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Chapter 6: Problem 701

The radii of two planets are respectively \(\mathrm{R}_{1}\) and \(\mathrm{R}_{2}\) and their densities are respectively \(\rho_{1}\) and \(\rho_{2}\) the ratio of the accelerations due to gravity at their surface is (A) $g_{1}: g_{2}=\left(\rho_{1} / R_{1}^{2}\right) \cdot\left(\rho_{2} / R_{2}^{2}\right)$ (B) $\mathrm{g}_{1}: \mathrm{g}_{2}=\mathrm{R}_{1} \mathrm{R}_{2}: \rho_{1} \rho_{2}$ (C) \(g_{1}: g_{2}=R_{1} \rho_{2} \cdot R_{2} p_{1}\) (D) \(g_{1}: g_{2}=R_{1} \rho_{1}: R_{2} \rho_{2}\)

Short Answer

Expert verified
The short answer is: The ratio of the accelerations due to gravity at the surface of the two planets is \(g_1 : g_2 = R_1 \rho_1 : R_2 \rho_2\). The correct option is (D).

Step by step solution

01

Recall the formula for gravitational acceleration on a planet's surface

The gravitational acceleration at the surface of a planet can be expressed as \(g = \frac{GM}{R^2}\), where G is the gravitational constant, M is the mass of the planet, and R is its radius. In this problem, we will need to find the mass of each planet in terms of their density and radius, and then plug those expressions into the formula for gravitational acceleration.
02

Find the mass of each planet using density and volume

The mass of a planet can be found using its density and volume. We know that the density is mass divided by volume, or \(\rho = \frac{M}{V}\). We can solve for mass by multiplying both sides by the volume: \(M = \rho V\). The volume of a sphere is given by \(\frac{4}{3}\pi R^3\), where R is the radius. Thus, the mass of each planet can be expressed as: - Planet 1: \(M_1 = \rho_1 \frac{4}{3}\pi R_1^3\) - Planet 2: \(M_2 = \rho_2 \frac{4}{3}\pi R_2^3\)
03

Find the gravitational acceleration at the surface of each planet

Now, we will substitute the mass expressions we found in Step 2 into the formula for gravitational acceleration: \(g = \frac{GM}{R^2}\) - Planet 1: \(g_1 = \frac{G(\rho_1 \frac{4}{3}\pi R_1^3)}{R_1^2} = \frac{4}{3}G\pi \rho_1 R_1\) - Planet 2: \(g_2 = \frac{G(\rho_2 \frac{4}{3}\pi R_2^3)}{R_2^2} = \frac{4}{3}G\pi \rho_2 R_2\)
04

Find the ratio of the gravitational accelerations

To find the ratio of the gravitational accelerations, we can divide \(g_1\) by \(g_2\): \[ \frac{g_1}{g_2} = \frac{\frac{4}{3}G\pi \rho_1 R_1}{\frac{4}{3}G\pi \rho_2 R_2} = \frac{\rho_1 R_1}{\rho_2 R_2} \] Therefore, the ratio of the accelerations due to gravity at the surface of the two planets is \(g_1 : g_2 = R_1 \rho_1 : R_2 \rho_2\). We can compare this result with the given options and conclude that the correct answer is: (D) \(g_1 : g_2 = R_1 \rho_1 : R_2 \rho_2\)

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Most popular questions from this chapter

3 particle each of mass \(\mathrm{m}\) are kept at vertices of an equilateral triangle of side \(L\). The gravitational field at center due to these particles is (A) zero (B) \(\left[(3 \mathrm{GM}) / \mathrm{L}^{2}\right]\) (C) \(\left[(9 \mathrm{GM}) / \mathrm{L}^{2}\right]\) (D) \((12 / \sqrt{3})\left(\mathrm{Gm} / \mathrm{L}^{2}\right)\)

The mass of a space ship is \(1000 \mathrm{~kg} .\) It is to be launched from earth's surface out into free space the value of \(\mathrm{g}\) and \(\mathrm{R}\) (radius of earth) are \(10 \mathrm{~ms}^{-2}\) and \(6400 \mathrm{~km}\) respectively the required energy for this work will be $=\ldots \ldots \ldots .$ J (A) \(6.4 \times 10^{11}\) (B) \(6.4 \times 10^{8}\) (C) \(6.4 \times 10^{9}\) (D) \(6.4 \times 10^{10}\)

A geostationary satellite is orbiting the earth at a height of \(5 \mathrm{R}\) above that of surface of the earth. \(\mathrm{R}\) being the radius of the earth. The time period of another satellite in hours at a height of $2 \mathrm{R}\( from the surface of earth is \)\ldots \ldots \ldots .$ hr (A) 5 (B) 10 (C) \(6 \sqrt{2}\) (D) \(6 / \sqrt{2}\)

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Energy required to move a body of mass \(\mathrm{m}\) from from an orbit of radius \(2 \mathrm{R}\) to \(3 \mathrm{R}\) is \(\ldots \ldots \ldots \ldots\) (A) \(\left[(\mathrm{GMm}) /\left(12 \mathrm{R}^{2}\right)\right]\) (B) \(\left[(\mathrm{GMm}) /\left(3 \mathrm{R}^{2}\right)\right]\) (C) \([(\mathrm{GMm}) /(8 \mathrm{R})]\) (D) \([(\mathrm{GMm}) /(6 \mathrm{R})]\)
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