Answer to Question #122809 in Physics for Susan Williams

Question

A rope 5 m long is fastened to two hooks 4.0 m apart on a horizontal ceiling. To the rope is attached a 10 kg mass so that the segments of the rope are 3.0 m and 2.0 m. Compute the tension in each segment.

Answer 1

First, let us find the angles between tensions T and the ceiling. This can be done by cosine rule. We know the sides - 2, 3, and 4 m. Let us find the angle between T1 and the blue line:

$$2^2=4^2+3^2-2\cdot4\cdot3\text{cos}\theta_1,\\\space\\ \text{cos}\theta_1=\frac{4^2+3^2+2^2}{2\cdot3\cdot4}=0.875,\\\space\\\theta_1=29°.$$

Let us find the angle between T2 and the blue line:

$$3^2=4^2+2^2-2\cdot4\cdot2\text{cos}\theta_2,\\\space\\\theta_2=46.6°.$$

Now write the equilibrium of forces along vertical and horizontal axes:

$$y:T_2\text{sin}\theta_2+T_1\text{sin}\theta_1-mg=0,\\ x:-T_2\text{cos}\theta_2+T_1\text{cos}\theta_2=0.\\ T_1=69.5\text{ N}.\\ T_2=88.5\text{ N}.$$

The equations were solved for T1 and T2 assuming that g=9.8 m/s²