What is the distance from the Earth's center to a point outside the Earth where the gravitational acceleration due to the Earth is 1/45 of its value at the Earth's surface?
g'/g = (r earth/r)^2
To find the distance from the Earth's center to a point outside the Earth where the gravitational acceleration due to the Earth is 1/45 of its value at the Earth's surface, we can use the concept of gravitational acceleration and the inverse square law.
The formula for gravitational acceleration is given by:
g = G * (M / r^2)
Where:
- g is the gravitational acceleration
- 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 the point
We are given that the gravitational acceleration at this point is 1/45th of its value at the Earth's surface. Therefore, we can set up the following equation:
g / (9.8 m/s^2) = 1/45
Simplifying the equation, we get:
g = (1/45) * 9.8 m/s^2
Now, we can substitute this value back into the original formula to solve for r:
(1/45) * 9.8 m/s^2 = G * (M / r^2)
Rearranging the equation, we have:
r^2 = (G * M) / ((1/45) * 9.8 m/s^2)
Simplifying further, we get:
r^2 = (45 * G * M) / 9.8 m/s^2
Finally, taking the square root of both sides, we can find the distance from the Earth's center to the desired point:
r = sqrt[(45 * G * M) / 9.8 m/s^2]
By substituting the values for M (approximately 5.972 × 10^24 kg) and G into the equation, you can calculate the distance from the Earth's center to the desired point.