A landing craft with mass 1.22 ×10^4 kg is in a circular orbit 5.90 ×10^5 m above the surface

step 1 Here in this given problem mass of the landing craft that has been given as m is equal to 1 .22

A landing craft with mass 1.22 ×10^4 kg is in a circular...

A landing craft with mass $1.22 \times 10^{4} \mathrm{~kg}$ is in a circular orbit $5.90 \times 10^{5} \mathrm{~m}$ above the surface of a planet. The period of the orbit is $5800 \mathrm{~s}$. The astronauts in the lander measure the diameter of the planet to be $9.50 \times 10^{6} \mathrm{~m}$. The lander sets down at the north pole of the planet. What is the weight of an $85.0 \mathrm{~kg}$ astronaut as he steps out onto the planet's surface?

Key Concepts

Newton's Law of Universal Gravitation

This principle states that any two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This fundamental law is used to understand both how objects remain bound in orbit around a planet and how gravitational forces determine the weight of objects at a planet's surface.

Circular Orbit Dynamics

In a circular orbit, the gravitational force acts as the centripetal force that keeps a satellite moving along its path. The balance between gravitational pull and the tendency of a body to move in a straight line enables us to relate orbital parameters, such as period and radius, to the mass of the central body through equations derived from Kepler’s laws.

Orbital Period and the Gravitational Parameter

The orbital period of a satellite is directly related to the radius of its orbit and the gravitational parameter (GM) of the planet. By applying the formula T² = (4?²/GM)r³, one can determine GM, which is crucial for deducing the mass of the planet and subsequently its surface gravitational acceleration.

Weight and Gravitational Acceleration

Weight is the force experienced by an object due to gravity, calculated as the product of its mass and the gravitational acceleration at the surface of a celestial body. Gravitational acceleration at the surface is determined by the planet's mass and radius using the formula g = GM/R², linking orbital mechanics with the everyday measurement of weight.

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