# An object with a mass of 2 kg is acted on by two forces. The first is F_1= <4 N , 6 N> and the second is F_2 = < 1 N, -4 N>. What is the object's rate and direction of acceleration?

Nov 16, 2017

The rate of acceleration is $= 2.69 m {s}^{-} 1$ in the direction ${21.8}^{\circ}$ anticlockwise from the x-axis

#### Explanation:

The resultant force is

$\vec{F} = {\vec{F}}_{1} + {\vec{F}}_{2} = < 4 , 6 > + < 1 , - 4 > = < 5 , 2 >$

Apply Newton's Second Law of motion

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

The mass is $m = 2 k g$

The acceleration is

$\vec{a} = \frac{\vec{F}}{m} = \frac{1}{2} < 5 , 2 > = < \frac{5}{2} , 1 > m {s}^{-} 2$

The rate of acceleration is

$= | | \vec{a} | | = | | < \frac{5}{2} , 1 >$

$= \sqrt{{\left(\frac{5}{2}\right)}^{2} + {1}^{2}} = \sqrt{\frac{25}{4} + 1} = \frac{\sqrt{29}}{2} = 2.69 m {s}^{-} 2$

The direction is

$\theta = \arctan \left(\frac{2}{5}\right) = \arctan \left(0.4\right) = {21.8}^{\circ}$ anticlockwise from the x-axis