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 27

Chapter 13, Problem 27

A satellite orbits the sun with a period of 1.0 day. What is the radius of its orbit?

Verified step by step guidance

Start by identifying the relevant formula for orbital motion. The period of an orbit is related to the radius of the orbit by Kepler's Third Law:

T^{2} = \frac{4}{π} \frac{r^{3}}{GM}

, where

T

is the orbital period,

r

is the orbital radius,

G

is the gravitational constant, and

M

is the mass of the Sun.

Rearrange the formula to solve for the orbital radius

r

. This gives:

r = {\frac{T^{2}}{4 π^{2} GM}}^{\frac{1}{3}}

Substitute the known values into the equation. The period

T

is 1.0 day, which must be converted to seconds:

T = 1.0 	imes 24 	imes 60 	imes 60

. The gravitational constant

G

is approximately

6.674 	imes 10^{-11}

N·m²/kg², and the mass of the Sun

M

is approximately

1.989 	imes 10^{30}

kg.

Perform the calculations step by step. First, calculate

T^{2}

. Then calculate the denominator

4 π^{2} GM

. Finally, divide

T^{2}

by the denominator and take the cube root to find

r

Express the final result for the orbital radius

r

in meters, ensuring the units are consistent throughout the calculation. This will give the radius of the satellite's orbit around the Sun.

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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. This law can be expressed mathematically as T² ∝ r³, where T is the orbital period and r is the radius of the orbit. This relationship allows us to determine the radius of an orbiting body when its period is known.

Gravitational Force

The gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. This force is what keeps satellites in orbit around larger bodies, such as the sun.

Circular Motion

In circular motion, an object moves along a circular path and experiences a centripetal force directed towards the center of the circle. For an object in orbit, this centripetal force is provided by gravity. The relationship between the orbital speed, radius, and period of the orbiting object is crucial for calculating the radius when the period is given.

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