# An object with a mass of 8 kg is acted on by two forces. The first is F_1= < -9 N , 2 N> and the second is F_2 = < -4 N, -3 N>. What is the object's magnitude and direction of acceleration?

Feb 12, 2018

The magnitude of acceleration is $\frac{\sqrt{170}}{8}$
And it makes an angle $\theta = \pi - \arctan \left(\frac{1}{13}\right)$ with the positive x-axis

#### Explanation:

The net force in x-direction is : $\left(\left(- 9\right) + \left(- 4\right)\right) N = - 13 N$
The net force in y-direction is : $\left(\left(+ 2\right) + \left(- 3\right)\right) N = - 1 N$

The magnitude of the net force: $F = \sqrt{{\left(13\right)}^{2} + {\left(1\right)}^{2}} = \frac{\sqrt{170}}{8}$

Thus the force makes an angle $\phi = \arctan \left(\frac{1}{13}\right)$ with the negative x-axis. Which can be written as it makes an angle $\theta = \pi - \arctan \left(\frac{1}{13}\right)$ with the positive x-axis

Hence the magnitude of acceleration: $a = \frac{F}{m} = \frac{\sqrt{170}}{8}$
And the direction will be same as that of the net force.