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Which of the following sets of concurrent forces may be in equilibrium?
(A) ${F_1} = 3N,\,{F_2} = 5N,\,{F_3} = 1N$
(B) ${F_1} = 3N,\,{F_2} = 5N,\,{F_{_3}} = 9N$
(C) ${F_1} = 3N,\,{F_2} = 5N,\,{F_3} = 6N$
(D) ${F_1} = 3N,\,{F_2} = 5N,\,{F_3} = 15N$

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Last updated date: 13th Jun 2024
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Answer
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Hint: The numerical problem given above can be solved to get the correct answer by using the triangle law of forces. We know that the triangle law of forces can be used when there are at least three concurrent forces.

Formula Used: For three concurrent forces to be in a state of equilibrium, the condition is as given below:
${F_1} + {F_2} \geqslant {F_3}$

Complete step by step answer:
We know that, according to the triangle law of forces, three forces are said to be in a state of equilibrium only if the sum of any two forces given is either greater than or equal to the third force. Thus, mathematically the triangle law of forces can be written in the following way:
${F_1} + {F_2} \geqslant {F_3}$

Now, we can solve the given numerical problem by substituting the forces in the given equation:
In case (A) ${F_1} = 3N,{\kern 1pt} \,{F_2} = 5N,\,{F_3} = 1N$now when we substitute these values in the above equation, we get:
$3N + 5N > 1N$

But, we see that ${F_1} + {F_3} \ne {F_2}$ as $3N + 1N < 5N$. Thus, we can conclude that in condition (A) the forces are not in a state of equilibrium. Thus, option (A) is an incorrect answer.
In case (B) ${F_1} = 3N,\,{F_2} = 5N,\,{F_{_3}} = 9N$ now when we substitute these values in the given mathematical expression, we get:
$3N + 5N < 15N$. Thus, we find that it does not satisfy the condition ${F_1} + {F_2} \geqslant {F_3}$.

Thus, we can conclude that in condition (B) is an incorrect answer as the forces are not in equilibrium.
Now for condition (C) we can clearly see that $3N + 5N > 6N$. Thus, we can say that in this case the three forces are in equilibrium in accordance with the triangle law of forces.
Thus we find that (C) is the correct answer to this question.

Note: It is important to note that the triangle law of forces apply only when there are at least three forces.These three forces in equilibrium can be represented in magnitude and direction by the three sides of a triangle taken in order. In case there are four forces the quadrilateral law of forces is used to determine if they would be in equilibrium.