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Vector Addition

When a force ...

Question

When a force F acts on a body of mass m,the acceleration produced in the body is a if three equal forces $F_{1} = F_{2} = F_{3} = F$ act on the same body as shown in figure. The acceleration produced is :

(\sqrt{2} - 1) a

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(\sqrt{2} + 1) a

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\sqrt{2} a

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a

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Solution

The correct option is A $(\sqrt{2} - 1) a$
$F = F_{1} = F_{2} = F_{3}$

Net force of $F_{1}$ and $F_{2} = F_{12} = \sqrt{F^{2} + F^{2}} = \sqrt{2} F$

$F_{12}$ is opposite to the direction of $F_{3}$

Net force of $F_{12}$ and $F_{3} = F_{n e t} = \sqrt{2} F - F = F (\sqrt{2} - 1) F_{n e t} = m a_{n e t} F_{n e t} / F = m a_{n e t} / m a \Rightarrow F (\sqrt{2} - 1) / F = \frac{a_{n e t}}{a} a_{n e t} = a (\sqrt{2} - 1)$

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