If a pendulum swings with the same period at the top of the mountain and at the bottom of the mine then the ratio between height H of a mountain and the depth `h` of a mine is :

Options

A. `(1)/(2)`
B. 1
C. `(1)/(4)`
D. 2

Answer

To solve the problem of finding the ratio between the height \( H \) of a mountain and the depth \( h \) of a mine when a pendulum swings with the same period at both locations, we can follow these steps:

### Step 1: Understand the Period of a Pendulum
The period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

### Step 2: Determine the Acceleration Due to Gravity
The acceleration due to gravity changes with height and depth. At a height \( H \) above the Earth's surface, the acceleration due to gravity \( g' \) is given by:
\[ g' = g \left(1 - \frac{2H}{R}\right) \]
where \( R \) is the radius of the Earth.

At a depth \( h \) below the Earth's surface, the acceleration due to gravity \( g'' \) is given by:
\[ g'' = g \left(1 - \frac{h}{R}\right) \]

### Step 3: Set the Periods Equal
Since the pendulum swings with the same period at both locations, we can set the expressions for the periods equal:
\[ 2\pi \sqrt{\frac{L}{g'}} = 2\pi \sqrt{\frac{L}{g''}} \]
This simplifies to:
\[ \sqrt{\frac{L}{g'}} = \sqrt{\frac{L}{g''}} \]
Squaring both sides gives:
\[ \frac{L}{g'} = \frac{L}{g''} \]
Since \( L \) is the same for both cases, we can cancel it out:
\[ g' = g'' \]

### Step 4: Substitute the Expressions for \( g' \) and \( g'' \)
Substituting the expressions for \( g' \) and \( g'' \):
\[ g \left(1 - \frac{2H}{R}\right) = g \left(1 - \frac{h}{R}\right) \]

### Step 5: Cancel \( g \) and Rearrange
Cancelling \( g \) from both sides (assuming \( g \neq 0 \)):
\[ 1 - \frac{2H}{R} = 1 - \frac{h}{R} \]
This simplifies to:
\[ -\frac{2H}{R} = -\frac{h}{R} \]
Multiplying through by \( -R \) gives:
\[ 2H = h \]

### Step 6: Find the Ratio \( \frac{H}{h} \)
From the equation \( 2H = h \), we can express the ratio of \( H \) to \( h \):
\[ \frac{H}{h} = \frac{1}{2} \]

### Conclusion
Thus, the ratio between the height \( H \) of the mountain and the depth \( h \) of the mine is:
\[ \frac{H}{h} = \frac{1}{2} \]