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

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?

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?