A body weights 63N on the surface of the earth. What is the gravitational force on it due to the erath at a height equal to half the radius of the earth?

Explanation

To find the gravitational force on a body at a height equal to half the radius of the Earth, we can follow these steps:

### Step 1: Understand the relationship between weight and gravitational force
The weight of the body on the surface of the Earth is given as 63 N. This weight is equal to the gravitational force acting on it at the surface, which can be expressed as:
\[ W = mg \]
where \( W \) is the weight, \( m \) is the mass of the body, and \( g \) is the acceleration due to gravity at the surface of the Earth.

### Step 2: Determine the gravitational acceleration at height
We need to find the gravitational acceleration at a height \( h \) which is equal to half the radius of the Earth. The formula for gravitational acceleration at a height \( h \) above the Earth's surface is:
\[ g' = g \cdot \frac{R^2}{(R + h)^2} \]
where \( g' \) is the acceleration due to gravity at height \( h \), \( R \) is the radius of the Earth, and \( g \) is the acceleration due to gravity at the surface.

### Step 3: Substitute the height into the formula
Given that \( h = \frac{R}{2} \), we can substitute this into the formula:
\[ g' = g \cdot \frac{R^2}{(R + \frac{R}{2})^2} \]
This simplifies to:
\[ g' = g \cdot \frac{R^2}{(\frac{3R}{2})^2} = g \cdot \frac{R^2}{\frac{9R^2}{4}} = g \cdot \frac{4}{9} \]

### Step 4: Calculate the new gravitational acceleration
Thus, we find:
\[ g' = \frac{4g}{9} \]

### Step 5: Calculate the mass of the body
From the weight on the surface, we can find the mass of the body:
\[ mg = 63 \, \text{N} \]
Thus, the mass \( m \) is:
\[ m = \frac{63}{g} \]

### Step 6: Calculate the weight at height \( h \)
The weight of the body at height \( h \) can be expressed as:
\[ W' = m \cdot g' \]
Substituting \( g' \):
\[ W' = m \cdot \frac{4g}{9} \]
Substituting \( m = \frac{63}{g} \):
\[ W' = \frac{63}{g} \cdot \frac{4g}{9} = \frac{63 \cdot 4}{9} \]

### Step 7: Simplify the expression
Calculating this gives:
\[ W' = \frac{252}{9} = 28 \, \text{N} \]

### Final Answer
The gravitational force on the body at a height equal to half the radius of the Earth is **28 N**.

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