# An object with a mass of 4 kg is acted on by two forces. The first is F_1= < -1 N , 8 N> and the second is F_2 = < -2 N, 5 N>. What is the object's rate and direction of acceleration?

Feb 21, 2018

The rate of acceleration is $= 3.34 m {s}^{-} 2$ in the direction $= {103}^{\circ} \text{ anticlockwise from the x- axis}$

#### Explanation:

The resultant force is

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

$= < - 1 , 8 > + < - 2 , 5 >$

$= < - 3 , 13 > N$

The mass of the object is $m = 4 k g$

According to Newton's Second Law

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

Therefore,

The acceleration is

$\vec{a} = \frac{\vec{F}}{m} = \frac{1}{4} \cdot < - 3 , 13 > = < - \frac{3}{4} , \frac{13}{4} >$

The rate of acceleration is

$| | \vec{a} | | = | | < - \frac{3}{4} , \frac{13}{4} > | |$

$= \sqrt{{\left(- \frac{3}{4}\right)}^{2} + {\left(\frac{13}{4}\right)}^{2}}$

$= 3.34 m {s}^{-} 2$

The direction is

$\theta = \arctan \left(- \frac{13}{3}\right) = {103}^{\circ} \text{in the direction anticlockwise from the x- axis}$