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

May 4, 2018

The rate of acceleration is $= 0.79 m {s}^{-} 1$ in the direction $= {18.4}^{\circ}$clockwise from the x-axis

#### Explanation:

The resultant force is

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

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

$= < 3 , - 1 > N$

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

According to Newton's Second Law

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

Therefore,

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

$= < \frac{3}{4} , - \frac{1}{4} > m {s}^{-} 2$

The rate of acceleration is

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

$= \sqrt{\frac{9}{16} + \frac{1}{16}} = \frac{\sqrt{10}}{4} = 0.79 m {s}^{-} 2$

The direction is

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