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 48b

Chapter 13, Problem 48b

The two stars in a binary star system have masses 2.0 x 10³⁰ kg and 6.0 x 10³⁰ kg. They are separated by 2.0 x 10¹² m. What are The speed of each star?

Verified step by step guidance

Step 1: Understand the problem. In a binary star system, the two stars orbit their common center of mass due to gravitational attraction. To find the speed of each star, we need to use the concept of circular motion and the gravitational force acting between the two stars.

Step 2: Calculate the center of mass of the system. The center of mass is given by the formula:

x = m

₁x₁+m₂x₂m₁+m₂. Here,

m

₁ and

m

₂ are the masses of the stars, and

x

₁ and

x

₂ are their positions. Use this formula to find the distance of each star from the center of mass.

Step 3: Apply Newton's law of gravitation to find the gravitational force between the two stars. The formula is:

F = G m

₁m₂r2, where

G

is the gravitational constant,

r

is the separation between the stars, and

F

is the gravitational force.

Step 4: Use the concept of circular motion to relate the gravitational force to the centripetal force for each star. The centripetal force is given by:

F = m v

²r, where

v

is the speed of the star and

r

is its distance from the center of mass. Equate the gravitational force to the centripetal force for each star to solve for their speeds.

Step 5: Solve the equations for the speed of each star. For the first star, substitute its mass and distance from the center of mass into the equation. Repeat the process for the second star. Ensure that the speeds are consistent with the orbital motion around the center of mass.

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Key Concepts

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

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every mass attracts every other mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This law is fundamental in understanding the gravitational forces acting between the two stars in a binary system.

Circular Motion and Centripetal Force

In a binary star system, each star orbits around a common center of mass, and their motion can be described as circular motion. The centripetal force required to keep each star in orbit is provided by the gravitational attraction between them. Understanding the relationship between gravitational force and centripetal force is essential for calculating the speeds of the stars.

Center of Mass

The center of mass of a binary star system is the point where the two stars balance each other due to their masses. The distance of each star from the center of mass is inversely proportional to its mass, which affects their orbital speeds. Knowing the position of the center of mass is crucial for determining how fast each star moves in relation to this point.

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