Question 279: Gravitation

Question

Water rises up to height ' $x$ ' in a capillary tube immersed vertically in water. When the whole arrangement is taken to a depth ' $d$ ' in a mine, the water level rises up to height Y . If R is the radius of the earth then the ratio $\mathrm{Y}: \mathrm{x}$ is

Accepted Answer

Step 1: Capillary Rise Formula
The height $h$ to which a liquid rises in a capillary tube is given by:
$$ h = \frac{2T \cos 	heta}{r ho g} $$
Where:
$T$ = Surface tension
$	heta$ = Angle of contact
$r$ = Radius of capillary
$ho$ = Density of liquid
$g$ = Acceleration due to gravity
Step 2: Effect of Gravitational Acceleration
At earth's surface:
Let capillary rise be $x$.
The value of gravity at the surface is $g$.
At depth $d$ below earth's surface:
The value of gravity $g'$ reduces according to:
$$   g' = g \left(1 - \frac{d}{R}ight)   $$
Where $R$ is the radius of the Earth.
Let the new height of water in the tube be $Y$.
Step 3: Write Heights at Surface and Depth
At surface:
$$ x = \frac{2T \cos 	heta}{r ho g} $$
At depth:
$$ Y = \frac{2T \cos 	heta}{r ho g'} $$
So,
$$ Y = \frac{2T \cos 	heta}{r ho \left[g \left(1-\frac{d}{R}ight)ight]}   = \frac{2T \cos 	heta}{r ho g} \cdot \frac{1}{1 - \frac{d}{R}} $$
So,
$$ Y = x \cdot \frac{1}{1 - \frac{d}{R}} $$
Step 4: Find the Ratio $\frac{Y}{x}$
$$ \frac{Y}{x} = \frac{1}{1 - \frac{d}{R}} $$
Rewrite denominator:
$$ \frac{Y}{x} = \frac{1}{\frac{R-d}{R}} = \frac{R}{R-d} $$
Step 5: Compare With Options
Thus, the correct ratio is:
$$ Y : x = R : (R - d) $$
Final Answer:
Option B
$\boxed{\mathrm{R} : (\mathrm{R} - \mathrm{d})}$

Answer

$\quad \mathrm{R}:(\mathrm{R}+\mathrm{d})$

Answer

$\mathrm{R}:(\mathrm{R}-\mathrm{d})$

Answer

$\mathrm{R}:(\mathrm{R}-\mathrm{d})^2$

Answer

$\mathrm{R}:(\mathrm{R}+\mathrm{d})^2$

Gravitation

Water rises up to height ' x ' in a capillary tube immersed vertically in water. When the whole arrangement is taken to a depth ' d ' in a mine, the water level rises up to height Y . If R is the radius of the earth then the ratio \mathrm{Y}: \mathrm{x} is

Choose the correct answer: