An earth satellite is moving round the earth in a circular orbit . For such a satellite which of the following statement is wrong ?

Options

A. It is a freely falling body
B. It is moving with a constant speed
C. It suffers no acceleration
D. Its angular momentum about the centre of the earth remains constant.

Answer

To determine which statement about an earth satellite moving in a circular orbit is wrong, we can analyze the properties of the satellite's motion. Let's break down the problem step by step.

### Step 1: Understand the nature of the satellite's motion
An earth satellite in a circular orbit is subject to gravitational force, which acts as the centripetal force required to keep it in orbit.

### Step 2: Analyze the gravitational force
The gravitational force acting on the satellite can be expressed using Newton's law of gravitation:
\[ F = \frac{G \cdot M \cdot m}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth to the satellite.

### Step 3: Relate gravitational force to centripetal force
For an object moving in a circular path, the centripetal force required is given by:
\[ F_c = \frac{m v^2}{r} \]
where \( v \) is the orbital speed of the satellite. For the satellite to maintain its orbit, the gravitational force must equal the centripetal force:
\[ \frac{G \cdot M \cdot m}{r^2} = \frac{m v^2}{r} \]

### Step 4: Derive the orbital speed
From the equation above, we can cancel \( m \) (mass of the satellite) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G \cdot M}{r^2} = \frac{v^2}{r} \]
Multiplying both sides by \( r \):
\[ \frac{G \cdot M}{r} = v^2 \]
Taking the square root gives us the orbital speed:
\[ v = \sqrt{\frac{G \cdot M}{r}} \]

### Step 5: Evaluate the statements
Now, let's evaluate the statements given in the question:

1. **The satellite is falling**: This is true. The satellite is in free fall, continuously falling towards the Earth but also moving forward, creating an orbit.

2. **The satellite is moving with constant speed**: This is true. The speed remains constant in a circular orbit, although the direction changes.

3. **There is no acceleration**: This is false. The satellite experiences centripetal acceleration directed towards the center of the Earth due to the change in direction of its velocity.

4. **Angular momentum about the center of the Earth remains constant**: This is true. Since there is no external torque acting on the satellite, its angular momentum remains constant.

### Conclusion
The statement that is wrong is: **"There is no acceleration."**