Nothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 ✕ 10⁸ m/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 5.0 times the mass of the sun? This distance is called the Schwarzschild radius.

A20.0 cmmassless spring with spring constant340 N/mis suspended from the ceiling. A student carefully hangs a450 gmass from the free end. How long is the spring now?

A spring with a spring constant of $100\raisebox{1ex}{$\mathrm{N}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{m}$}\right.$ is attached to a rubber block with mass 3.0 kg. The block is resting on a concrete surface (${\mu }_{\mathrm{s}}=1.0$). The spring is pulled to the side until the block begins to slip. What is the maximum extension of the spring before the block slips?

A 1.1 m length of string has a radius of 1.2 mm. It stretches by 1.2 cm when a 5.6 kg mass is suspended from it. What is Young's modulus of the rope?

What is the escape speed from Jupiter?

A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is placed a distance x from one end of the rod, along the axis of the rod (Fig. E13.34)

. (b) Use Fx = -dU>dx to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4). Show that your answer reduces to the expected result when x is much larger than L.

A uniform, spherical, 1000.0-kg shell has a radius of 5.00 m. (b) Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass m as a function of the distance r of m from the center of the sphere. Include the region from r = 0 to r -> ∞.

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 * 10^6 km from the earth and traveling at 1.20 * 10^4 km/h relative to the earth. At this time, what were (a) the spacecraft's kinetic energy relative to the earth and (b) the potential energy of the earth–spacecraft system?

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93 * 10^22 kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?