Torque to weight ratio for a circular shaft transmitting power is directly proportional to the diameter of the shaft.

Explanation:

Torque to weight ratio is a measure of the efficiency or performance of a shaft in transmitting power. It is calculated by dividing the torque (rotational force) exerted by the shaft by its weight.

Let's consider two shafts with different diameters but the same material and length. The torque transmitted by each shaft will depend on its diameter.

The torque transmitted by a shaft is directly proportional to its cross-sectional area. The cross-sectional area of a circular shaft is given by the formula A = πr^2, where r is the radius of the shaft.

Since the radius is directly proportional to the diameter (r = d/2), we can rewrite the formula as A = π(d/2)^2 = (π/4)d^2.

Therefore, the torque transmitted by a shaft is directly proportional to the square of its diameter.

The weight of a shaft is directly proportional to its volume. The volume of a circular shaft is given by the formula V = πr^2L, where L is the length of the shaft.

Using the same relationship between radius and diameter, we can rewrite the formula as V = π(d/2)^2L = (π/4)d^2L.

Therefore, the weight of a shaft is directly proportional to the square of its diameter.

To calculate the torque to weight ratio, we divide the torque by the weight:

Torque to weight ratio = (torque)/(weight) = [Aτ]/[ALρg] = (d^2)/(d^2Lρg) = 1/(Lρg),

where τ is the torque, ρ is the density of the material, and g is the acceleration due to gravity.

From the equation, we can see that the torque to weight ratio is inversely proportional to the length of the shaft and the density of the material. However, the torque to weight ratio is independent of the diameter of the shaft.

Therefore, the correct answer is option 'B': The torque to weight ratio for a circular shaft transmitting power is directly proportional to the diameter of the shaft.