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Chapter 6: Problem 668

Two identical solid copper spheres of radius \(R\) are placed in contact with each other. The gravitational force between them is proportional to (A) \(\mathrm{R}^{2}\) (B) \(\mathrm{R}^{-2}\) (C) \(\mathrm{R}^{-4}\) (D) \(\mathrm{R}^{4}\)

Short Answer

Expert verified
The gravitational force between the two identical solid copper spheres of radius R is proportional to \(R^{4}\). So, the correct answer is (D) \(\mathrm{R}^{4}\).

Step by step solution

01

Find the mass of a sphere

Given that both spheres are made of copper, the density of copper is: \[\rho = 8,960\ kg/m^3\] The volume of a sphere can be calculated using the formula: \[V=\frac{4}{3}\pi R^3\] Since the mass of an object is equal to its volume multiplied by its density, the mass of one of the copper spheres is: \[m = \rho V = 8960\cdot\frac{4}{3}\pi R^3\]
02

Calculate the distance between the centers of the spheres

Since the spheres are in contact, the distance between their centers is equal to the sum of their radii. Since they are identical, the distance between their centers is: \[r = 2R\]
03

Calculate the gravitational force between the spheres

Now, we can use the formula for gravitational force to find the force between the two spheres. Let m be the mass of one sphere and r be the distance between their centers. \[F = G\frac{m^2}{r^2} = G\frac{(8960\cdot\frac{4}{3}\pi R^3)^2}{(2R)^2}\]
04

Simplify the expression and find the proportionality

Simplify the gravitational force expression by canceling out some terms: \[F = G\frac{(8960\cdot\frac{4}{3}\pi R^3)^2}{(2R)^2} = G\frac{(8960\cdot\frac{4}{3}\pi)^2 R^6}{4R^2}\] Further simplify the expression: \[F = G\frac{(8960\cdot\frac{4}{3}\pi)^2}{4}R^4\] From the above expression, we can see that the gravitational force is proportional to \(R^{4}\), so the correct answer is (D) \(\mathrm{R}^{4}\).

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