Question

the bob of pendulum is a small sphere, filled with water.
now a small hole is made into the pendulum at the bottom because of which, water starts to flow out.
what will be the effect on the time period of the pendulum ?

Answer 1

None, the frequency of oscillation is independent of the mass of the object swinging.
\[f= \frac{ 1 }{2\pi }\sqrt{g/l}\]

where l is thelength of the string.

Answer 2

I am not so sure, because the system will lose momentum as it loses matter;

Answer 3

This analysis is simple minded I know but the problem describes a pendulum performing simple harmonic motion. The equation of motion given by:
\[d ^{ 2}\Theta/dt ^{2}= -(m(t)gl/I(t))\theta \]
for time dependent mass and moment of inertia. If the moment of inertia can be reasonably written\[I(t)= f(r,t)*m(r,t)\]

Just before the drop dissociate from the sphere nothing has changed. Unless f(r,t) changes significantly during the dissociation of the drop from the sphere I don't see how the change in mass affects the dynamics.

As the drop dissociates itself it has the velocity of the sphere and at that instant the momentums of the drop and sphere are equal to the the momentum of the sphere before the dissociation. Please let me know when you can show or explain why and how the changing mass has an affect on the period.

Answer 4

i got the ans.
it first increases, reaches a maximum, and then decreases back to initial most value
reason being as water drains out, center of gravity goes does, so effective length inc, which inc time period,
so COG first goes down and at some point in time starts going up, and comes back to the center, and hence the effect on the time period..

Answer 5

You are correct in recognizing the COM is lowered i.e. the length of the pendulum increases. Thus the frequency decreases. However I can't see it increasing at any point.

I was all wet in my initial assessment trying to over simplify the problem and not letting the math guide me . In fact the equation of motion is more accurately described by a differential equation for a damped harmonic oscillator giving a solution of a pendulum decreasing in frequency and amplitude with time. I can show more details if it is desired..

Answer 6

well it will increase after a certain point ofcorse..
water level goes down, so COM goes down initially,
a time will come when mass effect of sphere will dominate..

Answer 7

The COM begins to lower and the effective length of the pendulum increases the period. This continues until the water runs out then nothing changes and the period stabilizes.

Answer 8

am not able to explain this properly perhaps..the concept which i am stating here is absolutely correct..

Answer 9

can you tell me what is incorrect in what I have said?

Answer 10

you are saying COM doesnt come back up after it starts to go down.. ?

Answer 11

That's correct.

Answer 12

i'll give you a link
1 min..

Answer 13

http://forums.udacity.com/questions/9006925/simple-harmonic-motion-problem

this explains it well..

Answer 14

I see. thanks