Questions & Answers

Question

Answers

a. \[ 2\pi T{D^2}\]

b. \[ \pi T{D^2}\]

c. \[ \dfrac{{\pi T{D^2}}}{2}\]

d. \[ 4\pi T{D^2}\]

Answer

Verified

91.5k+ views

Change in energy \[ = \]\[(\]Change in Area \[ \times \] Surface Tension \[)\]

Volume of a sphere is \[ = \dfrac{4}{3}\pi {r^3}\], here \[r\] is the radius of the given sphere.

Area of a sphere is \[ = 4\pi {r^2}\].

A liquid drop is like a sphere.

So, the volume of the drop will be equal to the volume of the sphere.

Now we have to use the formula for volume of a sphere is \[ = \dfrac{4}{3}\pi {r^3}\], here \[r\] is the radius of the given sphere.

Let's say, the radius of each small drop is\[d\]unit.

The diameter of the large drop is \[ = D\] .

Radius of the large drop is \[ = \dfrac{D}{2}\] .

Volume of the large drop is \[ = \dfrac{4}{3}\pi {\left( {\dfrac{D}{2}} \right)^3}\] .

Volume of each small drop is \[ = \dfrac{4}{3}\pi {d^3}\].

Total volume of \[27\] drops will be\[ = \dfrac{4}{3}\pi {d^3} \times 27 = 9 \times 4\pi {d^3} = 36\pi {d^3}\] .

So, the total volume of \[27\]drops will be equal to the total volume of the large drop.

So, we derive the following equation:

\[ \Rightarrow 36\pi {d^3} = \dfrac{4}{3}\pi {\left( {\dfrac{D}{2}} \right)^3}\]

By simplifying the cubic form we get:

\[ \Rightarrow 36\pi {d^3} = \dfrac{4}{3}\pi \left( {\dfrac{{{D^3}}}{8}} \right)\]

Calculate the denominator parts in R.H.S:

\[ \Rightarrow 36\pi {d^3} = \pi \left( {\dfrac{{{D^3}}}{6}} \right)\]

Cancel out the \[\pi \]from both sides:

\[ \Rightarrow 36{d^3} = \left( {\dfrac{{{D^3}}}{6}} \right)\]

Divide R.H.S by \[36\]we get:

\[ \Rightarrow {d^3} = \left( {\dfrac{{{D^3}}}{6}} \right) \times \dfrac{1}{{36}}\]

Simplifying the above form:

\[ \Rightarrow {d^3} = \left( {\dfrac{{{D^3}}}{{{6^3}}}} \right)\]

Applying the rule of indices:

\[ \Rightarrow d = \left( {\dfrac{D}{6}} \right)\]

Initial area of the large liquid was \[ = 4\pi {\left( {\dfrac{D}{2}} \right)^2} = \pi {D^2}\] .

After division in \[27\] drops, the total area of all these drops becomes \[ = 27 \times 4\pi {d^2}\] .

But \[d = \left( {\dfrac{D}{6}} \right)\].

The changed area will be \[ = 27 \times 4\pi {\left( {\dfrac{D}{6}} \right)^2} = 3\pi {D^2}\] .

Change in total area will be \[ = (3\pi {D^2} - \pi {D^2}) = 2\pi {D^2}\] .

Resultant change in energy will be \[ = \]( Change in area \[ \times \] Surface Tension of the liquid) .

So, change in energy \[ = (2\pi {D^2} \times T) = 2\pi T{D^2}\].

Students Also Read