Investigating the damping of vibrational motion Imagine that a pendulum is set s

Question

Investigating the damping of vibrational motion Imagine that a pendulum is set swinging. What will happen to these vibrations as time passes? Now imagine an object hanging from a spring. The object is pulled downward and released so that it vibrates up and down. What will happen to these vibrations as time passes? They diminish, eventually stopping entirely. This phenomenon is called 'damping'. Design an experiment to investigate this damping effect quantitatively. Your goal is to find a pattern and represent it mathematically (i.e. construct a testable hypothesis). Available equipment: Ring stand, clamp, spring, object to hang from the spring, meter stick, balance, motion sensor connected to Logger Pro, stopwatch (software on desktop, website, or cell phone), masking tape, index cards WARNING: MAKE SURE EVERYTHING IS SECURE SO THAT NOTHING WILL FALL AND DAMAGE THE MOTION SENSOR. Determine which of the three kinds of experiments this is, then use the appropriate rubrics to evaluate your work as you design and perform the experiment. Testing your hypothesis Design an experiment to quantitatively test the hypothesis you constructed in the previous experiment. Available equipment: Same as above. Determine which of the three kinds of experiments this is, then use the appropriate rubrics to evaluate your work as you design and perform the experiment.

Explanation / Answer

Hi,

Well I will give you some theory about vibrational motion and what happens when you have a damping force.

As you know a vibrational motion can be described mathematically through an equation similar to the following:

x(t) = A cos (wt + ?) ; x is the displacement of an object from an equilibrium position, A is the amplitude of the movement which dictates the maximum and minimum value of the displacement while w and ? are parameters that depend on the initial conditions of the motion.

The previous equation can be applied to objects hanging from springs and, to certain extent, pendulums. However, as you can see, this equation implies that there are no forces that transform mechanical energy into internal one and therefore the movement will continue forever.

The previous situation is ideal. The reality is that the movement of those objects will stop eventually due to a damping force (usually friction).

Thanks to the materials they give you for the experiment I can say that the friction will occur between the hanging object and the air. The friction between them is small but enough to eventually stop the motion. If we consider that, a good mathematic model will be:

x(t) = A exp[(-b/2m)t] cos (wt + ?) ; where m is the mass of the object and b is a parameter called the damping coefficient.

Physically, what it is happening is that now we have a damping force acting over the object, and said force is equal to:

F = -bv ; where v is the speed of the object at a certain time

In the end, the equation we presented before is only a simplified solution of a much more complex one which is:

-kx - bv = ma (which results from applying the Second Law)

m (d2x/dt2) + b(dx/dt) + kx = 0 (this is a differential equation of second order)

But again, as you are doing this experiment with air as damping medium, you should be able to use the simplified equation.

To sum up, you should find in your experiment that the movement (the displacement) of the hanging object is reduced exponentially as time pass by.

Note: when you design this experiment you should: take note of the mass of a certain object and the spring, place the equilibrium position, measure a certain amplitude by stretching the spring, then take note of the maximum value of displacement reached by the object both up and down the equilibrium position. By repeating this with different values of amplitude, times and masses you could gather enough data to prove the mathematic model.

I hope it helps.

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