Acceleration due to gravity

I thought the acceleration due to gravity was constant if the distance between two objects was constant, i.e. an elephant will experience the same acceleration as a feather if you ignore air resistance. However, since

F=G\frac{m_1m_2}{r^2}

, surely the force and hence the acceleration will be greater when

m_2

is increased? So would an elephant actually accelerate slightly more quickly than a feather (if you ignore air resistance)?

Reply 1

Stonebridge

Original post by Chlorophile

F=G\frac{m_1m_2}{r^2}

, surely the force and hence the acceleration will be greater when

m_2

is increased? So would an elephant actually accelerate slightly more quickly than a feather (if you ignore air resistance)?

Yes, the force on an elephant will be greater. But to accelerate it you need a bigger force. This results in the acceleration being the same.

The maths

\frac{Gm_1m_2}{r^2} = m_1g

m₁ is the elephant and g his acceleration due to gravity.

As you can see, m₁ cancels, meaning that g is the same for all masses. It just depends on m₂ (earth mass) and how far you are from the centre (r).

Reply 2

Chlorophile

Original post by Stonebridge

Yes, the force on an elephant will be greater. But to accelerate it you need a bigger force. This results in the acceleration being the same.

The maths

\frac{Gm_1m_2}{r^2} = m_1g

Ah okay. Thank you!

Reply 3

Chlorophile

Original post by Stonebridge

Yes, the force on an elephant will be greater. But to accelerate it you need a bigger force. This results in the acceleration being the same.

The maths

\frac{Gm_1m_2}{r^2} = m_1g

Sorry, this might be a really stupid question but couldn't you also say

\frac{Gm_1m_2}{r^2} = m_1g

where m₁ is the mass of the earth, so the earth accelerates towards the elephant at gms^-2? How does that make sense?

Reply 4

Stonebridge

Original post by Chlorophile

Sorry, this might be a really stupid question but couldn't you also say

\frac{Gm_1m_2}{r^2} = m_1g

where m₁ is the mass of the earth, so the earth accelerates towards the elephant at gms^-2? How does that make sense?

So what would r be in your equation?

The Earth does accelerate towards the other mass. In reality they both accelerate towards each other.

But the formula is applied in the case where the one mass is very much larger than the other. That's the case where you can meaningfully talk about accelerations due to gravity (g) on the surface of a planet. You assume the larger body doesn't move to simplify the maths.

Reply 5

Stanno

Original post by Chlorophile

Sorry, this might be a really stupid question but couldn't you also say

\frac{Gm_1m_2}{r^2} = m_1g

where m₁ is the mass of the earth, so the earth accelerates towards the elephant at gms^-2? How does that make sense?

Not at 'traditional' g, no.

The elephant is accelerated at g = a[elephant] = G.m[earth]/r^2.
The earth is accelerated at a[earth] = G.m[elephant]/r^2.

From this, we can gather that the Earth's acceleration due to that of the elephant's mass; a[earth] = g.m[elephant]/m[earth]

An elephant is roughly 7x10^3 kg, say, where as the Earth is 6.0x10^24 kg, so: a[earth] = 7/6 * 10^-21 * g, or 0.000000000000000000117% of 'traditional' g.

To put that in perspective, if we are to pretend that the only object on Earth is the elephant then it would take 4.14 * 10^10 seconds, or 1312 years for the Earth to move 1m due to the mass of the elephant.

EDIT: Whoops, made a mistake, *ahem*...

(edited 10 years ago)