced the system, the total force will be:
-m * g + k * Xg + k * Xd = Flowest
And, at the highest point, the forces will be:
-m * g + k * Xg - k * Xh = Fhighest
Since the energy at the top must be the same as the energy at the bottom,
and since at the time the they are at the top and bottom, the mass is not
moving, all the energy is potential at these points.
-m * g + k * Xg + k * Xd = -m * g + k * Xg - k * Xh
Now we simplify. First we already know that -m*g + k * Xg = 0, so:
k * Xd = -k * Xh
And we can stop there and realize that the energy contributed by gravity
is constant throughout, and its only effect is to displace the entire
system downwards by a fixed amount. In other words, the displacement of
the spring by gravity reduces the potential energy due to gravity of the
entire system throughout its entire existence. So the mass is lower at
its lowest point that it otherwise would be, and it is lower at its
highest point than it otherwise would be.
I hope this helps,
Far be it from me to argue that text books never contain errors! Despite
the author's best efforts they do get there so it is always a
good practice to seek to understand the subject (as you and your
classmates are) rather than to accept information blindly.
In this case it sounds like a problem that you have been asked to work or
that is being presented as an example. Sometimes authors like to simplify
problems by removing factors from consideration. This is how a lot of
scientific work is done. We start by assuming certain factors do not have
a big impact on our results and come up with a theory that explains the
major effects, then we start finding the factors that result in minor
corrections until the theory has enough components to fully explain the
Now, in this case the change in potential energy due to gravity has been
neglected. In essence the author is assuming (or asking you to assume)
that the spring force constant is much greater than the gravitational
force. Have you worked the problem both ways? How much difference does
the gravitational force make? Was the author justified in neglecting
gravitational potential energy?
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Update: June 2012