# An object with a mass of 6 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?

Nov 7, 2017

The rate of acceleration is $= 1.18 m {s}^{-} 1$ in the direction $= {135}^{\circ}$ anticlockwise from the x-axis.

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2} = < - 6 , 2 > + < 1 , 3 > = < - 5 , 5 >$

The mass is $m = 6 k g$

According to Newton's Second Law of motion

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

The acceleration is

$\vec{a} = \frac{1}{m} \vec{F} = \frac{1}{6} < - 5 , 5 > = < - \frac{5}{6} , \frac{5}{6} >$

The rate of acceleration is

$| | \vec{a} | | = | | < - \frac{5}{6} , \frac{5}{6} > | | = \sqrt{{\left(- \frac{5}{6}\right)}^{2} + {\left(\frac{5}{6}\right)}^{2}} = \sqrt{\frac{25}{36} + \frac{25}{36}} = \frac{\sqrt{50}}{6} = 1.18 m {s}^{-} 2$

The direction of acceleration is

$\theta = \arctan \left(\frac{\frac{5}{6}}{- \frac{5}{6}}\right) = \arctan \left(- 1\right) = {135}^{\circ}$ antilockwise from the x-axis.