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Chapter 6: Problem 674

Mass \(M\) is divided into two parts \(\mathrm{xM}\) and $(1-\mathrm{x}) \mathrm{M}\(. For a given separation, the value of \)\mathrm{x}$ for which the gravitational force between the two pieces becomes maximum is (A) 1 (B) 2 (C) \(1 / 2\) (D) \(4 / 5\)

Short Answer

Expert verified
The value of x for which the gravitational force between the two pieces becomes maximum is 1/2. Therefore, the correct answer is (C) 1/2.

Step by step solution

01

Set up the gravitational force equation using the given masses

We are given two masses, xM and (1-x)M. We will plug these into the gravitational force equation: \(F = G \frac{(xM)((1-x)M)}{r^2}\)
02

Simplify the equation

We will now simplify the equation by expanding the terms in the numerator: \(F = G \frac{M^2(x-x^2)}{r^2}\)
03

Differentiate the equation with respect to x

In order to find the maximum value of F with respect to x, we will differentiate F with respect to x: \(\frac{dF}{dx} = G M^2 \frac{d(x-x^2)}{dx}\) \(\frac{dF}{dx} = G M^2 (1-2x)\)
04

Set the derivative equal to zero and solve for x

To find the maximum value, we will set the derivative equal to zero and solve for x: \(0 = G M^2 (1-2x)\) \(2x = 1\) \(x = \frac{1}{2}\)
05

Select the correct answer

The value of x for which the gravitational force between the two pieces becomes maximum is: x = 1/2 Therefore, the correct answer is (C) 1/2.

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