# An object with a mass of 8 kg is acted on by two forces. The first is F_1= < -9 N , 2 N> and the second is F_2 = < 6 N, -3 N>. What is the object's rate and direction of acceleration?

Jan 31, 2018

The object's rate of acceleration is $= 0.40 m {s}^{-} 2$ in the direction ${198.4}^{\circ}$ anticlockwise from the x-axis.

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2} = < - 9 , 2 > + < 6 , - 3 >$

$= < - 3 , - 1 >$

The mass of the object is $m = 8 k g$

According to Newton's Second Law

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

The acceleration is

$\vec{a} = \frac{1}{m} \vec{F} = \frac{1}{8} \cdot < - 3 , - 1 > = < - \frac{3}{8} , - \frac{1}{8} >$

The rate of acceleration is

$| | \vec{a} | | = | | < - \frac{3}{8} , - \frac{1}{8} > | | = \sqrt{{\left(- \frac{3}{8}\right)}^{2} + {\left(- \frac{1}{8}\right)}^{2}}$

$= \sqrt{\frac{9}{64} + \frac{1}{64}}$

$= \frac{1}{8} \sqrt{10}$

$= 0.40 m {s}^{-} 2$

The direction is

$\theta = {180}^{\circ} + \arctan \left(\frac{1}{3}\right) = 180 + 18.4 = {198.4}^{\circ}$ anticlockwise from the x-axis.