Calculating the Distance to Reach 7.5% Weight Reduction of a Uniform Sphere

[Charanjit]

1. Homework Statement :
It can be shown that for a uniform sphere the force of gravity at a point inside the sphere depends only on the mass closer to the center than that point. The net force of gravity due to points outside the radius of the point cancels.

Question: How far would you have to drill into the Earth, to reach a point where your weight is reduced by 7.5% ? Approximate the Earth as a uniform sphere.



2. Homework Equations :
F_g=(GMm)/r²



3. The Attempt at a Solution :

Well because the weight is reduced by 7.5 the inital weight has to be 0.925. This is a tough one. Thats why I need help. Get me started.

 

[rock.freak667]

You want the new weight to be 92.5% of the weight on the surface. Therefore you want


$$0.925 \frac{GMm}{R^2} = \frac{GMm}{r^2}$$

What should R be equal to in this case?

 

[Charanjit]

So is R the radius of the earth? So GMm cancel and all we are left with is 0.925, R2 and r. Solve for r?

 

[Charanjit]

I got it solved. Thanks.