# 011 Resultant of three forces acting in a ring

**Problem 011**

Three ropes are tied to a small metal ring. At the end of each rope three students are pulling, each trying to move the ring in their direction. If we look down from above, the forces and directions they are applying are shown in Fig. P-011. Find the net force on the ring due to the three applied forces.

**Solution 011**

$R_x = 30 \cos 37^\circ - 50 \cos 45^\circ - 80 \cos 60^\circ$

$R_x = -51.40 \, \text{ lb}$

$R_x = 51.40 \, \text{ lb to the left}$

$R_y = \Sigma F_y$

$R_y = 30 \sin 37^\circ + 50 \sin 45^\circ - 80 \sin 60^\circ$

$R_y = -15.87 \, \text{ lb}$

$R_y = 15.87 \, \text{ lb downward}$

$R = \sqrt{{R_x}^2 + {R_y}^2}$

$R = \sqrt{51.40^2 + 15.87^2}$

$R = 53.79 \, \text{ lb}$

$\tan \theta_x = \dfrac{R_y}{R_x} = \dfrac{15.87}{51.40}$

$\theta_x = 17.16^\circ$

Thus, the net force on the ring is 53.79 lb downward to the left at θ_{x} = 17.16°.