Static Deflection of a Shaft
The static deflection of a shaft refers to the amount by which the shaft bends or deforms under a static load. In this case, the static deflection of the shaft under a flywheel is given as 4 mm.

Calculating Critical Speed
The critical speed of a rotating shaft is the speed at which it begins to resonate or vibrate excessively. It is an important parameter to consider in the design of rotating machinery.

The formula to calculate the critical speed of a shaft is:

nc = (g / (2π)) * √(d / L)

Where:
- nc is the critical speed in revolutions per second (rps)
- g is the acceleration due to gravity (10 m/s^2)
- d is the static deflection of the shaft (4 mm or 0.004 m)
- L is the length of the shaft

Calculating Critical Speed in Rad/s
To convert the critical speed from rps to rad/s, we need to multiply it by 2π. Therefore, the formula to calculate the critical speed in rad/s is:

nc_rad = 2π * nc

Substituting the given values into the formula:

nc = (10 / (2π)) * √(0.004 / L)
nc_rad = 2π * (10 / (2π)) * √(0.004 / L)
nc_rad = 10 * √(0.004 / L)
nc_rad = 10 * √(0.004) / √L
nc_rad = 10 * 0.063 / √L
nc_rad = 0.63 / √L

Answer
To determine the critical speed, we need the length of the shaft (L). However, the length of the shaft is not given in the question. Therefore, we cannot calculate the exact value of the critical speed.

However, we can determine the relationship between the critical speed and the length of the shaft. From the equation above, we can see that the critical speed is inversely proportional to the square root of the length of the shaft. This means that as the length of the shaft increases, the critical speed decreases.

The correct answer is option 'A' - 50 rad/s. However, without the length of the shaft, it is impossible to determine the exact value.