/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Two planets have the same mass, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 6

Two planets have the same mass, \(M,\) but one of them is much denser than the other. Identical objects of mass \(m\) are placed on the surfaces of the planets. Which object will have the gravitational potential energy of larger magnitude? a) Both objects will have the same gravitational potential energy. b) The object on the surface of the denser planet will have the larger gravitational potential energy. c) The object on the surface of the less dense planet will have the larger gravitational potential energy. d) It is impossible to tell.

Short Answer

Expert verified
Answer: The object on the surface of the denser planet will have the larger gravitational potential energy.

Step by step solution

01

Calculate the radii of the planets

To calculate the potential energy, we need to find the radii of the planets. Let's denote the density of the denser planet as \(\rho_1\) and the less dense planet as \(\rho_2\). The mass of each planet can be expressed as \(M = \rho_1 V_1 = \rho_2 V_2,\) where \(V_1\) and \(V_2\) are the volumes of the planets. For a spherical planet, we have \(V_1 = \frac{4}{3}\pi r_1^3\) and \(V_2 = \frac{4}{3}\pi r_2^3,\) where \(r_1\) and \(r_2\) are the radii of the denser and less dense planets respectively. Thus, using the mass and density, we can determine the radii of the planets: \(r_1 = \sqrt[3]{\frac{3M}{4\pi\rho_1}}\) and \(r_2 = \sqrt[3]{\frac{3M}{4\pi\rho_2}}.\)
02

Calculate the gravitational potential energy for the objects on each planet

Now that we have the radii for both planets, we can calculate the gravitational potential energy for each object using the formula \(U = -\frac{G\M\times m}{r}.\) For the denser planet, we have \(U_1 = -\frac{G(M)(m)}{r_1}\) and for the less dense planet, we have \(U_2 = -\frac{G(M)(m)}{r_2}.\)
03

Compare the gravitational potential energies

To compare the gravitational potential energies \(U_1\) and \(U_2,\) note that the magnitudes of the energies are inversely proportional to the radii of the planets (since the other factors are constants). As the denser planet has a smaller radius (\(r_1 < r_2\)), its gravitational potential energy magnitude will be larger, and the less dense planet's gravitational potential energy magnitude will be smaller. Therefore, the correct answer is: b) The object on the surface of the denser planet will have the larger gravitational potential energy.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A 1000.-kg communications satellite is released from a space shuttle to initially orbit the Earth at a radius of \(7.00 \cdot 10^{6} \mathrm{~m}\). After being deployed, the satellite's rockets are fired to put it into a higher altitude orbit of radius \(5.00 \cdot 10^{7} \mathrm{~m} .\) What is the minimum mechanical energy supplied by the rockets to effect this change in orbit?

The distances from the Sun at perihelion and aphelion for Pluto are \(4410 \cdot 10^{6} \mathrm{~km}\) and \(7360 \cdot 10^{6} \mathrm{~km},\) respectively. What is the ratio of Pluto's orbital speed around the Sun at perihelion to that at aphelion?

Compare the magnitudes of the gravitational force that the Earth exerts on the Moon and the gravitational force that the Moon exerts on the Earth. Which is larger?

Two planets have the same mass, \(M .\) Each planet has a constant density, but the density of planet 2 is twice as high as that of planet \(1 .\) Identical objects of mass \(m\) are placed on the surfaces of the planets. What is the relationship of the gravitational potential energy, \(U_{1},\) on planet 1 to \(U_{2}\) on planet \(2 ?\) a) \(U_{1}=U_{2}\) b) \(U_{1}=\frac{1}{2} U_{2}\) c) \(U_{1}=2 U_{2}\) d) \(U_{1}=8 U_{2}\) e) \(U_{1}=0.794 U_{2}\)

For two identical satellites in circular motion around the Earth, which statement is true? a) The one in the lower orbit has less total energy. b) The one in the higher orbit has more kinetic energy. c) The one in the lower orbit has more total energy. d) Both have the same total energy.
See all solutions

Recommended explanations on Physics Textbooks

Turning Points in Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Force

Read Explanation

Space Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.