# An object with a mass of 5 kg is acted on by two forces. The first is F_1= < -1 N , 8 N> and the second is F_2 = < 6 N, -3 N>. What is the objects rate and direction of acceleration?

Jul 11, 2017

$a = 1.41$ ${\text{m/s}}^{2}$

$\theta = {45}^{\text{o}}$

#### Explanation:

We're asked to find the magnitude and direction of the object's acceleration, given its mass and two forces that act on it.

What we can do is split this problem up into the $x$- and $y$-components, and find the acceleration components from the net force components (and its mass).

(Remember, ${\overbrace{\sum \vec{F}}}^{\text{the net force}} = m \vec{a}$)

We need to calculate the components of the net force that acts on the object, using

$\sum {F}_{x} = {F}_{1 x} + {F}_{2 x} = - 1$ $\text{N}$ $+ 6$ $\text{N}$ = $5$ $\text{N}$

$\sum {F}_{y} = {F}_{1 y} + {F}_{2 y} = 8$ $\text{N}$ $- 3$ $\text{N}$ = $5$ $\text{N}$

The acceleration components are derived from the Newton's second law equation:

${a}_{x} = \frac{\sum {F}_{x}}{m} = \left(5 \textcolor{w h i t e}{l} \text{N")/(5color(white)(l)"kg}\right) = 1$ ${\text{m/s}}^{2}$

${a}_{y} = \frac{\sum {F}_{y}}{m} = \left(5 \textcolor{w h i t e}{l} \text{N")/(5color(white)(l)"kg}\right) = 1$ ${\text{m/s}}^{2}$

The magnitude of the acceleration is

a = sqrt((1color(white)(l)"m/s"^2)^2 + (1color(white)(l)"m/s"^2)^2) = color(red)("1.41 color(red)("m/s"^2

The direction of the acceleration is

theta = arctan((1color(white)(l)"m/s"^2)/(1color(white)(l)"m/s"^2)) = color(blue)(45^"o"