Answer to Question #102081 in Mechanics | Relativity for Jarmo Silander

Question

How to solve it Two teenagers are pulling on ropes attached to a tree. The angle between the ropes is 30.0° . David pulls with a force of 400.0 N and Stephanie pulls with a force of 300.0 N. (a) Find the component form of the net force. (b) Find the magnitude of the resultant (net) force on the tree and the angle it makes with David’s rope.

Answer 1

$F_1=400 \; N$

$F_2=300\; N$

$\alpha=30^{\circ}$

(a) $\overline {F_{net}}\; -?$

(b) $F_{net}, \;\theta\; - ?$

Solution:

$F_1:$

$F_{1x}=400\times \cos (0^{\circ}) =400, \;\; F_{1y}= 400\times \sin (0^{\circ}) =0$

$\overline{F_1}=400\hat{i}$

$F_2:$

$F_{2x}= 300\times \cos (30^{\circ}) =300\times \frac{\sqrt{3}}{2}= 259.8, \;\; F_{2y}= 300\times \sin (30^{\circ}) =150$

$\overline{F_2}=259.8\hat{i}+150\hat{j}$

$\overline{F_{net}}=\overline{F_1}+\overline{F_2}=659.8\hat{i}+150\hat{j}$

$F_{net}=\sqrt{F_x^2+F_y^2}=\sqrt{659.8^2+150^2}=676.6\; N$

$\tan \theta=\frac{F_y}{F_x}, \;\; \theta =\arctan \frac{150}{659.8}=12.8^{\circ}$

Answers:

(a) $\overline{F_{net}}=659.8\hat{i}+150\hat{j}$

(b) $F_{net}=676.6\; N, \;\;\theta= 12.8^{\circ}$