How will the time period of a simple pendulum will be affected if there is a hole at the bottom of the bob from which mercury falls down?
Answers
The formula for the time period of a pendulum (for small angles of displacement from mean position) is
T=2π√L/g
Now, L here is the length from the point of suspension to the center of mass of the bob. For illustration, assuming the bob is spherical, as the water leaks out, the center of mass will shift downwards, increasing L and hence increasing the time period.
As soon as all the water has leaked out, however, the center of mass will be back at the geometrical center (which was its original position when it was full of water), L will decrease and so will the time period. Therefore, 1 is the answer.
Of course, the effect the leakage of water will have on the change in time period and the point at which the time period again decreases (after the initial increase) depends on the density of the material of the bob and its geometry.
Hope this helps you ☺️☺️✌️✌️❤️❤️
The mass has nothing to do with it.
…well…unless there is air resistance (which there is). In that case, the mass of the bob does actually matter.
The real problem here is that what you say isn’t true.
The “length” of the pendulum is measured to the center of mass of the bob. If we’re talking about a simple spherical bob with a small hole in the bottom - then as the mercury drains out - the center of gravity of the bob would shift in some complicated way.
When the bob is completely full - the center of gravity would be in the center of the bob. When it’s completely empty - it’ll also be at the center. But when it’s half-full of mercury - then the center of gravity will be below the center of the spherical bob.
That would effectively lengthen the pendulum as the mercury starts to drain, then gradually shorten it again as the last of it goes away. So the period of the pendulum would actually slowly increase, then decrease again.
But as a “thought experiment” where a pendulum swinging in a vacuum has a zero-sized bob that changes slowly in mass - then indeed the period of the pendulum doesn’t change.