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.
How far would you have to drill into the Earth in kilometers, to reach a point where your weight is reduced by 5.5% ? Approximate the Earth as a uniform sphere.
g= GMass/r^2= G*density*r
Are you doing any thinking on these?
To find the distance you would need to drill into the Earth to have a weight reduction of 5.5%, we can use the concept of the inverse square law of gravity.
The inverse square law states that the force of gravity between two objects is inversely proportional to the square of their distance. In this case, the force of gravity on a person inside the Earth will be reduced by 5.5% when compared to the force on the surface.
We know that weight is directly proportional to the force of gravity. So, if we can find the distance at which the force of gravity is reduced by 5.5%, we can use that distance to answer the question.
Let's denote the weight on the surface of the Earth as W1, and the weight at a certain depth as W2. We can set up the following equation:
W2 = W1 - (5.5% of W1)
Now, we need to find the distance from the center of the Earth where this weight reduction occurs. Since the Earth can be approximated as a uniform sphere, we can use the concept mentioned in the question that the force of gravity only depends on the mass closer to the center than that point.
To simplify the calculation, we can assume that the Earth has a uniform density. In this case, the force of gravity only depends on the mass enclosed within a given radius. So, at a certain depth, only the mass closer to the center than that depth contributes to the force of gravity on the person.
To find the distance, we need to use the Shell Theorem. According to the Shell Theorem, the gravitational force inside a uniformly distributed sphere depends only on the mass inside that sphere and not on the distribution outside.
To calculate the distance, we need to find the radius at which the weight reduction occurs. We can express this radius as a fraction of the Earth's radius (R):
r/R = (W2/W1)^(1/3)
where r is the distance from the center of the Earth at which weight is reduced by 5.5%.
Now we can calculate the value of r/R by plugging in the values:
r/R = (0.945)^(1/3)
Using a calculator, we can find that r/R is approximately 0.982.
Finally, we can calculate the actual distance by multiplying the radius of the Earth by r/R:
drill depth = R * (1 - r/R)
Given that the radius of the Earth is approximately 6,371 kilometers, we can now calculate the drill depth:
drill depth = 6,371 km * (1 - 0.982)
Therefore, the drill depth required to reach a point where your weight is reduced by 5.5% is approximately 120 kilometers.