A simple pendulum has the same periodic time (T) at the top of a mountain of height (h) and at the bottom of a mine having a depth d . What is the relation between h and d?

Options

A. d=h
B. d=2h
C. `d=(h)/(2)`
D. d=3h

Explanation

To find the relationship between the height \( h \) of a mountain and the depth \( d \) of a mine, given that a simple pendulum has the same period \( T \) at both locations, we can follow these steps:

### Step 1: Understand the formula for the period of a simple 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: Define the acceleration due to gravity at height \( h \)
At the top of the mountain, the acceleration due to gravity \( g' \) can be expressed as:
\[ g' = g \left(1 - \frac{2h}{R}\right) \]
where \( R \) is the radius of the Earth and \( g \) is the standard acceleration due to gravity at the surface.

### Step 3: Define the acceleration due to gravity at depth \( d \)
At the bottom of the mine, the acceleration due to gravity \( g'' \) can be expressed as:
\[ g'' = g \left(1 - \frac{d}{R}\right) \]

### Step 4: Set the periods equal
Since the period of the pendulum is the same at both locations, we can set the expressions for the periods equal to each other:
\[ 2\pi \sqrt{\frac{l}{g'}} = 2\pi \sqrt{\frac{l}{g''}} \]
Since the length \( l \) is the same, we can simplify this to:
\[ \sqrt{\frac{1}{g'}} = \sqrt{\frac{1}{g''}} \]
Squaring both sides gives:
\[ \frac{1}{g'} = \frac{1}{g''} \]
This implies:
\[ g' = g'' \]

### Step 5: Substitute the expressions for \( g' \) and \( g'' \)
Substituting the expressions we derived for \( g' \) and \( g'' \):
\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

### Step 6: Cancel \( g \) from both sides
Assuming \( g \) is not zero, we can cancel \( g \) from both sides:
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

### Step 7: Simplify the equation
Now, simplifying the equation gives:
\[ -\frac{2h}{R} = -\frac{d}{R} \]
Multiplying through by \( -R \) results in:
\[ 2h = d \]

### Conclusion
Thus, the relationship between the height \( h \) of the mountain and the depth \( d \) of the mine is:
\[ d = 2h \]