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

In summary, for a uniform sphere, the force of gravity at a point inside the sphere only depends on the mass closer to the center than that point. This means that the net force of gravity due to points outside the radius of the point cancels. To determine how far one would have to drill into the Earth to reach a point where their weight is reduced by 7.5%, the formula Fg=(GMm)/r2 can be used. Setting the new weight to be 92.5% of the weight on the surface, the equation becomes 0.925(GMm)/R^2=(GMm)/r^2. Solving for r yields the answer.
  • #1
Charanjit
48
0
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 :
Fg=(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.
 
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  • #2
You want the new weight to be 92.5% of the weight on the surface. Therefore you want[tex]0.925 \frac{GMm}{R^2} = \frac{GMm}{r^2}[/tex]

What should R be equal to in this case?
 
  • #3
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?
 
  • #4
I got it solved. Thanks.
 

1. What is the formula for calculating the distance to reach 7.5% weight reduction of a uniform sphere?

The formula for calculating the distance to reach 7.5% weight reduction of a uniform sphere is: d = (V * (1-0.075))^(1/3) - r, where d is the distance from the center of the sphere to the outer surface, V is the initial volume of the sphere, and r is the initial radius of the sphere.

2. How do you determine the initial volume of the sphere?

The initial volume of the sphere can be determined by using the formula V = (4/3) * π * r^3, where V is the volume and r is the initial radius of the sphere.

3. What is the significance of 7.5% weight reduction in this calculation?

7.5% weight reduction is a commonly used benchmark in scientific studies, as it is considered a significant amount of weight loss. It is also a feasible and achievable goal for weight loss in many cases.

4. Can this formula be applied to non-uniform spheres?

No, this formula is specifically designed for calculating the distance to reach 7.5% weight reduction of a uniform sphere. For non-uniform spheres, a different formula would need to be used.

5. Are there any limitations to using this formula?

Yes, this formula assumes that the weight reduction is evenly distributed throughout the entire sphere. It also does not take into account any external factors that may affect the weight reduction process, such as changes in temperature or pressure. Additionally, this formula only applies to weight reduction and may not accurately calculate the distance for weight gain or other types of changes in the sphere's volume.

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