At the bottom of a well one quarter of radius of the earth deep your weight compare to your normal weight would be
To determine your weight at the bottom of a well that is one quarter of Earth's radius deep, we need to consider the concept of gravitational force and its relationship to distance.
First, let's define some values:
- Let R be the radius of the Earth.
- Let d be the depth of the well, which is one quarter of R (d = R/4).
- Let g be the acceleration due to gravity at the Earth's surface.
The gravitational force acting on an object is given by the equation:
F = (G * m1 * m2) / r^2
where:
- F is the gravitational force between two objects,
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
- m1 and m2 are the masses of the two objects, and
- r is the distance between the centers of the two objects.
In our case, the two objects are you and the Earth. At the Earth's surface, your weight can be calculated using the equation:
Weight = m * g
where:
- m is your mass,
- g is the acceleration due to gravity at the Earth's surface.
Now, to calculate your weight at the bottom of the well (d = R/4), we need to consider the distance r between you and the center of the Earth. Given that the radius of the Earth (R) is the distance from the center to the surface, the distance r at the bottom of the well can be expressed as:
r = R - d
Substituting r = R - d into the gravitational force equation, we find that the force (F) at the bottom of the well is:
F = (G * m * M) / (R - d)^2
where M is the mass of the Earth. Assuming your mass (m) remains constant, your weight at the bottom of the well (W_bottom) can be calculated using the equation:
W_bottom = F / g
Taking the ratio of your weight at the bottom of the well (W_bottom) to your weight at the Earth's surface (W_surface), we get:
W_bottom / W_surface = (F / g) / (m * g) = F / (m * g^2)
Now, if we substitute the expression for F from earlier, we get:
W_bottom / W_surface = ((G * m * M) / (R - d)^2) / (m * g^2) = (G * M) / (R - d)^2g
Simplifying further, we can cancel out the mass (m) term and express the weight ratio as:
W_bottom / W_surface = (G * M) / (R - d)^2g
Since we are interested in comparing the weights, we can ignore the mass term (m). Therefore, the weight ratio at the bottom of the well compared to your normal weight is given by:
W_bottom / W_surface = (G * M) / (R - d)^2g
By plugging in the values for the gravitational constant (G), the mass of the Earth (M), the radius of the Earth (R), and the depth of the well (d = R/4), you can calculate the weight ratio.