I have already solve the problems in many time, but still cannot get the correct answer, can someone kindly help me to solve the problem, please?
1. Suppose the Moon is at a (center-to-center) distance D from Earth. Find an exact expression for the magnitude of the tidal force applied by Earth to our rock of mass m in terms of D, m, the mass of Earth M_e, and the lunar radius R_m
2. Assuming that D is much greater than R_m find an approximate expression for the same tidal force, using Newton's approximation.
3. Set your answer in 2 equal to your answer in F = G*m*(M_m)/(R_m)^2 to find a value for D for which tidal forces can lift rocks off the Moon, in terms of Rm and the masses of Earth (M_e) and Moon (M_m).
Evaluate the expression you find, expressing the answer in km, using R_m=1737.5km and M_m=0.0123M_e.
Only these three problem, please help, thank you
Certainly! Let's break down each problem step by step:
1. To find the exact expression for the magnitude of the tidal force applied by Earth to our rock, we need to understand the concept of tidal forces and use Newton's law of universal gravitation.
Tidal forces arise because the gravitational force from a celestial body (in this case, Earth) on an extended object (the rock) is not uniform across the object due to the difference in distances.
The magnitude of the tidal force can be computed using the differential force equation:
F_tidal = dF = G * (M_e * dm) / (D - R_m)^2 - G * (M_e * dm) / (D + R_m)^2
where G is the gravitational constant, M_e is the mass of Earth, dm is the mass of the rock, D is the distance between Earth and the Moon (center-to-center), and R_m is the lunar radius.
2. Assuming that D is much greater than R_m, we can use Newton's approximation to simplify the tidal force expression. In this approximation, we neglect the difference between (D + R_m) and D in the denominator:
F_tidal ≈ 2G * M_e * dm * R_m / D^3
This approximation gets rid of the squared terms in the denominators.
3. Now, let's set the approximate tidal force expression equal to the gravitational force expression:
2G * M_e * dm * R_m / D^3 = G * m * M_m / R_m^2
Here, m is the mass of the rock, M_m is the mass of the Moon, and R_m^2 is the square of the lunar radius.
Rearranging this equation to solve for D:
D^3 = (2 * M_e * R_m^3) / (m * M_m)
Taking the cube root of both sides:
D = ((2 * M_e * R_m^3) / (m * M_m))^(1/3)
4. Now, let's evaluate the expression for D, using the given values of R_m = 1737.5 km and M_m = 0.0123 * M_e.
Substituting these values into the expression, we get:
D = ((2 * M_e * (1737.5 km)^3) / (m * 0.0123 * M_e))^(1/3)
D = (2 * (1737.5 km)^3 / (m * 0.0123))^(1/3)
Now you can calculate the value of D by plugging in the appropriate values for m and M_e.
I hope this explanation helps! If you have any further questions, feel free to ask.