Question

Three astronauts, propelled by jet backpacks, push and guide a 2kg asteroid toward a processing dock, exerting the forces shown in figure, with $F_1=4N,F_2=4N,F_3=4N,{\theta }_1={30}^{\circ }and{\theta }_3={60}^{\circ }$. What is the asteroid's acceleration in unit-vector notation ?

• A. $(\sqrt{3}+1)\hat{i}+2\hat{j}$
• B. $2\sqrt{3}\hat{i}+2\hat{j}$
• C. $(3+\sqrt{3})\hat{i}+(1-\sqrt{3})\hat{j}$
• D. $4\sqrt{3\hat{i}}+8\hat{j}$

Answer 1

The correct option is C

$(3+\sqrt{3})\hat{i}+(1-\sqrt{3})\hat{j}$

$F_1=4cos{30}^{\circ }\hat{i}+4sin{30}^{\circ }\hat{j}$

$=2\sqrt{3}\hat{i}+2\hat{j}$

$F_2=4\hat{i}$

$F_3=4cos{60}^{\circ }\hat{i}-4sin{60}^{\circ }\hat{j}$

$=2\hat{i}-2\sqrt{3}\hat{j}$

${\overrightarrow{F}}_{net}={\overrightarrow{F}}_1+{\overrightarrow{F}}_2+{\overrightarrow{F}}_3$

$=(6+2\sqrt{3})\hat{i}+(2-2\sqrt{3})\hat{j}$

${\overrightarrow{F}}_{net}=m{\overrightarrow{a}}_{net}$

${\overrightarrow{a}}_{net}=\frac{{\overrightarrow{F}}_{net}}{m}=\frac{(6+2\sqrt{3})\hat{i}+(2-2\sqrt{3})\hat{j}}{2}$

${\overrightarrow{a}}_{net}=(3+\sqrt{3})\hat{i}+(1-\sqrt{3})\hat{j}$