# An object with a mass of 4 kg is acted on by two forces. The first is F_1= < 8 N , -4 N> and the second is F_2 = < 2 N, 6 N>. What is the object's rate and direction of acceleration?

Mar 17, 2018

The rate of acceleration is $= 2.55 m {s}^{-} 2$ in the direction $= {11.3}^{\circ}$ anticlockwise from the x-axis

#### Explanation:

The resultant force is

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

$\vec{F} = < 8 , - 4 > + < 2 , 6 > = < 8 + 2 , - 4 + 6 >$

$= < 10 , 2 > N$

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

According to Newto's Second Law

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

Therefore,

$\vec{a} = \frac{1}{m} \cdot \vec{F} = \frac{1}{4} \cdot < 10 , 2 > = < \frac{10}{4} , \frac{2}{4} >$

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

The rate of acceleration is

$| | \vec{a} | | = | | < \frac{5}{2} , \frac{1}{2} > | | = \sqrt{{\left(\frac{5}{2}\right)}^{2} + {\left(\frac{1}{2}\right)}^{2}}$

$= \sqrt{\frac{25}{4} + \frac{1}{4}} = \frac{\sqrt{26}}{2} = 2.55 m {s}^{-} 2$

The direction is

$\theta = \arctan \left(\frac{\frac{1}{2}}{\frac{5}{2}}\right) = \arctan \left(0.2\right) = {11.3}^{\circ}$ anticlockwise from the x-axis