# An object with a mass of 2 kg is acted on by two forces. The first is F_1= < 9 N , 5 N> and the second is F_2 = < 4 N, -6 N>. What is the object's rate and direction of acceleration?

May 10, 2016

$a = 6.52 \text{ } \frac{m}{s} ^ 2$
$\tan \alpha = - \frac{1}{13}$

$\alpha = - {4.40}^{o} \text{(from clockwise)" " "or alpha=355.60 ^o"(from counterclockwise)}$

#### Explanation:

${F}_{1} = < 9 N , 5 N > \text{ } {\vec{F}}_{1} = 9 \hat{i} + 5 \hat{j}$
${F}_{2} = < 4 N , - 6 N > \text{ } {\vec{F}}_{2} = 4 \hat{i} - 6 \hat{j}$

${F}_{1} \hat{i} + {F}_{2} \hat{i} = 9 \hat{i} + 4 \hat{i} = 13 \hat{i}$
${F}_{1} \hat{j} + {F}_{2} \hat{j} = 5 \hat{j} - 6 \hat{j} = - \hat{j}$

$\text{The Resultant vector(green) :} \vec{R} = 13 \hat{i} - \hat{j}$

$\text{The magnitude of } \vec{R} = \sqrt{{13}^{2} + {1}^{2}} = \sqrt{169 + 1}$

$\vec{R} = \sqrt{170}$

$\vec{R} = 13.04 \text{ } N$

$\text{acceleration of object can be calculated using the Newton's second law:}$

$F = m \cdot a \text{ "a=F/m" "F=13.04N" } m = 2 k g$

$a = \frac{13.04}{2}$

$a = 6.52 \text{ } \frac{m}{s} ^ 2$

$\tan \alpha = - \frac{1}{13}$

$\alpha = - {4.40}^{o} \text{(from clockwise)" " "or alpha=355.60 ^o"(from counterclockwise)}$