A body weighs 64 N on the surface of the Earth. What is the gravitational force on it (in N) due to the Earth at a height equal to one-third of the radius of the Earth?

Explanation

To solve the problem of finding the gravitational force on a body at a height equal to one-third of the radius of the Earth, we can follow these steps:

### Step 1: Understand the given information
- The weight of the body on the surface of the Earth is given as \( W = 64 \, \text{N} \).
- The height \( h \) at which we need to find the gravitational force is \( h = \frac{R}{3} \), where \( R \) is the radius of the Earth.

### Step 2: Use the formula for gravitational force at a height
The gravitational force \( F \) at a height \( h \) above the surface of the Earth can be calculated using the formula:
\[ F = \frac{G \cdot m \cdot M}{(R + h)^2} \]
where:
- \( G \) is the gravitational constant,
- \( m \) is the mass of the body,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 3: Relate weight and mass
The weight of the body at the surface of the Earth is given by:
\[ W = m \cdot g \]
where \( g \) is the acceleration due to gravity at the surface of the Earth. Thus, we can express the mass \( m \) as:
\[ m = \frac{W}{g} \]
Given that \( W = 64 \, \text{N} \), we can substitute this into our equation.

### Step 4: Substitute values into the gravitational force formula
At a height \( h = \frac{R}{3} \):
\[ F = \frac{G \cdot \left(\frac{W}{g}\right) \cdot M}{\left(R + \frac{R}{3}\right)^2} \]
This simplifies to:
\[ F = \frac{G \cdot \left(\frac{64}{g}\right) \cdot M}{\left(\frac{4R}{3}\right)^2} \]

### Step 5: Simplify the expression
The denominator becomes:
\[ \left(\frac{4R}{3}\right)^2 = \frac{16R^2}{9} \]
Thus, we can rewrite \( F \):
\[ F = \frac{G \cdot 64 \cdot M \cdot 9}{g \cdot 16R^2} \]

### Step 6: Use the relationship between \( g \), \( G \), and \( M \)
We know that:
\[ g = \frac{G \cdot M}{R^2} \]
Substituting this into our expression for \( F \):
\[ F = \frac{64 \cdot 9}{16} \cdot g \]

### Step 7: Calculate the final value
Now, we can simplify:
\[ F = 64 \cdot \frac{9}{16} = 64 \cdot 0.5625 = 36 \, \text{N} \]

### Final Answer
The gravitational force on the body at a height equal to one-third of the radius of the Earth is \( 36 \, \text{N} \).

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