Question

A landing craft with mass 1.22×104 kg is in a circular orbit a distance 5.70×105 m above the surface of a planet. The period of the orbit is 5600 s . The astronauts in the lander measure the diameter of the planet to be 9.80×106 m . The lander sets down at the north pole of the planet.

A. What is the weight w of an astronaut of mass 86.0 kg as he steps out onto the planet's surface?

Answer 1

Gravitational constant = G = 6.67 x 10^-11 N.m²/kg²

Mass of the planet = M₁

Mass of the landing craft = M₂ = 1.22 x 10⁴ kg

Orbital period of the landing craft = T = 5600 sec

Diameter of the planet = D_p = 9.8 x 10⁶ m

Radius of the planet = R_p = D_p/2 = (9.8x10⁶)/2 = 4.9 x 10⁶ m

Altitude of the landing craft above the surface = H = 5.7 x 10⁵ m

Radius of the orbit of the landing craft = R

R = R_p + H

R = 4.9x10⁶ + 5.7x10⁵

R = 5.47 x 10⁶ m

Orbital speed of the landing craft = V

VT = 2R

V(5600) = 2(5.47x10⁶)

V = 6137.325 m/s

The centripetal force for the circular motion of the landing craft is provided by the gravitational force of the planet on the landing craft.

M₁ = 3.089 x 10²⁴ kg

Mass of the astronaut = m = 86 kg

Weight of the astronaut on the planet's surface = W

The weight of the astronaut on the surface of the planet is equal to the gravitational force exerted by the planet on the astronaut.

W = 737.99 N

A) Weight of the astronaut on the planet's surface = 737.99 N