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Chapter 6: Problem 691

The moon's radius is \(1 / 4\) that of earth and its mass is \(1 / 80\) times that of the earth. If g represents the acceleration due to gravity on the surface of earth, that on the surface of the moon is (A) \(g / 4\) (B) \(\mathrm{g} / 5\) (c) \(\mathrm{g} / 6\) (D) \(\mathrm{g} / 8\)

Short Answer

Expert verified
The acceleration due to gravity on the surface of the moon (g') is given by: \[g' = \frac{1}{5}g\]. The correct answer is (B) \(g / 5\).

Step by step solution

01

Write the formula for gravitational force

We know that the gravitational force between two objects is given by the formula: \[F = G \frac{m_1 m_2}{r^2}\] Where F is the gravitational force, G is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects, and r is the distance between their centers.
02

Write the formula for acceleration due to gravity

Acceleration due to gravity is defined as the gravitational force acting on an object divided by the object's mass. Since only the mass, radius and acceleration due to gravity of Earth and the moon are given, we can assume an object with mass m is placed on the surface of each celestial body. We get the following equation for the acceleration due to gravity (g) on Earth: \[g = \frac{F_{earth}}{m}\] Similarly, for the moon, we have: \[g' = \frac{F_{moon}}{m}\]
03

Find the ratio of gravitational forces on Earth and the moon

We will find the ratio of the gravitational forces acting on the object on the surfaces of the Earth and the moon using the given information about the radius and mass. Since the ratio of the moon's mass to Earth's mass is 1/80 and their radius ratio is 1/4, we can write the ratio of their gravitational forces as: \[\frac{F_{moon}}{F_{earth}} = \frac{G \frac{M_{moon} m}{R_{moon}^2}}{G \frac{M_{earth} m}{R_{earth}^2}} = \frac{\frac{1}{80}M_{earth}}{\left(\frac{1}{4} R_{earth}\right)^2}\]
04

Simplify the ratio of gravitational forces

We can now simplify the ratio of gravitational forces: \[\frac{F_{moon}}{F_{earth}} = \frac{\frac{1}{80}M_{earth}}{\left(\frac{1}{16} R_{earth}^2\right)} = \frac{1}{5}\]
05

Find the ratio of acceleration due to gravity

Since we know the ratio of gravitational forces, we can find the ratio of acceleration due to gravity on the moon and Earth using the equations we found in Step 2: \[\frac{g'}{g} = \frac{\frac{F_{moon}}{m}}{\frac{F_{earth}}{m}} = \frac{F_{moon}}{F_{earth}} = \frac{1}{5}\] So, the acceleration due to gravity on the surface of the moon (g') is given by: \[g' = \frac{1}{5}g\] The correct answer is (B) \(g / 5\).

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Most popular questions from this chapter

The Gravitational P.E. of a body of mass \(\mathrm{m}\) at the earth's surface is \(-\mathrm{mgRe}\). Its gravitational potential energy at a height \(\operatorname{Re}\) from the earth's surface will be \(=\ldots \ldots \ldots\) here (Re is the radius of the earth) (A) \(-2 \mathrm{mgRe}\) (B) \(2 \mathrm{mgRe}\) (C) \((1 / 2) \mathrm{mg} \mathrm{Re}\) (D) \(-(1 / 2) \mathrm{mg} \operatorname{Re}\)

He period of revolution of planet \(\mathrm{A}\) around the sun is 8 times that of \(\mathrm{B}\). The distance of A from the sun is how many times greater than that of \(\mathrm{B}\) from the sun. (A) 2 (B) 3 (C) 4 (D) 5

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) \(g_{1}: g_{2}=R_{1} \rho_{1}: R_{2} \rho_{2}\)

The earth revolves round the sun in one year. If distance between then becomes double the new period will be years. (A) \(0.5\) (B) \(2 \sqrt{2}\) (C) 4 (D) 8

let \(\mathrm{V}\) and \(\mathrm{E}\) denote the gravitational potential and gravitational field at a point. Then the match the following \(\begin{array}{ll}\text { Table }-1 & \text { Table }-2\end{array}\) (A) \(\mathrm{E}=0, \mathrm{~V}=0\) (P) At center of spherical shell (B) \(\mathrm{E} \neq 0, \mathrm{~V}=0\) (Q) At center of solid sphere (C) \(\mathrm{V} \neq 0, \mathrm{E}=0\) (R) at centre of circular ring (D) \(\mathrm{V} \neq 0, \mathrm{E} \neq 0\) (S) At centre of two point masses of equal magnitude (T) None
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