Question

One thousand water droplets of equal size combined to form a single drop. The ratio of the initial energy of the droplets to the final energy of the single droplet is: (\[T = 70dyne/cm\])
\[(A)10:1\]
\[(B)1:100\]
\[(C)100:1\]
\[(D)1000:1\]

Answer 1

Hint: The ratio of the energy of the big drops to the energy of the small drops is the formula that can be used to find the ratio of the initial surface energy of the droplets to the final surface energy of the single droplet. The one thousand similar drops are connected to make a big drop. Where \[n\] is taken as the number of small drops and \[r\] is taken as the radius of the small drops.

Formula used:
\[R = {n^{\dfrac{1}{3}}}\left( r \right)\]


Complete answer:
When \[1000\] identical drops are combined, the radius of the big drop formed will be-
\[R = {n^{\dfrac{1}{3}}}\left( r \right)\]
\[R = 10\left( r \right)\]
Here,
\[n\]is the number of small drops
\[r\]is the radius of a small drop.
So, \[\dfrac{{Energy{\text{ }}of{\text{ }}big{\text{ }}drops}}{{Energy{\text{ }}of{\text{ }}small{\text{ }}drop}}\]
\[ = \dfrac{{4\pi {{\left( {10r} \right)}^2}}}{{1000\left( {4\pi {r^2}} \right)}}\]
\[ = \dfrac{{100{{\left( r \right)}^2}}}{{1000{{\left( r \right)}^2}}}\]
\[ = \dfrac{1}{{10}}\]
Here, the Initial surface energy of the droplet\[ = 10\], Final surface energy of the droplet \[ = 1\]
So, the ratio of the initial energy of the droplets to the final energy of the single droplet is\[10:1\].

Note: The unit traditionally used to measure surface tension and interfacial tension is defined as a dyne per centimeter.
The initial mechanical energy of a system equals the final mechanical energy for a system where no work is done by non-conservative forces
Force is a unit of a dyne per centimeter. A dyne is a force needed to accelerate a body of mass of one gram at a rate of one centimeter per second squared.