At what altitude above the Earth's surface is the gravitational field strength equal to g/7?
.981 = -(1.0003322(6,371/6,371+h))^2)
To find the altitude above the Earth's surface at which the gravitational field strength is equal to g/7 (where g = 9.81 m/s^2), we can use the formula for gravitational field strength and solve for the altitude.
The formula for gravitational field strength is given by:
g = G * (M / r^2),
where g is the gravitational field strength, G is the gravitational constant (approximately 6.67430 x 10^-11 m^3/(kg*s^2)), M is the mass of the Earth (approximately 5.972 x 10^24 kg), and r is the distance from the center of the Earth.
By rearranging the formula, we have:
r^2 = G * M / g.
To find the altitude above the Earth's surface, we need to calculate the difference between the distance from the center of the Earth to the surface (which is the radius of the Earth, r_earth) and the distance from the center of the Earth to the desired altitude (r_altitude). Therefore, we can write:
(r_earth + r_altitude)^2 = G * M / (g/7).
Next, we solve for r_altitude:
r_altitude^2 + 2 * r_earth * r_altitude + r_earth^2 = G * M / (g/7).
Substituting the values, we have:
(6,371,000 + r_altitude)^2 = (6.67430 x 10^-11) * (5.972 x 10^24) / (9.81/7).
Simplifying the equation:
(6,371,000 + r_altitude)^2 = 10.6736 x 10^24 m^3 / kg * r_altitude.
Now, let's solve the equation by finding the value of r_altitude.
Note: Since the equation involves squaring on both sides, there may be two possible solutions. We will consider the positive solution, as the negative solution does not make physical sense in this context.
(6,371,000 + r_altitude)^2 = 10.6736 x 10^24 m^3 / kg * r_altitude.
Simplifying the equation further, we have:
r_altitude^2 + 2 * 6371000 * r_altitude + (6371000)^2 = (10.6736 x 10^24 m^3 / kg) * r_altitude.
r_altitude^2 + 2 * 6371000 * r_altitude + (6371000)^2 - (10.6736 x 10^24 m^3 / kg) * r_altitude = 0.
Now, we have a quadratic equation that can be solved to find the value of r_altitude.