Spring Force (Hooke's Law) Practice Problems

A regular round bar has a mass M and a length L. A point mass is placed at a point parallel to the bar's length (on its axis), an arbitrary distance d from the nearest end of the bar. Using the relationship between force and energy (F = -dU/dr) determine the magnitude and direction of the gravitational force due to the bar on the point mass. For d >> L, prove that your answer agrees with the expected result.

A 750 kg rocket on its return journey from Mars to Earth reaches its cruise speed soon after launch. At 10⁵ km from Mars' surface, the speed of the rocket is 36000 km/h. At that position, determine i) the rocket's kinetic energy relative to Mars, and ii) the rocket-Mars potential energy. Mars has a radius of 3390 km and a mass of 6.42 × 10²³ kg.

During a scientific experiment on Earth, objects are ejected 10 km into the atmosphere from a high-speed launcher. If gravity is constant over the 10 km range of the objects' heights, when the experiment is reproduced with the same launch speed on Mars how high would these materials go? Mars has a mass of 6.42 × 10²³ kg and a radius of 3390 km.

A thin rogue planet in the form of a homogeneous spherical shell of mass M and radius R floats in space without orbiting any star. Draw a graph that illustrates the variation of the gravitational force on a point-like object of mass m moving away radially from the center of the planet to a point at infinity.

Consider an imaginary planet having the shape of a uniform spherical shell in our Milkway galaxy. The planet's mass is 3.0 × 10²¹ kg and the shell radius is 100 km. Determine the gravitational force exerted by the planet on a 200 kg point-like alien situated at i) 101 km,  ii) 99 km. and iii) 200 m from its center.

Titania and Oberon are the largest Uranian moons. The radius of Titania is 788.9 km, and its mass is 35.2 × 10²⁰ kg. The radius of Oberon is 761.4 km, and its mass is 30.1 × 10²⁰ kg. a) What is the escape speed of a vertically launched rocket on the surface of i) Titania and ii) Oberon? b) Why is the escape speed independent of the escaping object's mass? 

A large black hole has been revealed to be the center of our own Milky Way galaxy. A newly discovered star is orbiting this black hole at an orbital speed of 2.1 × 10⁸ m/s and a radius of approximately 1.12 × 10⁸ m. Calculate the radius of its event horizon (R_S)

Let's assume that a chunk of ionized matter is observed by astronomers orbiting a black hole once every 12 days at 190 km/s. How much does this black hole weigh? Your answer should be given in i) kilograms and ii) as a multiple of the mass of the sun.

The first black hole, Cygnus X-1, was identified by several researchers in 1971. They observed a chunk of matter orbiting around Cygnus X-1 once every 5 ms at a speed of 1.49 × 10⁸ m/s. What is the distance r between the chunk of matter and the center of the black hole?

Astronomers have detected a supermassive black hole, known as Sagittarius A, at the center of our galaxy. A particular star, orbiting Sagittarius A, has an orbital speed of 2.1 × 10⁸ m/s. The radius of the star's orbit around Sagittarius A is approximately 0.08 AU. Theories indicate that a single star cannot have a mass greater than about 50 solar masses. i) Determine the mass of  Sagittarius A ii) would Sagittarius A has been formed from only one star?

Mini black holes, with the same diameter as the nucleus of a hydrogen atom, are thought to have been formed during the Big Bang. The radius of the hydrogen nucleus is approximately one fermi. What is the mass of a mini black hole?