The weight of a body at the surface of earth is 18 N. The weight of the body at an altitude of 3200 km above the earth `s (given,radius of earth `R_e` = 6400km)

Options

A. 19.6N
B. 9.8 N
C. 4.9N
D. 8 N

Answer

To solve the problem of finding the weight of a body at an altitude of 3200 km above the Earth's surface, we can follow these steps:

### Step 1: Understand the relationship between weight and gravitational force
The weight of a body is given by the formula:
\[ W = mg \]
where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.

### Step 2: Determine the effective gravitational acceleration at altitude
The gravitational acceleration at a distance \( r \) from the center of the Earth is given by:
\[ g' = \frac{g}{(1 + \frac{h}{R_e})^2} \]
where:
- \( g \) is the gravitational acceleration at the Earth's surface (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( h \) is the altitude above the Earth's surface (3200 km),
- \( R_e \) is the radius of the Earth (6400 km).

### Step 3: Substitute the values into the formula
We need to calculate the effective gravitational acceleration at an altitude of 3200 km:
- Convert \( h \) and \( R_e \) into the same units (kilometers):
\[ h = 3200 \, \text{km}, \quad R_e = 6400 \, \text{km} \]

Now, substitute these values into the formula:
\[ g' = \frac{g}{(1 + \frac{3200}{6400})^2} \]

### Step 4: Simplify the expression
Calculate the term inside the parentheses:
\[ 1 + \frac{3200}{6400} = 1 + 0.5 = 1.5 \]

Now, square this value:
\[ (1.5)^2 = 2.25 \]

So we have:
\[ g' = \frac{g}{2.25} \]

### Step 5: Calculate the weight at altitude
The weight at the surface of the Earth is given as 18 N. We can express the weight at altitude using the ratio of gravitational accelerations:
\[ \frac{W_a}{W_b} = \frac{g}{g'} \]

Substituting \( g' \):
\[ \frac{W_a}{W_b} = \frac{g}{\frac{g}{2.25}} = 2.25 \]

### Step 6: Solve for \( W_b \)
We know \( W_a = 18 \, \text{N} \):
\[ \frac{18}{W_b} = 2.25 \]

Rearranging gives:
\[ W_b = \frac{18}{2.25} \]

Calculating this:
\[ W_b = 8 \, \text{N} \]

### Final Answer
The weight of the body at an altitude of 3200 km above the Earth's surface is **8 N**.

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