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Ch 13: Gravitation
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 13: GravitationProblem 44a
Chapter 13, Problem 44a

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. How far are these clumps from the center of the black hole?

Verified step by step guidance
1
Start by identifying the key information given in the problem: the orbital period of the clumps is 27 hours, and their orbital speed is 30,000 km/s.
Convert the orbital period from hours to seconds to ensure consistency in units. Since 1 hour is 3600 seconds, multiply 27 hours by 3600 seconds/hour to get the period in seconds.
Use the formula for the orbital velocity of an object in circular motion: \( v = \frac{2\pi r}{T} \), where \( v \) is the orbital speed, \( r \) is the radius of the orbit, and \( T \) is the orbital period.
Rearrange the formula to solve for the radius \( r \): \( r = \frac{v \cdot T}{2\pi} \). Substitute the values for \( v \) and \( T \) into the equation.
Calculate the radius \( r \) using the rearranged formula. This will give you the distance of the clumps from the center of the black hole.

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Key Concepts

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

Gravitational Force

Gravitational force is the attractive force between two masses, such as a black hole and orbiting matter. It is described by Newton's law of universal gravitation, which states that the force is proportional to the product of the masses and inversely proportional to the square of the distance between them. This concept is crucial for understanding the dynamics of objects orbiting a black hole.
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Orbital Mechanics

Orbital mechanics involves the study of the motion of objects in space under the influence of gravitational forces. Kepler's laws of planetary motion and Newton's laws of motion are fundamental to calculating orbital parameters, such as the distance of clumps of matter from a black hole, based on their orbital period and velocity.
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Relativistic Speeds

Relativistic speeds refer to velocities that are a significant fraction of the speed of light, affecting the physics of objects moving at such speeds. In this context, the clumps of matter orbiting the black hole at 30,000 km/s require consideration of relativistic effects, as these speeds can alter mass, time, and distance calculations compared to classical physics.
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Related Practice
Textbook Question

Consider the ringshaped body of Fig. E13.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use Fx = -dU/dx to find the magnitude and direction of the force on the particle (see Section 7.4). (d) Show that your answer to part (c) reduces to the expected result when x is much larger than a. (e) What are the values of U and Fx when x = 0? Explain why these results make sense.

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Textbook Question

You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of 395.0 N in the rope. If Sneezy hangs from a similar rope while delivering presents at the earth's equator, what will the tension in it be? (Recall that the earth is rotating about an axis through its north and south poles.)

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Textbook Question

Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star?

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Textbook Question

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass.

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Textbook Question

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. What is the radius of its event horizon?

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