17. Periodic Motion

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. (b) 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.

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. (a) How far are these clumps from the center of the black hole?

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. (b) 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?

Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be 1.0 * 10^-15 m, what would be the mass of a mini black hole?

The Parker Solar Probe, launched in 2018, was the first spacecraft to explore the solar corona, the hot gases and flares that extend outward from the solar surface. The probe is in a highly elliptical orbit that, using the gravity of Venus, will be nudged ever closer to the sun until, in 2025, it reaches a closest approach of 6.9 million kilometers from the center of the sun. Its maximum speed as it whips through the corona will be 192 km/s.(b) The probe's highly elliptical orbit carries it out to a maximum distance of 160 Rₛ with a period of 88 days. What is its slowest speed, in km/s?

Two stars, one twice as massive as the other, are 1.0 light year (ly) apart. One light year is the distance light travels in one year at the speed of light, 3.00 ✕ 10⁸ m/s . The gravitational potential energy of this double-star system is ─8.0 ✕ 10³⁴ J. What is the mass of the lighter star?

A rocket is launched straight up from the earth's surface at a speed of 15,000 m/s. What is its speed when it is very far away from the earth?

A space station orbits the sun at the same distance as the earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

What is the total gravitational potential energy of the three masses in FIGURE P13.35?

September 2015 saw the historic discovery of gravitational waves, almost exactly 100 years after Einstein predicted their existence as a consequence of his theory of general relativity. Gravitational waves are a literal stretching and compressing of the fabric of space. Even the most sensitive instruments—capable of sensing that the path of a 4-km-long laser beam has lengthened by one-thousandth the diameter of a proton—can detect waves created by only the most extreme cosmic events. The first detection was due to the collision of two black holes more than 750 million light years from earth. Although a full description of gravitational waves requires knowledge of Einstein's general relativity, a surprising amount can be understood with the physics you've already learned. (d) Two black holes collide and merge when their Schwarzchild radii overlap; that is, they merge when their separation, which we've defined as 2r, equals 2RSch . Find an expression for ΔE=Ef−Ei , where Ei ≈ 0 because initially the black holes are far apart and Ef is their total energy at the instant they merge. This is the energy radiated away as gravitational waves. Your answer will be a fraction of Mc², and you probably recognize that this is related to Einstein's famous E=mc² . The quantity Mc² is the amount of energy that would be released if an entire star of mass M were suddenly converted entirely to energy.

You have been visiting a distant planet. Your measurements have determined that the planet's mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large.(b) To get back to earth, you need to escape the planet. What minimum speed does your rocket need?

FIGURE CP13.71 shows a particle of mass m at distance 𝓍 from the center of a very thin cylinder of mass M and length L. The particle is outside the cylinder, so 𝓍 > L/2 .(a) Calculate the gravitational potential energy of these two masses.

The 15 g head of a bobble-head doll oscillates in SHM at a frequency of 4.0 Hz.b. The amplitude of the head's oscillations decreases to 0.5 cm in 4.0 s. What is the head's damping constant?

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.10 g) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil.b. What is the disk's maximum speed at this amplitude?

Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant of the musculature that holds the eyeball in the socket?