Question

The cranes in the figure are lifting

an object that weighs 18,278 lb. Find the tensions $T_1$ and $T_2$.

Short answer

The tension $T_1$and$T_2$in the cranes are:$-9126.2i+3743j$and$9126.2i+8074.2j$respectively.

Step-by-step solution

Step 1. Given information.

Lifting weight is 18278lb

Step 2. Write the concept.

Tension $T_2$makes an angle of$41.5°$with horizontal

$T_2=T_2\cos 41.5°i+T_2\sin 41.5°j$ ……(i)

$T_1$ makes an angle of$22.3°$

$T_1=T_1\cos 22.3°(-i)+T_1\sin 22.3°j$ …….(ii)

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

Lifting weight W = 18278(-j) in downward

Resultant force at lifting point

$T_1+T_2+W=(-T_1\cos 22.3°i+T_1\sin 22.3°j)+(T_2\cos 41.5°i+T_2\sin 41.5°j)+(-18278j)$

$=(-T_1\cos 22.3°+T_1\cos 41.5°)i+(T_2\sin 22.3°+T_2\cos 41.5°-18278)j$

Step 3. Determining the values.

At equilibrium horizontal force will be zero

$-T_1\cos 22.3°+T_2\cos 41.5°=0$

$T_1\cos 22.3°=T_2\cos 41.5°$ ……..(iii)

the vertical forces

$T_1\sin 22.3°+T_2\sin 41.5°-18278=0$ ……….(iv)

$1.5T_2=18278$

$T_2=12185.3$

Substitute in equation (iii)

$T_1=9864$

Calculating tension from equation (i) & (ii)

$\begin{array}{l}{T}_2\approx 9126.2i+8074.2j\\ T_1\approx -9126.2i+3743j\end{array}$