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.

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?

A simple pendulum has a period of 4.0 s in a vacuum. When it is moving to a region with air resistance, the amplitude decreases to 40 % of its initial value after 45 s. How many additional oscillations will it take for the amplitude to decrease to 20% of its initial value?

If a spacecraft is launched from a planet's surface at 20000 m/s, what would be its velocity at an infinite distance from its gravitational field? Consider the mass of the planet is 5.98 × 10²⁴ kg and the radius of the planet is 6.37 × 10⁶ m.

Three masses are placed at the vertices of an equilateral triangle. Determine the total gravitational potential energy of the system.

In a binary star system, one of the stars is four times heavier than the other. It is known that the gravitational potential energy of the two stars is -1.2 × 10³⁴ J. Find the mass of the heavier star if the distance between the two stars is 1.5 light years.

Scientists expect the release of gravitational waves and energy when two black holes merge. The merger of two black holes is conceivable when they are so close to one another in such a way that they cannot escape their gravitational attraction. A sophomore is asked to use the concepts of classical mechanics to find the energy variation ΔE of two black holes of mass M_b each during their merger. The energy variation is defined as the difference between the energy at the instant the merge occurs and the energy at the instant black holes were very far from each other. What does the sophomore report in terms of M_bc²? Hint: The merger occurs when the separation distance between the two black holes is twice the Schwarzschild radius and the black holes can be treated as point-like objects.

A space agency is planning to send a spacecraft to explore the atmosphere of Venus and study its surface characteristics. Its aim is to put the spacecraft into a 24-hour elliptical orbit. Based on the calculations, the spacecraft will reach an altitude of 300 kilometers from Venus when it is at its closest distance to the planet and 700 kilometers when it is at its farthest distance. The spacecraft's maximum speed, at its closest distance, is 400 km/h. Calculate the spacecraft's speed at its farthest distance from Venus.

While exploring space, a spaceship lands on an exoplanet. As the conditions on the exoplanet are extremely dangerous, the crew decides to escape the exoplanet as soon as possible. Using the spaceship instruments, they measured the gravitational acceleration of the exoplanet to be half that of Earth and its mass to be a third of that of Earth. Calculate the minimum speed necessary to escape the planet's gravity.

Suppose that two twin exoplanets separated by a distance of d=2 × 10⁸ m are released from rest and approach each other under mutual gravitational attraction until they collide together.  Find the speeds at the collision if the mass and the radius of each exoplanet are  M= 2 × 10²⁵ Kg and r = 5 × 10⁵ m respectively.

A uniform rod of length a and mass m_r is placed along the y-axis with its center at the origin. A tiny sphere of mass m_s is placed at point (0, y_s) with y_s > a/2. Derive the expression of the potential energy between the rod and the sphere.

A 300 g ball is thrown horizontally at a speed of 15 m/s and collides with a spring attached to a wall. The ball is in contact with the spring for 0.5 s before bouncing back. Find the spring constant.

A 2 kg object is pushed with an initial speed of 20 cm/s. The object then collides with a horizontal spring having a spring constant of 50 N/m. The object is in contact with the spring for 0.2 seconds before bouncing back. Determine the maximum compression experienced by the spring during the collision.

A person standing on the bridge notices that the bridge starts to vibrate with a frequency of 3.5 Hz when a large truck passes by. If the mass of the bridge is 5000 kg, what is the effective spring constant of the bridge?

A thin metal plate having a mass of 0.25 g is experiencing oscillations. Determine the maximum speed attained by the plate if the frequency of oscillations is 1.5 MHz. The metal plate shows a maximum acceleration of 1.40 × 10⁸ m/s².

A mass of 25 g attached to a spring undergoes SHM with a frequency of 2.0 Hz and an amplitude of 10.0 cm. If the amplitude decreases to 6.0 cm after 5.0 s due to damping, what is the damping constant of the system?

A spring-mass oscillator is used in a mechanical clock to be exhibited during the school's science fair. It consists of two springs, with spring constants k₁ and k₂, where k₁=k₂/4, and bob of mass m, as shown below. Derive an expression for the bob’s oscillation frequency f in terms of the frequencies f₁ and f₂ at which it would oscillate if attached to spring 1 or spring 2 alone. 

A young astrophysicist is studying the atmosphere of a newly discovered gas giant planet and is interested in creating a model of acetylene gas. Acetylene consists of two carbon (C) atoms and two hydrogen (H) atoms, as shown below. The C–C and H–C bonds can be considered to be ideal springs with unknown spring constants k_C-C and k_H-C, respectively. The symmetric stretch frequency of the carbon-carbon bond in acetylene is measured to be 9.94 x 10¹³ Hz. If the atomic masses of carbon and hydrogen are known to be m_C = 12.0 amu and m_H = 1.0 amu, respectively, calculate the effective spring constant of the carbon-carbon bond in acetylene.

What is the stiffness constant of the spring that is compressed by 6.0 m when a 960-kg vehicle collides with it at a speed of 26 m/s?

How long does a 960-kg truck remain in contact with a heavy-duty buffer after crashing into it at 26 m/s and compressing it by 6.0 m before it rebounds?

A 5.42 kg dartboard is attached to the free end of a spring that is oriented horizontally. The other end of the spring is fixed to a wall. A student throws a dart of mass 0.0423 kg at the dartboard and it gets stuck in the dartboard. As a result, the spring gets compressed by a maximum distance of 5.20 cm. Given that the spring constant of the spring is k = 122 N/m, find the speed of the dart v_o which strikes the dartboard. [Assume that there is no friction involved.]