Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 50

Chapter 13, Problem 50

A 4000 kg lunar lander is in orbit 50 km above the surface of the moon. It needs to move out to a 300-km-high orbit in order to link up with the mother ship that will take the astronauts home. How much work must the thrusters do?

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Determine the gravitational potential energy of the lunar lander in its initial orbit (50 km above the moon's surface). Use the formula for gravitational potential energy:

U = - \frac{G}{M} r

, where

G

is the gravitational constant,

M

is the mass of the moon,

m

is the mass of the lander, and

r

is the distance from the center of the moon to the lander (sum of the moon's radius and the altitude of the orbit).

Calculate the gravitational potential energy of the lunar lander in its final orbit (300 km above the moon's surface) using the same formula as in step 1, but with the new distance

r

(moon’s radius + 300 km).

Find the change in gravitational potential energy by subtracting the initial potential energy from the final potential energy:

ΔU = U

_final - U_initial. This represents the work required to move the lander to the higher orbit.

Account for the kinetic energy difference between the two orbits. The total energy in an orbit is the sum of kinetic and potential energy, and for a circular orbit, the total energy is given by

E = - \frac{G}{M} 2 r

. Calculate the total energy for both the initial and final orbits.

Determine the total work done by the thrusters by finding the difference in total energy between the final and initial orbits:

W = E

_final - E_initial. This value represents the work required to move the lander to the higher orbit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gravitational Potential Energy

Gravitational potential energy (U) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula U = -G(m1*m2)/r, where G is the gravitational constant, m1 and m2 are the masses involved, and r is the distance from the center of the mass creating the gravitational field. In this scenario, understanding how potential energy changes with altitude is crucial for calculating the work done by the thrusters.

Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In the context of the lunar lander, the work done by the thrusters must overcome the change in gravitational potential energy as the lander moves from a lower orbit to a higher one. This principle helps in quantifying the energy required for the lander to achieve its new orbit.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. It involves understanding how altitude affects gravitational force and orbital speed. For the lunar lander, transitioning from a 50 km orbit to a 300 km orbit requires knowledge of how these factors interact, which is essential for calculating the necessary work to achieve the desired altitude.

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