Understanding Gravitational Acceleration at Different Depths on Earth

[Deebu R]

Homework Statement

If we take the gravitational acceleration at the Earth's surface as 10,/s^2 and the radius of the Earth as 6400km, decrease in the value of the gravitational acceleration g at a depth of 64 km from its surface would be ...,/s^2

2. The attempt at a solution
I was never really good in physics or maths so forgive me if this sound too stupid.
I tried using acceleration due to gravity at a distence d to the center,

Gd= Go [ 1-d/R], where Go is the acceleration due to gravity at the surface of earth.
But when I did this the solution became very messed up. Thank you very much for your help.

 

[Merlin3189]

Perhaps you could show your calculation. Your formula seems reasonable and when I substitute your values in, I get a simple answer without any complications.

 

[drvrm]

I was never really good in physics or maths so forgive me if this sound too stupid.
I tried using acceleration due to gravity at a distence d to the center,

Gd= Go [ 1-d/R], where Go is the acceleration due to gravity at the surface of earth.
But when I did this the solution became very messed up. Thank you very much for your help.

i think your equation is good -try to understand the relation...see
https://en.wikipedia.org/wiki/Gravity_of_Earth#Depth
as you move inside the Earth's surface g decreases from the surface value and is now proportional to (R-d) and then you can proceed further..
this decrease is due tl less mass being enclosed by the spherical volume of radius (R-d).

 

[Deebu R]

Perhaps you could show your calculation. Your formula seems reasonable and when I substitute your values in, I get a simple answer without any complications.

I got this,
Gd= 10 [1-64/6400]
Gd= 10 [99/100] = 9.9

 

[Merlin3189]

Your G₀ must be wrong, since we know it to be around 9.8 m/sec². This is probably what you spotted originally.

Looking at the values substituted in your formula, I can't see anything which could be the mass of the earth.
You seem to have put M=6.13 x 10^-4kg - a bit light for an earth!

Also I don't agree with your value for R = 6400 m. You are mixing your units. You were told R = 6400 km. You must convert everything to a consistent set of units. Here that needs to be metres to match the G constant. So R= 6.4 x 10⁶ m

I think the gravitational constant G=6.67 x 10^-11 Nm²/kg²
and Earth mass M=5.97 x 10²⁴ kg

Of course, once you have calculated g₀ your calculation for g_p is ok. But you started with a bad value for g₀

BUT you were told to use g₀ = 10 m/sec² so you just have to put this into your second equation and you'll get the right answer.

Well done for spotting that your G_p must be wrong with an acceleration of 10^-22 m/sec²