You push a 3-kg mass against a spring and release it from rest. Its maximum acceleration is 10m/s² when pushed back 0.5m. What is the (a)spring constant and (b) restoring force at this point?

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.

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). Use F_x = -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, $1000.0\text{ kg}$ shell has a radius of $5.00\text{ m}$. 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 $r\to \infty$.

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 × 10²² 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?