Question

A 100-lb weights hangs from a string as shown in figure. Find the tensions $T_1$ and $T_2$ in the strings.

Short answer

The tension $T_1$and$T_2$in the strings are:$-56.5i+67.4j$and$56.5i+32.6j$respectively.

Step-by-step solution

Step 1. Given information.

Lifting weight is 100lb

Step 2. Write the concept.

Tension $T_2$makes an angle of$30°$with horizontal

$T_2=T_2\cos 30°i+T_2\sin 30°j$ ……(i)

$T_1$ makes an angle of$50°$

$T_1=T_1\cos 50°(-i)+T_1\sin 50°j$ …….(ii)

The horizontal component is in negative direction so we use -i

Lifting weight W = 100(-j) in downward

Resultant force at lifting point

$T_1+T_2+W=(-T_1\cos 50°i+T_1\sin 50°j)+(T_2\cos 30°i+T_2\sin 30°j)+(-100j)$

$=(-T_1\cos 50°+T_2\cos 30°)i+(T_2\sin 50°+T_2\sin 30°-100)j$

Step 3. Determining the values.

At equilibrium horizontal force will be zero

$-T_1\cos 50°+T_2\cos 30°=0$

$T_1\cos 50°=T_2\cos 30°$ ……..(iii)

the vertical forces

$T_1\sin 50°+T_2\sin 30°-100=0$ ……….(iv)

$1.53T_2=100$

$T_2=65.36$

Substitute in equation (iii)

$T_1=88$

Calculating tension from equation (i) & (ii)

$\begin{array}{l}{T}_1\approx -56.6i+67.4j\\ T_2\approx 56.5i+32.6j\end{array}$