How far would you have to drill into the Earth, to reach a point where your weight is reduced by 8.0 ? Approximate the Earth as a uniform sphere.
8.0 what? Percent? m/s^2?
I answered the same question yesterday for 5.5%. Use the same method.
http://www.jiskha.com/display.cgi?id=1271306511
See also my answer to the followup question of Mikhail.
whats the distance from the center of the earth?
To determine how far you would have to drill into the Earth to experience an 8.0% reduction in weight, we can use the concept of the gravitational field strength.
The formula for gravitational field strength is given by:
g = (G * M) / r^2
Where:
g is the gravitational field strength (referred to as weight per unit mass).
G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2 / kg^2).
M is the mass of the Earth.
r is the distance from the center of the Earth.
To achieve an 8.0% reduction in weight, we need to find the distance r2, where the gravitational field strength is 8.0% lower than the surface value (g0). This can be expressed as:
g2 = (0.92 * g0)
Now, we equate these two expressions and solve for r2:
(G * M) / r2^2 = (0.92 * G * M) / r1^2
Simplifying and rearranging the equation:
r2^2 = (0.92 / 1) * r1^2
r2 = sqrt((0.92 / 1) * r1^2)
r2 = 0.96 * r1
Therefore, you would have to drill approximately 4% of the Earth's radius deeper to experience an 8.0% reduction in weight.
To determine how far you would have to drill into the Earth to experience a specific reduction in weight, you need to understand the concept of gravitational force and how it changes as you move towards the center of the Earth.
The gravitational force acting on you is given by the equation:
F = (G * m * M) / r^2
Where:
F = gravitational force
G = gravitational constant (approximately 6.67 × 10^-11 Nm^2/kg^2)
m = mass of the object (your weight)
M = mass of the Earth
r = distance between the center of the Earth and your location
Since we are approximating the Earth as a uniform sphere, its mass remains constant as you drill deeper. Therefore, we can cancel out the mass of the Earth (M) from the equation.
The equation for weight is:
W = m * g
Where:
W = weight
g = acceleration due to gravity near the surface of the Earth (approximately 9.8 m/s^2)
Therefore, we can rewrite the gravitational force equation as:
F = G * (m * M) / r^2 = (m * g)
Now, let's rearrange the equation to solve for r:
r = √((G * M) / g)
To calculate the depth at which your weight is reduced by 8.0, we need to multiply your initial weight (W_initial) by 8.0 to get the reduced weight (W_reduced):
W_reduced = W_initial - 8.0
Now, plug in the values into the weight equation:
m * g = W_reduced
Solving for m:
m = W_reduced / g
Now we can substitute these values back into the rearranged equation to find the distance:
r = √((G * M) / (W_reduced / g))
By substituting the appropriate values for G, M, g, and W_reduced into this equation, you can calculate the approximate distance you would need to drill to experience a reduction in weight by 8.0.