# An object with a mass of 5 kg is acted on by two forces. The first is F_1= <2 N , 6 N> and the second is F_2 = <4 N, -3 N>. What is the objects rate and direction of acceleration?

Apr 2, 2018

The rate of acceleration is $= 1.34 m {s}^{-} 2$ in the direction $= {26.6}^{\circ}$ anticlockwise from the x-axis

#### Explanation:

The resultant force is

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

$\vec{F} = < 2 , 6 > + < 4 , - 3 > = < 2 + 4 , 6 - 3 >$

$= < 6 , 3 > N$

The mass of the object is $m = 5 k g$

According to Newto's Second Law

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

Therefore,

$\vec{a} = \frac{1}{m} \cdot \vec{F} = \frac{1}{5} \cdot < 6 , 3 > = < \frac{6}{5} , \frac{3}{5} > m {s}^{-} 2$

The rate of acceleration is

$| | \vec{a} | | = | | < \frac{6}{5} , \frac{3}{5} > | | = \sqrt{{\left(\frac{6}{5}\right)}^{2} + {\left(\frac{3}{5}\right)}^{2}}$

$= \sqrt{\frac{36}{25} + \frac{9}{25}} = \frac{\sqrt{45}}{5} = 1.34 m {s}^{-} 2$

The direction is

$\theta = \arctan \left(\frac{\frac{3}{5}}{\frac{6}{5}}\right) = \arctan \left(0.5\right) = {26.6}^{\circ}$ anticlockwise from the x-axis