Acceleration due to gravity Does acceleration due to gravity increase when an object at a said rad

Azzalictpdv

Azzalictpdv

Answered question

2022-05-08

Acceleration due to gravity
Does acceleration due to gravity increase when an object at a said radius is doubled? Like for instance, if I said the radius of the earth is 5 k m (not to scale obviously) if I doubled that, so 10 k m will acceleration due to gravity double? Or maybe g / 2? Or would the acceleration stay the same no matter distance?

Answer & Explanation

priffEmipsy4i37m

priffEmipsy4i37m

Beginner2022-05-09Added 17 answers

The force on a particle of mass m on the surface of a spherically symmetric mass distribution (the Earth) is
F = G M m R 2 ,
where G is a constant, M is the body mass and R is the radius of the distribution. By Newton's second law, that force equals m a, where a is the acceleration of particle m. As you can see, this acceleration, which we call gravity acceleration, reads
a = G M R 2 .
If you double the radius (keeping the mass M fixed), acceleration is decreased by a factor of four.
Matilda Webb

Matilda Webb

Beginner2022-05-10Added 3 answers

This depends on exactly what you mean by this.
Acceleration due to gravity is given by :
g = G M r 2
And note that G is a universal constant, M is the mass doing the attracting and r is the distance from the object's center of mass.
Let's assume you simply move twice as far away as r, then the gravitational acceleration you feel will be quartered - that power of two !
But if you made the Earth twice and big kept it's density the same - so you filled the extra space, what would happen ?
Well :
M = 4 3 π r 3 ρ
Where ρ is average density.
That makes :
g = 4 3 G π ρ r
So in this case ( doubling the radius and filling the space ), the gravitational acceleration at the new radius would be twice as large - a completely different result from the first interpretation of your question.
So, as always in physics, you need to be very precise in describing what you mean.

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