# Advantages of a cone-shaped spring?

• KTBMedia
KTBMedia
TL;DR Summary
For measuring period of oscillation, what are the advantages of using a cone spring over a cylindrical spring?
I'm doing a personal experiment where I take a conical spring (that is, a spring with two different diameters on either end), hang it from the ceiling, and measure the period of oscillation for different masses hanging below the spring. I do this for two different orientations of the spring; one in which the larger radius is facing upwards, and one where the smaller radius is facing upwards.

My results have determined that, for the case in which the larger radius is facing the ceiling, the periods of oscillation measured are consistently longer than the other way around. Why might this be the case?

On a semi-related note, I've noticed that many descriptions of spring-based lab experiments in high schools and colleges opt to use conical springs (always with the larger radius facing the floor, interestingly) over a more standard cylindrical spring. Is there some advantage to using conical springs in general over a simple cylindrical one?

Thanks!

Rive
KTBMedia said:
Is there some advantage to using conical springs in general over a simple cylindrical one?
Perhaps an opposing force when pushed off axis and less like likely to buckle in compression? I'd rather try to balance on top of a conical spring than a cylindrical one.

Also, for some applications they can be squished flatter.

KTBMedia said:
My results have determined that, for the case in which the larger radius is facing the ceiling, the periods of oscillation measured are consistently longer than the other way around. Why might this be the case?
Hand waving explanation here, so interpret accordingly. A conical spring is a coil spring with one end wound to a smaller radius than the other end as in the image below.

The spring rate of a coil spring is proportional to the cube of the mean diameter. The mean diameter is equal to the outer diameter minus the wire diameter. The mean diameter of a conical spring is variable - each turn has a different mean diameter. The spring in the image above has a large end mean diameter about 2.5 times larger than the small end mean diameter. Since the spring rate is proportional to the cube of the mean diameter, the spring rate of the small end is about 15 times higher than the spring rate of the large end.

The period of a mass hanging from a spring and oscillating in the direction shown below (not swinging back and forth) is proportional to the square root of the ratio of the mass to the spring constant.

In a introductory physics class, the mass of the spring is neglected. In the real world, part of the mass of the spring is added to the oscillating mass.

A conical spring adds another complication. The small end is so much stiffer than the large end that it behaves more like a mass than a spring. When the small end is attached to the ceiling, the mass of that end is barely moving, so contributes almost nothing to the oscillating mass. When the large end is attached to the ceiling, almost all of the mass of the small end is part of the moving mass. The moving mass is different depending on which end of the conical spring is fixed, but the total spring constant is the same in both cases.

Normally, the moving mass is much larger than the mass of the spring, so the difference in natural frequency is small. The OP's observation is the result of careful measurements, thinking to turn the spring around and measure again, and recognizing that the results were real and not some experimental error. Good work!

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