That's an interesting point you've brought up regarding the motion of satellites and the concept of non-inertial frames of reference. Let's break this down step by step to clarify how we derive the formula for the orbital speed of a satellite and where pseudo forces come into play.
The Basics of Orbital Motion
First, it's essential to understand that when we talk about a satellite orbiting the Earth, we are considering the gravitational interaction between the Earth and the satellite. The satellite moves in a circular path due to the gravitational force acting as the centripetal force required for circular motion.
Understanding Frames of Reference
In physics, a frame of reference is a perspective from which we observe and measure phenomena. The Earth is indeed a non-inertial frame of reference because it is rotating and also moving in its orbit around the Sun. However, when we analyze the motion of a satellite, we often simplify our calculations by considering the Earth as an inertial frame for the following reasons:
- Gravitational Force Dominance: The gravitational force acting on the satellite is significantly stronger than any pseudo forces that might arise from the Earth's rotation.
- Relative Motion: From the satellite's perspective, it is in free fall, and the only force acting on it is gravity, which simplifies our calculations.
Deriving the Orbital Speed Formula
To derive the formula for the orbital speed of a satellite, we start with the balance of forces. The gravitational force provides the necessary centripetal force for the satellite's circular motion. The gravitational force \( F \) acting on the satellite can be expressed as:
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.
- r is the distance from the center of the Earth to the satellite.
The centripetal force required to keep the satellite in circular motion is given by:
F = \frac{m \cdot v^2}{r}
Where \( v \) is the orbital speed of the satellite. Setting these two expressions for force equal to each other gives us:
\frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r}
We can simplify this equation by canceling out the mass of the satellite \( m \) (assuming \( m \neq 0 \)) and rearranging it to solve for \( v \):
v^2 = \frac{G \cdot M}{r}
Taking the square root of both sides, we find the formula for the orbital speed:
v = \sqrt{\frac{G \cdot M}{r}}
Considering Pseudo Forces
Now, regarding your point about applying a pseudo force due to the Earth's rotation, while it is true that in a rotating frame, we would consider such forces, they do not significantly affect the satellite's motion in this context. The gravitational force is much more substantial than any pseudo force that might arise from the Earth's rotation. Therefore, for practical purposes, we can ignore the pseudo force when deriving the orbital speed of a satellite.
In summary, while the Earth is a non-inertial frame, the gravitational forces at play dominate the dynamics of satellite motion, allowing us to treat the situation as if we were in an inertial frame for the sake of simplicity and accuracy in our calculations.
