University Physics Volume 1
18th Edition
ISBN: 9781938168277
Author: William Moebs, Samuel J. Ling, Jeff Sanny
Publisher: OpenStax - Rice University
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Chapter 13, Problem 81CP
A tunnel is dug through the center of a perfectly spherical and airless planet fo radius R. Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute
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A tunnel is dug through the center of a perfectly spherical and airless planet of radius R. Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute simple harmonic motion. Deduce the period of oscillation of m and show that it has the same period as an orbit at the surface.
The free-fall acceleration on the surface of Mercury is about three eighths that on the surface of the Earth. The radius of Mercury is about 0.375 RE (RE = Earth's radius = 6.4 x 106 m). Find
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IIISuppose a large spherical object, such as a planet, with radius R and mass M has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance x less than or equal to R from thecenter. It can be shown that the net gravitational force on the particlei sdue entirely to the sphere of mass with radius r ≤ ;x there is no netgravitational force from the mass in the spherical shell with r > .x a. Find an expression for the gravitational force on the particle,assuming the object has uniform density. Your expression willbe in terms of ,x R, m, M, and any necessary constants.b. You should have found that the gravitational force is a linear restoring force. Consequently, in the absence ofair resistance, objects in the tunnel will oscillate with SHM. Suppose an in- trepid astronaut exploring a 150-km-diameter, 3.5 X 1 0 kgasteroid discovers a tunnel through the center. If she jumps into the hole, how long will ti take her to fall all the way…
Chapter 13 Solutions
University Physics Volume 1
Ch. 13 - Check Your Understanding What happens to force and...Ch. 13 - Check Your Understanding How does your weight at...Ch. 13 - Check Your Understanding Why not use the simpler...Ch. 13 - Check Your Understanding If we send a probe out of...Ch. 13 - Check Your Understanding Assume you are in a...Ch. 13 - Check Your Understanding By what factor must the...Ch. 13 - Check Your Understanding There is another...Ch. 13 - Check Your Understanding Galaxies are not single...Ch. 13 - Check Your Understanding The nearly circular orbit...Ch. 13 - Check Your Understanding Earth exerts a tidal...
Ch. 13 - Check Your Understanding Consider the density...Ch. 13 - Action at a distance, such as is the case for...Ch. 13 - In the law of universal gravitation, Newton...Ch. 13 - Must engineers take Earth’s rotation into account...Ch. 13 - It was stated that a satellite with negative total...Ch. 13 - It was shown that the energy required to lift a...Ch. 13 - One student argues that a satellite in orbit is in...Ch. 13 - Many satellites are placed in geosynchronous...Ch. 13 - Are Kepler’s laws purely descriptive, or do they...Ch. 13 - In the diagram below for a satellite in an...Ch. 13 - As an object falls into a black hole, tidal forces...Ch. 13 - The principle of equivalence states that all...Ch. 13 - As a person approaches the Schwarzschild radius fo...Ch. 13 - Evaluate the magnitude of gravitational force...Ch. 13 - Estimate the gravitational force between two sumo...Ch. 13 - Astrology makes much of the position of the...Ch. 13 - A mountain 10.0 km from a person exerts a...Ch. 13 - The International Space Station has a mass of...Ch. 13 - Asteroid Toutatis passed near Earth in 2006 at...Ch. 13 - (a) What was the acceleration of Earth caused by...Ch. 13 - (a) Calculate Earth’s mass given the acceleratioln...Ch. 13 - (a) What is the acceleration due to gravity on the...Ch. 13 - (a) Calculate the acceleration due to gravity on...Ch. 13 - The mass of a particle is 15 kg. (a) What is its...Ch. 13 - On a planet whose radius is 1.2107m , the...Ch. 13 - The mean diameter of the planet Saturn is 1.2108m...Ch. 13 - The mean diameter of the planet Mercury is...Ch. 13 - The acceleration due to gravity on the surface of...Ch. 13 - A body on the surface of a planet with the same...Ch. 13 - Find the escape speed of a projectile from the...Ch. 13 - Find the escape speed of a projectile from the...Ch. 13 - What is the escape speed of a satellite located at...Ch. 13 - (a) Evaluate the gravitational potential energy...Ch. 13 - An average-sized asteroid located 5.0107km from...Ch. 13 - (a) What will be the kinetic energy of the...Ch. 13 - (a) What is the change in energy of a 1000-kg...Ch. 13 - If a planet with 1.5 times the mass of Earth was...Ch. 13 - Two planets in circular orbits around a star have...Ch. 13 - Using the average distance of Earth from the Sun,...Ch. 13 - What is the orbital radius of an Earth satellite...Ch. 13 - Calculate the mass of the Sun based on data for...Ch. 13 - Find the mass of Jupiter based on the fact that I0...Ch. 13 - Astronomical observatrions of our Milky Way galaxy...Ch. 13 - (a) In order to keep a small satellite from...Ch. 13 - The Moon and Earth rotate about their common...Ch. 13 - The Sun orbits the Milky Way galaxy once each...Ch. 13 - A geosynchronous Earth satellite is one that has...Ch. 13 - Calculate the mass of the Sun based on data for...Ch. 13 - I0 orbits Jupiter with an average radius of...Ch. 13 - The “mean” orbital radius listed for astronomical...Ch. 13 - The perihelion of Halley’s comet is 0.586 AU and...Ch. 13 - The perihelion of the comet Legerkvist is 2.61 AU...Ch. 13 - What is the ratio of the speed at perihelion to...Ch. 13 - Eros has an elliptical orbit about the Sun, with a...Ch. 13 - What is the difference between the force on a...Ch. 13 - If the Sun were to collapse into a black hole, the...Ch. 13 - Consider Figure 13.23 in Tidal Forces. This...Ch. 13 - What is the Schwarzschild radius for the black...Ch. 13 - What would be the Schwarzschild radius, in light...Ch. 13 - A neutron star is a cold, collapsed star with...Ch. 13 - (a) How far from the center of Earth would the net...Ch. 13 - How far from the center of the Sun would the net...Ch. 13 - Calculate the values of g at Earth’s surface for...Ch. 13 - Suppose you can communicate with the inhabitants...Ch. 13 - (a) Suppose that your measured weight at the...Ch. 13 - A body of mass 100 kg is weighed at the North Pole...Ch. 13 - Find the speed needed to escape from the solar...Ch. 13 - Consider the previous problem and include the fact...Ch. 13 - A comet is observed 1.50 AU from the Sun with a...Ch. 13 - An asteroid has speed 15.5km/s when it is located...Ch. 13 - Space debris left from old satellites and their...Ch. 13 - A satellite of mass 1000 kg is in circular orbit...Ch. 13 - After Cares was promoted to a dwarf planet, we now...Ch. 13 - (a) Using the data in the previous problem for the...Ch. 13 - What is the orbital velocity of our solar system...Ch. 13 - (a) Using the information in the previous problem,...Ch. 13 - Circular orbits in Equation 13.10 for conic...Ch. 13 - Show that for eccentricity equal to one in...Ch. 13 - Using the technique shown in Satellite Orbits and...Ch. 13 - Given the perihelion distance, p , and aphelion...Ch. 13 - Comet P/1999 R1 has a perihelion of 0.0570 AU and...Ch. 13 - A tunnel is dug through the center of a perfectly...Ch. 13 - Following the technique used in Gravitation Near...Ch. 13 - Show that the areal velocity for a circular orbit...Ch. 13 - Show that the period of orbit for two masses, m1...Ch. 13 - Show that for small changes in height h, such that...Ch. 13 - Using Figure 13.9, carefull sketch a free body...Ch. 13 - (a) Show that tidal force on a small object of...Ch. 13 - Find the Hohmann transfer velocities,...
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- Show that the areal velocity for a circular orbit of radius r about a mass M is At=12GMr . Does your expression give the correct value for Earth’s areal vilocity about the Sun?arrow_forwardFollowing the technique used in Gravitation Near Earth’s Surface, find the value of g as a function of the radius r from the center of a spherical shell planet of constant density with inner and outer radii Rin and Rout . Find g for both eq and for RinrRout . Assuming the inside of the shell is kept airless, describe travel inside the spherical shell planet.arrow_forwardWhat is the orbital radius of an Earth satellite having a period of 1.00 h? (b) What is unreasonable about this result?arrow_forward
- Compute directly the gravitational force on a unit mass at a point exterior to a homogeneous sphere of matter.arrow_forwardCheck Your Understanding Why not use the simpler expression U=mg(y2y1) ? How significant would the error be? (Recall the previous result, in Example 13.4, that the value g at 400 km above the Earth is 8.67m/s2 .)arrow_forwardFind the Hohmann transfer velocities, vEllipseEarth and vEllipseMars ,needed for a trip to Mars. Use Equation 13.7 to find the circular orbital velocities for Earth and Mars. Using Equation 13.4 and the total energy of the ellips (with semi-major asix a), given by E=GmMs2a , find the velocities at Earth (perihelion) and at Mars (aphelion) required to be on the transfer ellipse. The difference, v , at each point is the velocity boost or transfer velocity needed.arrow_forward
- Using Figure 13.9, carefull sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass,m. Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal accleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle , defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. tan(+)=gg2REtan , where is the angular velocity of Earth.arrow_forwardGiven the perihelion distance, p , and aphelion distance, q , for an elliptical orbit, show that the velocity at perihelion, vp , is given by vp=2GMSun(q+p)qp . (Hint: Use conservation of angular momentum to relate vp and vq , and then substitute into the conservation fo energy equation.)arrow_forwardOn a planet whose radius is 1.2107m , the acceleration due to gravity is 18m/s2 . What is the mass of the planet?arrow_forward
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