What is the magnitude of the apparent weight

Question

of a 83kg astronaut 2500km from the center of the Earth s Moon in a space vehicle accelerating toward the Moon at 2.0m/s2 ?

Answer 1

Mass of astronaut m = 83 kg

Distance from the center of the Earth's Moon r = 2500 km

r = 2500000 m

Acceleration a = 2 m/s²

Apparent weight W_net = ?

Assuming that the Earth, astronaut, and Moon are momentarily all in perfect alignment.

Net force Fg = G[m1m2] / r^2

G = gravitational constant (6.673E-11)

m1 = mass of object one in kg (the Moon)

m2 = mass of object two in kg (the astronaut)

r = distance from the center-to-center of the two objects in meters

G = 6.67428×10^-11 m^3/kg-s^2

m1 = 83 kg

m2 = 7.3477×10²² kg

r = 2500000

Fg = (6.67428×10^-11)(83)(7.3477×10²²) / (2500000)²

Fg = 65.126 N

Gravitional acceleration due to moon g_moon =  Gm1/r²

g_moon =  (6.67428×10^-11)(7.3477×10²²) / (2500000)²

g_moon = 0.784 m/s²

The astronaut is accelerating towards moon so net acceleration g_r = 2 - 0.784 - 2 = 1.216 m/s²

Apparent weight W_net = Fg + mg_r

W_net = 65.126 + 83*1.216

W_net = 65.126 + 100.928

W_net = 166.054 N

The apparent weight is 166.054 N