# An object with a mass of 2 kg is acted on by two forces. The first is F_1= <5 N , 7 N> and the second is F_2 = < 2 N, -4 N>. What is the object's rate and direction of acceleration?

Nov 1, 2017

The rate of acceleration is $= 3.81 m {s}^{-} 2$ in the direction $= {23.2}^{\circ}$ anticlockwise from the x-axis

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2} = < 5 , 7 > + < 2 , - 4 > = < 7 , 3 >$

The mass is $m = 2 k g$

According to Newton's Second Law of motion

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

The acceleration is

$\vec{a} = \frac{1}{m} \vec{F} = \frac{1}{2} < 7 , 3 > = < \frac{7}{2} , \frac{3}{2} >$

The rate of acceleration is

$| | \vec{a} | | = | | < \frac{7}{2} , \frac{3}{2} > | | = \sqrt{{\left(\frac{7}{2}\right)}^{2} + {\left(\frac{3}{2}\right)}^{2}} = \sqrt{\frac{49}{4} + \frac{9}{4}} = \frac{\sqrt{58}}{2} = 3.81 m {s}^{-} 2$

The direction of acceleration is

$\theta = \arctan \left(\frac{\frac{3}{2}}{\frac{7}{2}}\right) = \arctan \left(\frac{3}{7}\right) = {23.2}^{\circ}$ anticlockwise from the x-axis.