An object with a mass of 3 kg is acted on by two forces. The first is F_1= < -2 N , -5 N> and the second is F_2 = < 7 N, 3 N>. What is the object's rate and direction of acceleration?

Aug 19, 2016

rate : $\frac{\sqrt{29}}{3} m \text{/} {s}^{2}$

direction : $\arctan \left(- \frac{2}{5}\right)$

Explanation:

Newton's Second Law

$\Sigma \vec{F} = m \vec{a}$

Here

$\Sigma \vec{F} = \left(\begin{matrix}- 2 \\ - 5\end{matrix}\right) + \left(\begin{matrix}7 \\ 3\end{matrix}\right) = \left(\begin{matrix}5 \\ - 2\end{matrix}\right) = \textcolor{red}{m \vec{a}}$

$\implies \left(\begin{matrix}5 \\ - 2\end{matrix}\right) = 3 \left(\begin{matrix}{a}_{x} \\ {a}_{y}\end{matrix}\right)$

$\implies \vec{a} = \frac{1}{3} \left(\begin{matrix}5 \\ - 2\end{matrix}\right)$

$\left\mid \vec{a} \right\mid = \frac{1}{3} \sqrt{{5}^{2} + {\left(- 2\right)}^{2}} = \frac{\sqrt{29}}{3}$

$\tan \varphi = \frac{- 2}{5} \implies \varphi = \arctan \left(- \frac{2}{5}\right)$