How damping constant of pendulum varies with length of wire? (experiment result confusion)

Question

I'm a high school student doing an experiment for my physics course. The experiment aims to find out the relationship between the damping constant and the length of the wire. The constant is calculated with this formula:

$$\ln \left(\frac{θ}{θ_0}\right)=-\frac{k\,t}{2}$$

From this paper http://dx.doi.org/10.4236/jamp.2017.51013

Since I don't have any advance equipments, I put the pendulum in water to make the damping more observable. My hypothesis was that the longer the length, the bigger the damping constant will be, since the velocity of the bob will be faster while damping being proportional to velocity. However, my experiment shows completely opposite result: the shorter the wire, the bigger the damping constant. I can't figure out how to explain this result, is there any possible logical explanation? Or is my experiment just wrong?

Answer 1

It was a good idea, but a pendulum in water will behave differently than a pendulum in air. Looking in section $2$ of your link, Reynolds number is given by

$$Re = \frac{\rho L |v|}{\mu}$$

The derivation that follows is valid for sufficiently low Re. But $\rho_{water} \approx 1000 \space \rho_{air}$.

There are two sources of drag. One is friction. Not just from an object passing through a fluid, but the object makes nearby fluid move faster than far away fluid. Parts of the fluid moving at different speeds exert friction forces on each other. This is called viscosity.

The other is the the object must push fluid out of the way, and fluid must flow in to fill the space behind the object. The fluid is accelerated up to a speed and then back to a stop. The moving fluid has kinetic energy which comes from the kinetic energy of the object. The object must exert a force on the fluid to accelerate it. These are called inertial forces.

The definition of Re is the ratio of inertial forces to viscous forces. The formula is an approximation. In most objects, one force is much larger than the other. You usually can ignore the smaller force. An approximate is all you need to decide which one is way bigger. Reynolds number is a useful rule of thumb, more than anything exact.

Things that make inertial forces big (and therefore Re big) are things that make $1/2 m v^2$ big. For example, a dense fluid, a high speed, and a large object (which pushes a lot of fluid around). Things that make viscous forces big (Re small) are a viscous fluid.

So your pendulum in water is dominated by inertial forces, where the experiment is expecting viscous forces to be the big thing.

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You can modify your experiment to run in air, but have bigger viscous forces.

Try using yarn for the string, which has a lot of little hairs that create more friction without adding much mass.

Try keeping the speed low by using a smaller amplitude.

Try using a light mass on your pendulum.

This last one is a little misleading. This doesn't change the inertial forces. That comes from the mass of the moving fluid. The shape of the object, not the mass of the object, determines how much air is pushed around.

But given forces that slow the pendulum, you have $F = ma$. A small mass means a larger acceleration. The forces slow a light mass more effectively than a large mass. That will make damping easier to measure. Think of using a balloon for your mass vs a water balloon.

Answer 2

The increase in velocity for the longer pendulum does, while increasing the total dampening force, not increase the damping constant. This can easily be shown by analyizing the mechanical problem.

Damping that leads to $e^{-kt}$-like decay of the oscillations is due to friction forces that are proportional to the velocity (such as the drag in a fluid at sufficiently low speeds, for a sphere this is given by the Stokes formula $F = 6\pi \eta r v$, with the viscosity $\eta$, the radius of the sphere $r$ and the velocity $v$). As is explained in the answer by @mmesser314 this formula is only valid if the velocities are low enough and the cutoff depends roughly on the Reynolds number – but we will assume in this answer that the formula is valid for the case of the water (which it may well be for slow oscillations).

The differential equation describing the motion of a pendulum with this kind of friction then is: $$ m l \frac{d^2\phi(t)}{dt^2} = m g l \sin(\phi) + 6\pi\eta r l \frac{d\phi(t)}{dt} $$ (This is just Newtons $F = ma$ written down for the pendulum, and writing the acceleration and velocity in terms of the angle $\phi(t)$ – which then becomes the function we try to determine.)

As you can see, all terms are proportional to the length $l$ of the pendulum, so this parameter can't affect the result of $\phi(t)$ under the given assumptions!

Possibilities for this to go wrong:

• My best guess on the behavior you observe is due to a friction component that's not proportional to the velocity, but constant (such as the friction of the bearing where your wire is suspended – dry friction of solids on solids are typically independent of the velocity). This would mean an extra term in the equation above, that does not change with the length of the wire, so as the wire gets longer and longer, this terms influence will get smaller and smaller.

• Experimental limitations. For example, the period of a pendulum is no longer constant when the elongation gets too large (we typically say larger than $5°$) – this in turn leads to a higher velocity of the pendulum and therefore to larger friction losses. (The same linear elongation leads to a larger angular elongation when the wire gets shorter).

• Yet another possible explanation is that it is purely a measurement error. It is quite challenging to measure maximal elongation precisely (especially when the wire is short).

A good way to identify, or at least restrict the possible causes, is to do multiple measurements of the maximal elongation for one run of the experiment (e.g. every ten periods), and then graphing the $\ln \theta_{\text{max}}$ over $t$.

This way you can see the following:

• Does the oscillation actually follow the expected $e^{-kt}$-behaviour – then the points should lie on a straight line (and its slope gives you $k$)?

• Inaccuracies of the individual time and elongation measurements can potentially be compensated by combining several measurements.

• By also the initial elongation and then graphing the number of oscillations over time, you can check whether you really are in the regime of small angles that the analysis assumes (Are the periods the same for the different string lengths? Does the period time remain constant?)