Wouldn’t Newton’s second law mean the acceleration is indirectly proportional to the mass?
In: Physics
This comes back to the concept that if you drop objects of different masses in a gravitational field they will all fall at the same rate.
It might not seem intuitive that is how it works, because as you point out something with a greater mass will have a greater attraction under the force of gravity. That is why more massive things are heavier, right? But more massive things also have greater inertia or resistance to acceleration.
It turns out that the increased inertia exactly cancels out the increased attraction of gravity, so a feather and a dumbbell fall at the same speed in a vacuum.
The force is proportional to the mass. An object that weights 2kg will be subject to twice as much force as an object that weighs 1kg(assuming they’re the same distance from the Earth). But force is mass * acceleration. 1 newton is the force needed to accelerate 1 kg 1m/s^2. So even though it’s twice as much force, it needs to move twice as much mass, so it cancels out and ends up being the same acceleration.
>Wouldn’t Newton’s second law mean the acceleration is indirectly proportional to the mass?
Yes, acceleration would be inversely proportional to the mass if the force is constant.
In this case the force is not constant. It is gravity and according to Newton’s law of universal gravitation f= G m1 m2/r^2 where G is the universal constant, m1 is the mass of the object, m2 is the mass of earth and r is the distance to the center of earth.
So the force is directly proportional to the mass. The result is the acceleration from gravity is independent of mass. If you look at formaulas for acceleration the mass will be cancled out.
Because f = m a => a=f/m or in this case a= f/m1 we can create it as a= Gm2 /r^2 Close to the earth’s surface Gm2 /r^2 is equal to about 9.8m/s^2. This is freefall acceleration on earth, that is when gravity is the only force. On an inclined plane, only a percentage of that force will accelerate the object. The force will still be directly proportional to the mass.
There are two ways to think about how gravity affects objects; The Newtonian way and the Einsteinian way:
Newton would say what the other commenters all have already said: More massive objects experience a larger *Force* proportional to their mass, but the *acceleration* itself is force divided by the mass again, so it cancels out in the end. Basically, if you’re heavier, gravity pulls on you more, but also you need to be pulled on more for the same acceleration because you’re harder to move.
Einstein’s approach to gravity is very different: According to him, gravity is not really a force, instead space just accelerates downwards at a constant acceleration, and an object that isn’t accelerated by an actual force moves along with it, downwards. This also explains very well why you’re weightless when you’re falling; you’re not accelerating in space, so there are no net forces acting on you. Standing on the ground, it pushes up against you and accelerates you upwards, so by the principle of action and reaction, you feel a downwards force.
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