Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 68a

Chapter 13, Problem 68a

A satellite in a circular orbit of radius r has period T. A satellite in a nearby orbit with radius r + Δr, where Δr ≪ r, has the very slightly different period T + ΔT. Show that ΔT/T = (3/2) (Δr/r)

Verified step by step guidance

Start by recalling Kepler's Third Law, which states that the square of the orbital period (T) of a satellite is proportional to the cube of the radius (r) of its orbit. Mathematically, this is expressed as:

T^2 \(\propto\) r^3

or equivalently

T^2 = k r^3

, where k is a constant of proportionality.

Differentiate both sides of the equation

T^2 = k r^3

with respect to r to find the relationship between changes in T and r. Using the chain rule, we get:

2T \(\frac{dT}{dr}\) = 3k r^2

Rearrange the differentiated equation to isolate

\(\frac{dT}{T}\)

\(\frac{dT}{T}\) = \(\frac{3}{2}\) \(\frac{dr}{r}\)

. Here,

\(\frac{dT}{T}\)

represents the fractional change in the period, and

\(\frac{dr}{r}\)

represents the fractional change in the radius.

Recognize that for small changes in the radius and period,

\(\Delta\) r

and

\(\Delta\) T

can be used in place of

dr

and

dT

, respectively. Thus, the equation becomes:

\(\frac{\Delta T}{T}\) = \(\frac{3}{2}\) \(\frac{\Delta r}{r}\)

Conclude that the fractional change in the orbital period is proportional to the fractional change in the orbital radius, with a proportionality constant of

\(\frac{3}{2}\)

. This completes the derivation:

\(\frac{\Delta T}{T}\) = \(\frac{3}{2}\) \(\frac{\Delta r}{r}\)

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Key Concepts

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

Kepler's Third Law

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. For circular orbits, this can be expressed as T² ∝ r³, which implies that as the radius increases, the period increases, but at a specific rate determined by the gravitational force acting on the satellite.

Differential Calculus

Differential calculus is a branch of mathematics that deals with the rates at which quantities change. In this context, it helps us understand how small changes in the radius (Δr) affect the orbital period (ΔT) of the satellite. By applying calculus, we can derive relationships between these small changes and express them in terms of ratios.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. It encompasses the principles governing satellite motion, including the relationship between radius, velocity, and period. Understanding these principles is crucial for analyzing how changes in orbital radius affect the period of a satellite's orbit.

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