Gravitational field intensity at the center of the earth

[I Like Pi]

Homework Statement

what is the gravitational field intensity at the center of the earth

Homework Equations

The Attempt at a Solution

i was told that it was zero because you have the Earth's mass surrounding it, and graivty moves to the center of its mass (center of radius) therefore the net force would be zero? is this correct?

also if you sub in 0 in the gravitational field intensity formula for the radius, you get a value over zero, meaning it is undefined... so i can't really use that to support why it is zero at the center...

thanks

 

[NobodySpecial]

i was told that it was zero because you have the Earth's mass surrounding it, and graivty moves to the center of its mass (center of radius) therefore the net force would be zero? is this correct?

Yes - in simple terms, ignoring the moon and tidal effects

also if you sub in 0 in the gravitational field intensity formula for the radius, you get a value over zero, meaning it is undefined

You also have zero mass inside that radius so it's zero/zero.
In fact you should be thinking about 'r' as the distance to each bit of mass above you - and since the Earth is all around you there is an equal force in every direction out from the centre - hence zero overall field

 

[D H]

i was told that it was zero because you have the Earth's mass surrounding it, and graivty moves to the center of its mass (center of radius) therefore the net force would be zero? is this correct?

You presumably are asking about gravitational acceleration, and yes, this is correct. A reasonably good model for the Earth is that it is a sphere whose density is a function of distance from the center of the Earth (and nothing else). With this model, the gravitation acceleration at some point inside the surface of the Earth depends only on that portion of the Earth that is closer to the center of the Earth than is the point in question. Imagine dividing the Earth into two parts, one part being the portion of the Earth closer to the center of the Earth than the point in question and the other part being the portion of the Earth from from the center of the Earth than the point in question. The gravitational acceleration at the point in question is the sum of the gravitational acceleration toward the inner part plus the gravitational acceleration toward the outer part. The contribution of the outer portion is identically zero because the gravitational acceleration anywhere inside a spherical shell of mass is identically zero. All that remains is the contribution from the inner portion,

$$a(r) = \frac{GM_{\text{inner}}}{r^2}$$

where M_inner is the mass of that inner portion.

also if you sub in 0 in the gravitational field intensity formula for the radius, you get a value over zero, meaning it is undefined... so i can't really use that to support why it is zero at the center...

That's because you are using Newton's law incorrectly. The correct usage is to only use the mass at or below the point in question.

If the density of the Earth is finite at every point inside the Earth, including at the center, then as r approaches zero, the inner mass will approach zero much faster than will r². That in turn means that a(r)→0 as r→0.

 

[I Like Pi]

Thank you very much! You have both helped! I really appreciate it!

 

[rwisz]

for any help

I can confirm that the gravitational field intensity at the center of the Earth is indeed zero. This is because at the center of the Earth, the gravitational forces from all directions cancel out, resulting in a net force of zero. This is in accordance with Newton's Law of Universal Gravitation, which states that the force of gravity between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them. At the center of the Earth, the distance to all points on the surface is equal, resulting in an equal but opposite force of gravity, canceling each other out. Furthermore, the Earth's mass surrounding the center also contributes to the cancellation of gravitational forces. Therefore, the gravitational field intensity at the center of the Earth is zero.