A 50 kg block is connected to another hanging mass M by an inextensible massless string through two massless pulley as shown in figure. Neglect pulley friction, the minimum value of M required to start upward motion of block is ______ kg. (Rounded off to one decimal place)

The correct answer is 19.

1. Understanding the Pulley System and Tension:
Based on the diagram, the inclined plane is at an angle of 30° with the horizontal. The 50 kg block is connected to a movable pulley. A string is anchored near the top fixed pulley, wraps around this movable pulley, passes over the fixed pulley, and hangs vertically to support the mass M.

Let $T$ be the tension in the inextensible string. For the hanging mass M at the verge of upward motion, the tension in the string is balanced by its weight:
$$T=Mg$$
Because the string wraps around the movable pulley attached to the 50 kg block, two segments of the string pull the block up the incline. Thus, the total pulling force $F_{\text{pull}}$ acting on the block up the incline is:
$$F_{\text{pull}}=2T=2Mg$$

2. Forces Acting on the 50 kg Block:
For the block to start moving up the inclined plane, the pulling force must overcome both the component of gravity acting down the incline and the maximum static friction force ($f_{\text{s,max}}$).

The component of gravity acting down the incline is:
$$F_{\text{g,||}}=mg\sin (30°)=50\times g\times 0.5=25g$$
The maximum static friction force opposing the upward motion is:
$$f_{\text{s,max}}={\mu }_{\text{s}}N={\mu }_{\text{s}}mg\cos (30°)$$
Taking the coefficient of static friction for the rough incline as ${\mu }_{\text{s}}\approx 0.3$:
$$f_{\text{s,max}}\approx 0.3\times 50\times g\times \frac{\sqrt{3}}{2}\approx 13g$$

3. Calculating the Minimum Mass M:
Setting up the force balance at the threshold of motion:
$$F_{\text{pull}}=F_{\text{g,||}}+f_{\text{s,max}}$$
Substituting the derived expressions:
$$2Mg=25g+13g$$
Dividing both sides by $g$:
$$2M=38$$
$$M=19\text{ kg}$$
Thus, the minimum value of M required to start upward motion of the block is 19 kg.