This might be stupid but I need an answer.
Newton's second law stated that the force exerted on a body equals the mass of the body times the acceleration of the body. And the law of gravitation states that:
$$F = G\frac{m_1m_2}{r^2}$$
My question is: if I take two identical masses and place one higher than the other then one experiences greater gravitational force than the other. Both accelerate at the same rate $g$. If this is so, according to Newton's second law the forces on the bodies must be equal.
Where am I wrong?
Best Answer
The answer is that falling objects do not all accelerate towards the Earth at the same rate of $9.8 \text{ m/s}^2$.
All objects, at the surface of the Earth, accelerate the same, regardless of their mass. Also, all objects at the same distance from the center of the Earth accelerate at the same rate. But objects at different heights do not accelerate exactly the same. Otherwise how could you ever escape the Earth's gravity?
The more correct way to calculate the acceleration is to do it the way you have done, using Newton's 2nd Law and Newton's Law of Universal Gravitation.
Physics teachers often teach their classes that all object's accelerate at the same rate and then don't emphasize the limits on that statement.