What is the centripetal acceleration of a space vehicle moving round the earth, assuming its orbit to be circular ? What is the apparent weight of an astronaut ?

Answer

To solve the question step by step, we will break it down into two parts: calculating the centripetal acceleration of the space vehicle and determining the apparent weight of the astronaut.

### Step 1: Calculate the Centripetal Acceleration

1. **Definition of Centripetal Acceleration**:
Centripetal acceleration (\(a_c\)) is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle.

2. **Formula for Centripetal Acceleration**:
The formula for centripetal acceleration is given by:
\[ a_c = \frac{v^2}{R} \]
where \(v\) is the tangential velocity of the object and \(R\) is the radius of the circular path.

3. **Determine the Values**:
- The radius \(R\) of the orbit can be approximated as the distance from the center of the Earth to the space vehicle. If we assume the vehicle is in low Earth orbit, \(R\) is approximately \(6,371 \text{ km} + h\), where \(h\) is the altitude of the orbit.
- The tangential velocity \(v\) can be calculated using the formula \(v = \sqrt{\frac{GM}{R}}\), where \(G\) is the gravitational constant and \(M\) is the mass of the Earth.

4. **Substituting Values**:
After calculating \(v\), substitute it back into the centripetal acceleration formula to find \(a_c\).

### Step 2: Calculate the Apparent Weight of the Astronaut

1. **Understanding Apparent Weight**:
The apparent weight of an astronaut in a space vehicle is the normal force acting on them, which is influenced by the centripetal acceleration.

2. **Real Weight**:
The real weight \(W\) of the astronaut is given by:
\[ W = mg \]
where \(m\) is the mass of the astronaut and \(g\) is the acceleration due to gravity.

3. **Apparent Weight Formula**:
The apparent weight \(W'\) can be expressed as:
\[ W' = W - ma_c \]
where \(a_c\) is the centripetal acceleration.

4. **Considering Weightlessness in Orbit**:
In a circular orbit, the effective gravitational force experienced by the astronaut is zero (as they are in free fall). Hence, \(g\) can be considered negligible, leading to:
\[ W' = 0 - ma_c = -ma_c \]
This indicates that the astronaut feels weightless.

### Final Answers:
- The centripetal acceleration of the space vehicle can be calculated using the values of \(v\) and \(R\).
- The apparent weight of the astronaut is effectively zero due to the conditions of free fall in orbit.