# An object with a mass of 3 kg is acted on by two forces. The first is F_1= < -6 N , -5 N> and the second is F_2 = < 3 N, 3 N>. What is the object's rate and direction of acceleration?

Apr 19, 2018

The rate of acceleration is $= 1.20 m {s}^{-} 2$ in the direction $= {213.7}^{\circ}$ anticlockwise from the x-axis

#### Explanation:

The resultant force is

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

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

$= < - 3 , - 2 > N$

The mass of the object is $m = 3 k g$

According to Newton's Second Law

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

Therefore,

$\vec{a} = \frac{1}{m} \cdot \vec{F} = \frac{1}{3} \cdot < - 3 , - 2 >$

$= < - 1 , - \frac{2}{3} > m {s}^{-} 2$

The rate of acceleration is

$| | \vec{a} | | = | | < - 1 , - \frac{2}{3} > | | = \sqrt{{\left(- 1\right)}^{2} + {\left(- \frac{2}{3}\right)}^{2}}$

$= \sqrt{1 + \frac{4}{9}} = \frac{\sqrt{13}}{3} = 1.20 m {s}^{-} 2$

The direction is

$\theta = \arctan \left(\frac{- \frac{2}{3}}{- 1}\right) = {180}^{\circ} + \arctan \left(\frac{2}{3}\right) = {213.7}^{\circ}$ anticlockwise from the x-axis