0

If we have a stone of 2 lb that falls vertically we have $F_a = m \times g = 2 \times 32 = 64 $ poundals
If we have a stone of 4lb that falls vertically we have $F_b = m \times g = 4 \times 32 = 128 $ poundals
So $F_a \gt F_b$
I was also reading about Newton's universal law of gravitation. If I understand correctly the force of attraction of a stone and the force of attraction of a moon by the earth is the same i.e. $F = \frac{G \times E}{r^2}$ where $E$ is the earth's mass and $r$ is either the distance of the moon to the earth's center or the distance of the stone to the earth's center.
I am confused how these 2 formulas combine.
I assume that the $F_a = m \times g = 2 \times 32 = 64 $ of the stone is caused by earth's attractional force $\frac{G \times E}{r^2}$ (or at least somehow related). But that force is the same for both stone's of different size at the same place. But we know the $F_b > F_a$
So how is the second law of motion and acceleration and universal gravitation relate in the formulas?
I guess that the attractional force is the same and the difference is the larger weight has larger acceleration. Is that right?

smith
  • 181

1 Answers1

3

You are confusing the concept of gravitational force and the accelration due to gravity.

For example, the gravitational force on an object of mass $m_a$ is given by $$F_a={GM_em_a\over r^2},$$ similarly the gravitational force on another object of differing mass $m_b$ is: $$F_b={GM_em_b\over r^2}.$$ Clearly then we have that $F_a\neq F_b$.

Now from Newton's second law we have that given a certain force, one can find the corresponding acceleration: $$F=ma\implies\;\;a={F\over m}.$$ So that if we were to compute the acceleration of the two earlier contemplated masses $m_a$ and $m_b$ we get from the second law: $$a_a={GM_em_a\over m_ar^2}={GM_e\over r^2},\;\text{and}\;a_b={GM_em_b\over m_br^2}={GM_e\over r^2}.$$ Clearly these quantities are the same even for different masses, i.e. this is the acceleration due to gravity: $$g={GM_e\over r^2}.$$ Thus, each massive object has a distinct weight or gravitational force of attraction to the earth despite the fact that each is accelerated towards the earth at the same rate, i.e. $a_a=a_b=g$.

This explains why two objects each of different weight, if dropped from the same height, each reach the Earth at the same time.

Albertus Magnus
  • 5,652