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Chapter 6: Problem 681

Two point masses \(\mathrm{A}\) and \(\mathrm{B}\) having masses in the ratio $4: 3\( are separated by a distance of \)\operatorname{lm}$. When another point mass of mass \(\mathrm{M}\) is placed in between \(\mathrm{A}\) and \(\mathrm{B}\) the forces \(\mathrm{A}\) and is \((1 / 3 \mathrm{rd})\) of the force between \(\mathrm{B}\) and \(\mathrm{C}\), Then the distance \(\mathrm{C}\) from \(\mathrm{A}\) is \(=\ldots \ldots \ldots \mathrm{m}\) (A) \((2 / 3)\) (B) \(1 / 3\) (C) \(1 / 4\) (D) \(2 / 7\)

Short Answer

Expert verified
The distance between A and C is 2/7 meters. Therefore, the correct answer is (D) 2/7.

Step by step solution

01

Let mA be the mass of A and mB the mass of B. Since their masses are in the ratio 4:3, we can write mA = 4m and mB = 3m, where m is some arbitrary mass unit. #Step 2: Write down the forces between A and C, and B and C#

According to the universal law of gravitation, the force F between two point masses is given by \[F = G\frac{m_{1}m_{2}}{r^2}\], where G is the gravitational constant, m1 and m2 are the masses and r is the distance between them. Therefore, the forces between A and C, and B and C can be written as \[F_{AC} = G\frac{mA \times M}{x^2}\] and \[F_{BC} = G\frac{mB \times M}{(1-x)^2}\], where M is the mass of point C and x is the distance between A and C. #Step 3: Formulate the given condition#
02

We are given that the force between A and C is 1/3 of the force between B and C, so we can write the condition as \[F_{AC} = \frac{1}{3}F_{BC}\]. Substitute the expressions for F_AC and F_BC from Step 2 and we get \[\frac{m_{A}\times M}{x^{2}} = \frac{1}{3}\frac{m_{B} \times M}{(1-x)^{2}}\] #Step 4: Solve for the distance x#

We can cancel out the mass M in the equation, and then substitute mA = 4m and mB = 3m from step 1, which gives us: \[\frac{4m}{x^2} = \frac{1}{3}\frac{3m}{(1-x)^2}\]. After canceling out the mass m and solving for x, we get: \[4(1-x)^2 = 3x^2 \Rightarrow 4 - 8x + 4x^2= 3x^2 \Rightarrow x^2 = 8x - 4 \Rightarrow x = \frac{2}{7}\] So, the distance between A and C is x = 2/7 meters. Therefore, the correct answer is (D) 2/7.

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Most popular questions from this chapter

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