How to derive an equation for the mass of a pendulum bob?
If anyone knows of a resource that shows this (I've looked) or could explain how to derive an equation for the mass of a pendulum would be much appreciated.
EDIT:
I'm talking about a simple pendulum. Some more context may be given with the following example:
Some mass $M$ is release from rest at a horizontal position. $M$ reaches the bottom of its path (so directly under the pivot) at a velocity of 2.2m/s. I understand that I can derive for instance, the length of the rope attached to the bob as $L = v^2/2g$. So I can find the length of the rope. Now with no further information is it possible to derive an equation that can be used to describe the mass of this bob?
Best Answer
The velocity of the bob is due to the conversion of gravitational potential energy to kinetic energy.
When you displace the bob sideways by some angle $\theta$ it moves upwards by a distance $d$ given by:
$$ d = \ell(1 - \cos\theta) $$
where $\ell$ is the length of the string. This increases its gravitational potential energy by $Mgd$. At the lowest point of the swing the kinetic energy will be $\tfrac{1}{2}Mv^2$ and equating the two gives us:
$$ Mgd = \tfrac{1}{2}Mv^2 $$
or:
$$ v = \sqrt{2gd} $$
The mass cancels out and doesn't appear in the equation for the velocity, so there isn't any way to calculate the mass of the bob.
This happens for the same reason that all objects fall at the same speed (ignoring air resistance etc). The gravitational force on a mass is proportional to the mass, but the acceleration is inversely proportional to the mass. That means the mass term cancels out and objects of all masses experience the same gravitational acceleration.