# An object with a mass of 2 kg is acted on by two forces. The first is F_1= < -6 N , 2 N> and the second is F_2 = < 1 N, 3 N>. What is the object's rate and direction of acceleration?

Feb 23, 2017

The net acceleration is $= 3.54 m {s}^{-} 2$ in the direction of $= \left(\frac{3}{4} \pi\right) r a d$

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2}$

$= < - 6 , 2 > + < 1 , 3 >$

$= < - 5 , 5 >$

we use Newton's second Law

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

$\vec{a} = \frac{1}{m} \cdot \vec{F}$

$= \frac{1}{2} < - 5 , 5 > = < - \frac{5}{2} , \frac{5}{2} >$

The magnitude of the acceleration is

$| | \vec{a} | | = | | < - \frac{5}{2} , \frac{5}{2} > | |$

$= \sqrt{{\left(- \frac{5}{2}\right)}^{2} + {\left(\frac{5}{2}\right)}^{2}}$

$= \sqrt{\frac{25}{4} + \frac{25}{4}}$

$= \frac{\sqrt{50}}{2} = 3.54 m {s}^{-} 2$

The direction is $\theta = \arctan \left(\frac{\frac{5}{2}}{- \frac{5}{2}}\right)$

The angle is in the 2nd quadrant

$\theta = \arctan \left(- 1\right) = \left(\frac{3}{4} \pi\right) r a d$