Question

A weight 'W' is supported by two cables as shown in the figure below. The tension in the cable making an angle $\theta$ will be the minimum when the value of $\theta$ is

• A. $0^o$
• B. ${30}^o$
• C. ${45}^o$
• D. ${60}^o$

Answer 1

The correct option is B ${30}^o$



From the diagram after resolving forces we get


$T_{1}sin\theta +T_{2}sin60=W$ ......(i)

$T_{1}cos\theta =T_{2}cos60$ ......(ii)

From (ii)

$T_2=2T_{1}cos\theta$

Putting in (i)

$T_{1}sin\theta +2T_{1}cos\theta \times \frac{\sqrt{3}}{2}=W$

$T_{1}sin\theta +\sqrt{3}cos\theta T_1=W$


$T_1=\frac{W}{\sqrt{3}cos\theta +sin\theta }S$

Fir $T_1$ to be minimum

$f\left(\theta \right)=\sqrt{3}cos\theta +sin\theta$ is maximum

$=2[\frac{\sqrt{3}}{2}cos\theta +\frac{1}{2}sin\theta ]$

$=2[cos\theta cos30+sin\theta sin30]$

$=2\left[cos\right(\theta -30\left)\right]$

$\Rightarrow$ $\theta -30=0\Rightarrow \theta ={30}^o$