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Chapter 6: Problem 687

A spherical planet far out in space has mass \(\mathrm{M}_{0}\) and diameter \(\mathrm{D}_{0}\). A particle of \(\mathrm{m}\) falling near the surface of this planet will experience an acceleration due to gravity which is equal to (A) $\left[\left(\mathrm{GM}_{0}\right) /\left(\mathrm{D}_{\circ}^{2}\right)\right]$ (B) $\left[\left(4 \mathrm{mGM}_{0}\right) /\left(\mathrm{D}_{0}^{2}\right)\right]$ (C) $\left[\left(4 \mathrm{GM}_{0}\right) /\left(\mathrm{D}_{0}^{2}\right)\right]$ (D) $\left[\left(\mathrm{GmM}_{0}\right) /\left(\mathrm{D}_{\circ}^{2}\right)\right]$

Short Answer

Expert verified
The acceleration due to gravity acting on a particle of mass m near the surface of a spherical planet with mass M0 and diameter D0 can be found using the gravitational force formula and simplification. The acceleration due to gravity (g) is given by: \[\boxed{g = \frac{4GM_0}{D_0^2}}\] This corresponds to option (C).

Step by step solution

01

Understand the gravitational force formula

The gravitational force between two objects is given by Newton's Law of Universal Gravitation: \[F = G \frac{m_1 m_2}{r^2}\] Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of mass of the two objects.
02

Rewrite the formula to find the acceleration due to gravity

We can rewrite the gravitational force formula by solving for the acceleration due to gravity acting on the particle (g) instead of the gravitational force. Since F = m * g, we have: \[g = \frac{F}{m} = G \frac{m_1 m_2}{m r^2}\] We can now substitute the planet's mass (M0) and particle's mass (m) and the distance between their centers of mass (D0): \[g = G \frac{M_0 m}{r^2}\]
03

Calculate the distance between the centers of masses

Since the particle is near the surface of the planet, the distance between their centers of mass (r) is approximately equal to the radius of the planet. Since diameter D0 = 2 * radius, we can find the radius: \[r = \frac{D_0}{2}\]
04

Substitute the distance into the equation for acceleration due to gravity

Now, let's substitute the found distance (r) into the equation for acceleration due to gravity (g): \[g = G \frac{M_0 m}{\left(\frac{D_0}{2}\right)^2} = G \frac{M_0 m}{\frac{D_0^2}{4}}\]
05

Simplify the equation

By simplifying the equation, we can cancel out the mass m of the particle and get the expression for the acceleration due to gravity only in terms of the gravitational constant (G), mass of the planet (M0), and diameter of the planet (D0): \[g = \frac{4GM_0}{D_0^2}\]
06

Compare with the given options

Comparing our found expression for the acceleration due to gravity (g) with the given options, we see that it corresponds to option (C): \[\boxed{g = \left[\left(4GM_0\right)/\left(D_0^2\right)\right]}\]

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Most popular questions from this chapter

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