# An object with a mass of 2 kg is acted on by two forces. The first is F_1= < 2 N , 3 N> and the second is F_2 = < -7 N, 1 N>. What is the object's rate and direction of acceleration?

Apr 23, 2018

The rate of acceleration is $= 3.2 m {s}^{-} 1$ in the direction $= {141.3}^{\circ}$anticlockwise from the x-axis

#### Explanation:

The resultant force is

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

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

$= < - 5 , 4 > N$

The mass of the object is $m = 2 k g$

According to Newton's Second Law

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

Therefore,

$\vec{a} = \frac{1}{m} \cdot \vec{F} = \frac{1}{2} \cdot < - 5 , 4 >$

$= < - \frac{5}{2} , 2 > m {s}^{-} 2$

The rate of acceleration is

$| | \vec{a} | | = | | < - \frac{5}{2} , 2 > | | = \sqrt{{\left(- \frac{5}{2}\right)}^{2} + {\left(2\right)}^{2}}$

$= \sqrt{\frac{25}{4} + 4} = \frac{\sqrt{41}}{2} = 3.20 m {s}^{-} 2$

The direction is

$\theta = \arctan \left(\frac{- 4}{5}\right) = {180}^{\circ} - \arctan \left(\frac{4}{5}\right) = {141.3}^{\circ}$ anticlockwise from the x-axis