# An object with a mass of 2 kg is acted on by two forces. The first is F_1= <-3 N , -5 N> and the second is F_2 = < 6 N, 7 N>. What is the object's rate and direction of acceleration?

Apr 25, 2016

Adding the vectors gives a magnitude of $3.6$ $N$, and an angle of ${34}^{o}$.

Using Newton's Second Law:

$a = \frac{F}{m} = \frac{3.6}{2} = 1.8$ $m {s}^{-} 2$ at ${34}^{o}$ to the positive x-axis.

#### Explanation:

First we find the resultant force acting on the object, by adding the components of the force vectors:

${F}_{1} + {F}_{2} = < - 3 , - 5 > + < 6 , 7 >$
$= < \left(- 3 + 6\right) , \left(- 5 + 7\right) > = < 3 , 2 >$

So the resultant force vector is 3 in the x direction and 2 in the y direction. We can use Pythagoras' theorem to find the magnitude of the resultant: $\sqrt{{3}^{2} + {2}^{2}} = \sqrt{13} \approx 3.6$ $N$.

To find the direction, we use trig: 3 and 2 are the opposite and adjacent sides of a right-angled triangle, so:

$\tan \theta = \frac{o p p}{a \mathrm{dj}} = \frac{2}{3}$ so $\theta \approx {34}^{o}$

Using Newton's Second Law, the acceleration is given by:

$a = \frac{F}{m} = \frac{3.6}{2} = 1.8$ $m {s}^{-} 2$ at ${34}^{o}$ to the positive x-axis.