Amazing Pendulum!!!

The effective length of the pendulum is: $l_{eff} = l+r$ , where $l$ is the length of string and $r$ is the distance of the centre of mass (C.O.M.) from the point where the string ends on the sphere.

The $l_{eff}$ w.r.t. time is shown below: image

Time period of a pendulum is $2 \pi \sqrt\frac{l_{eff}}{g}$ .

$l_{eff}$ first increases and then decreases. Thus, the time period will first increase and then gradually decrease to its initial value.

We know that time period $T=2\pi \sqrt{\frac{L}{g}} \Rightarrow T \alpha L^{\frac{1}{2}}$

Here, $L$ is the length of the pendulum measured from the point of suspension to the center of mass of the bob .

Since mercury is dripping out, the center of mass will keep shifting down $\Rightarrow L$ will increase and hence time period will increase. But after mercury has dripped out, the center of mass will shift back to middle of the bob and hence the time period will decrease to its initial value.

Since mercury is dripping out, the center of mass will keep shifting down will increase and hence time period will increase. But after mercury has dripped out, the center of mass will shift back to middle of the bob and hence the time period will decrease to its initial value.

BECAUSE EFFECTIVE LENGTH FIRST INCREASES THEN DECREASES AND THEN FINALLY BECOMES EQUAL TO INITIAL LENGTH

time period of a pendulum is given by :

T = 2*3.14 \sqrt{l/g}

as the mercury drops out , the centre of mass lowers down , hence the length of pendulum increase,

hence the time period.

Then the center of mass starts climbing up ( just think when sphere gets empty, it climbs up back to its original pos. ) hence the length starts decreasing and hence the time period.

centre of mass will come down and then at last come up again thus legth of rope will be considered first larger gradually and then small thus incresing the t first and then decreasing

The time period of a pendulum is given as _ $2*pi$ $sqrt{(L/g)}$ _ where L is the effective length of the pendulum and g is acceleration due to gravity. Here L is the distance of the center of mass of glass sphere( with mercury ). When the sphere is completely filled with mercury the center of mass is at the center of the sphere. When the level of mercury decreases, the center of mass goes down and the effective length of pendulum increase.Hence the time period increases( T is directly proportional to L). But when the mercury drains out more and more, the center of mass starts to go up. So the effective length of pendulum decreases and hence the time period decreases.
Initially the CM of bulb and mercury was at center of sphere and at the end too. So the L remains same as before. Since other factors change, the time period remains unchanged.