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 26

Chapter 13, Problem 26

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0 earth years. What are the asteroid's orbital radius and speed?

Verified step by step guidance

Use Kepler's Third Law, which states that the square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis (r) of its orbit:

T^{2} = \frac{r^{3}}{K}

, where K is a constant for objects orbiting the Sun. For simplicity, in astronomical units (AU) and Earth years, K = 1.

Rearrange Kepler's Third Law to solve for the orbital radius (r):

r = {\sqrt[2]{T}}^{3}

. Substitute T = 5.0 years into the equation.

Convert the orbital radius from astronomical units (AU) to meters if needed, using the conversion factor: 1 AU =

1.496 \times 10^{11}

meters.

To find the orbital speed (v), use the formula for circular orbital motion:

v = \frac{2}{π} \frac{r}{T}

. Ensure that the orbital radius (r) is in meters and the period (T) is in seconds (convert years to seconds using 1 year =

3.156 \times 10^{7}

seconds).

Substitute the values of r and T into the orbital speed formula to calculate the asteroid's speed. Ensure all units are consistent (meters for r and seconds for T).

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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 average distance from the sun. This relationship allows us to determine the orbital radius of the asteroid based on its period.

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 crucial for understanding how celestial bodies, like asteroids, maintain their orbits around the sun.

Orbital Speed

Orbital speed refers to the velocity at which an object travels along its orbit. It can be calculated using the formula v = 2πr/T, where v is the orbital speed, r is the orbital radius, and T is the orbital period. Understanding this concept is essential for determining how fast the asteroid moves in its orbit around the sun.

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