# An object with a mass of 2 kg is acted on by two forces. The first is F_1= < 2 N , 5 N> and the second is F_2 = < 1 N, -6 N>. What is the object's rate and direction of acceleration?

Oct 29, 2017

The rate of acceleration is $= \frac{\sqrt{10}}{2} m {s}^{-} 2$ in the direction $= {18.4}^{\circ}$ clockwise from the x-axis.

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2} = < 2 , 5 > + < 1 , - 6 > = < 3 , - 1 >$

The mass is $m = 2 k g$

According to Newton's Second Law of motion

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

The acceleration is

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

The rate of acceleration is

$| | \vec{a} | | = | | < \frac{3}{2} , - \frac{1}{2} > | | = \sqrt{{\left(\frac{3}{2}\right)}^{2} + {\left(\frac{1}{2}\right)}^{2}} = \sqrt{\frac{9}{4} + \frac{1}{4}} = \frac{\sqrt{10}}{2} m {s}^{-} 2$

The direction of acceleration is

$\theta = \arctan \left(\frac{- \frac{1}{2}}{\frac{3}{2}}\right) = \arctan \left(- \frac{1}{3}\right) = {18.4}^{\circ}$ clockwise from the x-axis.