A 50 kg satellite circles the earth in an orbit with a period of 110 min. What minimum energy E is required to change the orbit to one with a period of 320 min? Assume both the initial and final orbits are circular. The Earth’s radius is 6.37 × 10^6 m, its mass is 5.98 × 10^24 kg.

To find the minimum energy required to change the orbit, we'll use the principle of conservation of energy.

We know that the gravitational potential energy (U) of an object in orbit is given by the equation U = -G * (m1 * m2) / r, where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between the centers of the objects.

Initially, the satellite is in a circular orbit with a period of 110 min. We can find the initial radius (r1) of this orbit using the equation for orbital period (T) T = 2 * π * sqrt(r^3 / G * M), where r is the radius of the orbit and M is the mass of the planet. Solving for r, we have r1 = (T^2 * G * M) / (4 * π^2).

Using the given values, we can calculate the initial radius r1:
r1 = (110^2 * 6.674 * 10^-11 * 5.98 * 10^24) / (4 * π^2) = 7.28 * 10^6 m.

Now, we want to change the orbit to one with a period of 320 min. Let's call the new radius r2. Using the same equation for orbital period, we can solve for r2:
r2 = (320^2 * 6.674 * 10^-11 * 5.98 * 10^24) / (4 * π^2) = 1.66 * 10^7 m.

The minimum energy required to change the orbit is equal to the difference in gravitational potential energy between the two orbits. So, ΔU = U2 - U1.

Calculating U1 and U2:
U1 = -G * (m1 * M) / r1 = -6.674 * 10^-11 * (50 * 5.98 * 10^24) / (7.28 * 10^6) = -3.47 * 10^9 J.
U2 = -G * (m1 * M) / r2 = -6.674 * 10^-11 * (50 * 5.98 * 10^24) / (1.66 * 10^7) = -1.25 * 10^10 J.

ΔU = U2 - U1 = -1.25 * 10^10 - (-3.47 * 10^9) = -8.05 * 10^9 J.

The minimum energy required to change the orbit to one with a period of 320 min is 8.05 * 10^9 J.