# An object with a mass of 4 kg is acted on by two forces. The first is F_1= < -7 N , 4 N> and the second is F_2 = < 8 N, -2 N>. What is the object's rate and direction of acceleration?

Jan 19, 2018

The rate of acceleration is $= \frac{\sqrt{5}}{4} m {s}^{-} 2$ in the direction $= {63.4}^{\circ}$ anticlockwise from the x-axis

#### Explanation:

The resultant force is

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

${\vec{F}}_{1} = < - 7 , 4 >$

${\vec{F}}_{2} = < 8 , - 2 >$

$\vec{F} = < - 7 , 4 > + < 8 , - 2 > = < 1 , 2 > N$

The mass is $m = 4 k g$

According to Newton's Second Law of motion

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

The acceleration is

$\vec{a} = \frac{1}{m} \times \vec{F} = \frac{1}{4} \cdot < 1 , 2 > = < \frac{1}{4} , \frac{1}{2} > m {s}^{-} 2$

The rate of acceleration is

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

$= \sqrt{\frac{1}{16} + \frac{1}{4}} = \sqrt{\frac{5}{16}}$

$= \frac{\sqrt{5}}{4} m {s}^{-} 2$

The direction is

$\theta = \arctan \left(\frac{2}{1}\right) = {63.4}^{\circ}$ anticlockwise from the x-axis