How much below the surface of the earth does the acceleration due to gravity (i) reduced to `36%` (ii) reduces by `36% ` , of its value on the surface of the earth ? Radius of the earth = 6400 km .

Options

A. `(R)/(4)`
B. `(R)/(2)`
C. `(16R)/(25)`
D. 4 R

Answer

To solve the problem of how much below the surface of the Earth the acceleration due to gravity (g) reduces to 36% of its value at the surface, we can follow these steps:

### Step 1: Understand the relationship between g at the surface and g at depth
The acceleration due to gravity at a depth \( d \) below the surface of the Earth can be expressed as:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
where:
- \( g' \) is the acceleration due to gravity at depth \( d \),
- \( g \) is the acceleration due to gravity at the surface (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( R \) is the radius of the Earth (given as \( 6400 \, \text{km} \)).

### Step 2: Set up the equation for 36% of g
We need to find \( d \) when \( g' = 0.36g \). Substituting this into the equation gives:
\[ 0.36g = g \left(1 - \frac{d}{R}\right) \]

### Step 3: Simplify the equation
Dividing both sides by \( g \) (assuming \( g \neq 0 \)):
\[ 0.36 = 1 - \frac{d}{R} \]

### Step 4: Solve for \( \frac{d}{R} \)
Rearranging the equation:
\[ \frac{d}{R} = 1 - 0.36 = 0.64 \]

### Step 5: Calculate the depth \( d \)
Now, we can find \( d \):
\[ d = 0.64R \]
Substituting \( R = 6400 \, \text{km} \):
\[ d = 0.64 \times 6400 \, \text{km} = 4096 \, \text{km} \]

### Final Answer for Part (i)
The depth at which the acceleration due to gravity reduces to 36% of its value at the surface is \( 4096 \, \text{km} \).

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### Part (ii): How much below the surface does g reduce by 36%?

### Step 1: Set up the equation for a reduction of 36%
For the second part, we need to find \( d \) when \( g' = g - 0.36g = 0.64g \). Using the same formula:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
we set up the equation:
\[ 0.64g = g \left(1 - \frac{d}{R}\right) \]

### Step 2: Simplify the equation
Dividing both sides by \( g \):
\[ 0.64 = 1 - \frac{d}{R} \]

### Step 3: Solve for \( \frac{d}{R} \)
Rearranging gives:
\[ \frac{d}{R} = 1 - 0.64 = 0.36 \]

### Step 4: Calculate the depth \( d \)
Now, we can find \( d \):
\[ d = 0.36R \]
Substituting \( R = 6400 \, \text{km} \):
\[ d = 0.36 \times 6400 \, \text{km} = 2304 \, \text{km} \]

### Final Answer for Part (ii)
The depth at which the acceleration due to gravity reduces by 36% of its value at the surface is \( 2304 \, \text{km} \).

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