# An object with a mass of 3 kg is acted on by two forces. The first is F_1= < 2 N , 8 N> and the second is F_2 = < 2 N, -3 N>. What is the object's rate and direction of acceleration?

May 5, 2017

The rateof acceleration is $= 2.13 m {s}^{-} 2$ in the direction $= 51.3$º

#### Explanation:

The resultant force is

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

$= < 2 , 8 > + < 2 , - 3 >$

$= < 4 , 5 >$

We apply Newton's second Law

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

Mass, $m = 3 k g$

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

$= \frac{1}{3} < 4 , 5 > = < \frac{4}{3} , \frac{5}{3} >$

The magnitude of the acceleration is

$| | \vec{a} | | = | | < \frac{4}{3} , \frac{5}{3} > | |$

$= \sqrt{{\left(\frac{4}{3}\right)}^{2} + {\left(\frac{5}{3}\right)}^{2}}$

$= \frac{\sqrt{41}}{3} = 2.13 m {s}^{-} 2$

The direction is $\theta = \arctan \left(\frac{5}{4}\right)$

The angle is in the 1st quadrant

$\theta = \arctan \left(\frac{5}{4}\right) = 51.3$º

May 5, 2017

We are given two forces
${F}_{1} = 2 \hat{i} + 8 \hat{j}$
${F}_{2} = 2 \hat{i} - 3 \hat{j}$
$\therefore$Net force $F = {F}_{1} + {F}_{2}$
$\implies F = 4 \hat{i} + 5 \hat{j}$
Now by Newton's second law
$F = m \times a$
$\implies 4 \hat{i} + 5 \hat{j} = 3 \times a$
$\implies a = \frac{1}{3} \left(4 \hat{i} + 5 \hat{j}\right)$