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Chapter 6: Problem 745

3 particle each of mass \(\mathrm{m}\) are kept at vertices of an equilateral triangle of side \(L\). The gravitational field at center due to these particles is (A) zero (B) \(\left[(3 \mathrm{GM}) / \mathrm{L}^{2}\right]\) (C) \(\left[(9 \mathrm{GM}) / \mathrm{L}^{2}\right]\) (D) \((12 / \sqrt{3})\left(\mathrm{Gm} / \mathrm{L}^{2}\right)\)

Short Answer

Expert verified
The gravitational field at the center of the equilateral triangle formed by three particles, each of mass m and located at the vertices of the triangle with a side of length L, is zero. \(g_{total} = 0\).

Step by step solution

01

Gravitational field formula

To calculate the gravitational field, we can use Newton's law of gravitation: \(g = G * \frac{m}{r^2}\) where: - g is the gravitational field - G is the gravitational constant - m is the mass of the particle causing the gravitational field - r is the distance between the particle and the point where we are calculating the field
02

Calculate the gravitational field from each particle

At the center of the equilateral triangle, each particle will result in an equal gravitational field, and their directions will form a 120-degree angle due to symmetry. So, we can denote this common gravitational field as: \(g_1=g_2=g_3=G * \frac{m}{(\frac{L}{2\sqrt{3}})^{2}}=12\sqrt{3} * G * \frac{m}{L^{2}}\)
03

Sum the gravitational fields

Now, we will add the vector sum of gravitational fields due to each of the particles: \(g_{total} = g_1 + g_2 + g_3\) As mentioned earlier, the angle between the directions of the gravitational fields due to the individual particles is 120 degrees. So, we can use the formula of vector addition and the cosine rule: \(g_{total}^2 = (g_1)^2 + (g_2)^2 + (g_3)^2 + 2g_1g_2\cos(120^\circ)+ 2g_1g_3\cos(120^\circ) + 2g_2g_3\cos(120^\circ)\) Since the gravitational field is the same for each particle, we can rewrite the equation as: \(g_{total}^2 = 3(g_1)^2 - 3(g_1)^2 = 0\)
04

Finalize the answer and choose the correct option

From our calculation, we can conclude that: \(g_{total} = 0\) So, the correct answer is: (A) zero

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