An experimental spacecraft is found to have weight W when assembled before launching from a rocket site. It is placed in a circular orbit at a height h = 6R above the surface of the earth (of radius R). The gravitational force acting on the spacecraft in this orbit is ________.

Options

A. W/6
B. W/7
C. W/36
D. W/49

Answer

To find the gravitational force acting on the spacecraft in a circular orbit at a height \( h = 6R \) above the surface of the Earth, we can follow these steps:

### Step 1: Understand the Weight on the Surface
When the spacecraft is on the surface of the Earth, its weight \( W \) is given by the formula:
\[ W = mg \]
where \( m \) is the mass of the spacecraft and \( g \) is the acceleration due to gravity at the Earth's surface.

### Step 2: Determine the Height Above the Earth's Surface
The spacecraft is in a circular orbit at a height \( h = 6R \) above the surface of the Earth. The radius of the Earth is \( R \), so the total distance from the center of the Earth to the spacecraft in orbit is:
\[ d = R + h = R + 6R = 7R \]

### Step 3: Calculate the Gravitational Acceleration at Height \( h \)
The gravitational acceleration \( g' \) at a distance \( d \) from the center of the Earth is given by:
\[ g' = \frac{GM}{d^2} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth. Substituting \( d = 7R \):
\[ g' = \frac{GM}{(7R)^2} = \frac{GM}{49R^2} = \frac{g}{49} \]
Here, \( g \) is the acceleration due to gravity at the Earth's surface.

### Step 4: Calculate the Weight in Orbit
Now, the weight of the spacecraft in orbit \( W' \) can be calculated using the new gravitational acceleration \( g' \):
\[ W' = mg' = m \left(\frac{g}{49}\right) = \frac{mg}{49} = \frac{W}{49} \]
where we have replaced \( mg \) with \( W \) since \( W \) is the weight of the spacecraft at the Earth's surface.

### Conclusion
Thus, the gravitational force acting on the spacecraft in this orbit is:
\[ W' = \frac{W}{49} \]

### Final Answer
The gravitational force acting on the spacecraft in this orbit is \( \frac{W}{49} \).

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