# Question related to g

Gold Member
We know that the force of gravitation is F=GMm/r^2. The acceleration of the body of any mass m is a=GM/r^2 which we call g. So same acceleration regardless of any weight(ignoring air resistance). But when we solve laws of motion problems we take a downward force of -mg. So force depend on mass. Heavier body has more pull towards the earth. So shouldn’t it move faster than the lighter body?

weirdoguy
So force depend on mass.
There is also m in Newton's second law, so it disappears.

Hamiltonian
Gold Member
There is also m in Newton's second law, so it disappears.
So when we do problems related to tension or inclined plane friction problems, we take -mg. The force is always mg. And sometimes we need to find acceleration. But that never is equal to g. Is it because the body is not under free fall and the acceleration will not be g. Only when body is free to move the acceleration will always be g and bodies of any weight will come together.

Lnewqban
weirdoguy
Is it because the body is not under free fall and the acceleration will not be g.

Yes.

Gold Member
Yes.
It’s weird that when in free fall the acceleration doesn’t depend on force but when we take system of bodies like a pulley and attached masses suddenly the acceleration does depend on net force.

weirdoguy
the rotation of the Earth is also included into the vector constant ##\boldsymbol g##

Mentor
It’s weird that when in free fall the acceleration doesn’t depend on force ….
but it does. The gravitational force on the more massive object is greater and that’s how we can get it to accelerate at the same rate as a less massive object subjected to a smaller force.

russ_watters
Gold Member
but it does. The gravitational force on the more massive object is greater and that’s how we can get it to accelerate at the same rate as a less massive object subjected to a smaller force.
Ok. To produce the same acceleration to a more massive body we need more force. Thats Newtons second law F=ma. Gravitational force is F=GMm/r^2 but then the mass is also more for more massive bodies so a=F/m, a=g. No matter the weight all bodies does same acceleration. But when we arrange a pulley and two masses system force on each mass changes. Force on more massive body becomes less than mg so a<g. even though acceleration is downwards but its not g. For lighter body the force of tension is much more than mg. So its direction of acceleration changes. Its not g again.
I made a mistake. Force and acceleration are always connected. Its the g that doesn't depend on mass.

Gold Member
Take a deeper look at the definition of force.
$$F=\frac{d(mv)}{dt}$$
The force is the variation of momentum. Momentum could be seen as the amount of resistance to change in motion, i.e. the principle of inertia. In the case of planets interacting together, we assume mass doesn't vary such that:
$$F=m\frac{dv}{dt}$$
Or:
$$m\frac{GM}{r^2}=m\frac{dv}{dt}$$
So what does it mean? The force ##F## that wants to change the velocity and the inertia that wants to resist that change are both proportional to the mass ##m##. The effect of the mass becomes then effectively irrelevant.

But imagine the mass ##m## would change as time goes on, then:
$$m\frac{GM}{r^2}=\frac{d(mv)}{dt}$$
$$m\frac{GM}{r^2}=m\frac{dv}{dt} + v\frac{dm}{dt}$$
$$\frac{dv}{dt}= \frac{GM}{r^2} - \frac{v}{m}\frac{dm}{dt}$$
Not only the mass is relevant to determine the acceleration, but also the velocity and the rate of change of that mass.

Lnewqban
Homework Helper
Gold Member
It’s weird that when in free fall the acceleration doesn’t depend on force but when we take system of bodies like a pulley and attached masses suddenly the acceleration does depend on net force.

All bodies are weightless while in free fall.