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 . Calculate the gravitational potential energy of these two masses.

While visiting Planet Physics, you toss a rock straight up at 11 m/s and catch it 2.5 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet's radius every 230 min. What are the (a) mass and (b) radius of Planet Physics?

Let’s look in more detail at how a satellite is moved from one circular orbit to another. FIGURE CP13.70 shows two circular orbits, of radii r₁ and r₂, and an elliptical orbit that connects them. Points 1 and 2 are at the ends of the semimajor axis of the ellipse. Consider a 1000 kg communications satellite that needs to be boosted from an orbit 300 km above the earth to a geosynchronous orbit 35,900 km above the earth. Find the velocity v'₁ on the inner circular orbit and the velocity v'₁ at the low point on the elliptical orbit that spans the two circular orbits.

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. 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 2R_Sch . Find an expression for ΔE=E_f−E_i , where E_i ≈ 0 because initially the black holes are far apart and E_f 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.

Let's look in more detail at how a satellite is moved from one circular orbit to another. FIGURE CP13.70 shows two circular orbits, of radii r₁ and r₂, and an elliptical orbit that connects them. Points 1 and 2 are at the ends of the semimajor axis of the ellipse. How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit?