The Relationship Between Power, Speed, and Diameter in Shaft Design

Introduction:
In this question, we are given two shafts, A and B, made of the same material. Both shafts transmit 100 kW of power. However, Shaft A rotates at 250 rpm while Shaft B rotates at 300 rpm. We are asked to determine which shaft has a greater diameter.

Understanding the Relationship:
To solve this problem, we need to understand the relationship between power, speed, and diameter in shaft design. The power transmitted by a shaft can be calculated using the following formula:

Power (P) = Torque (T) x Angular Speed (ω)

Torque is the twisting force applied to the shaft, and angular speed is the rate at which the shaft rotates. The torque can be expressed as:

Torque (T) = Force (F) x Radius (r)

The radius of the shaft is half of its diameter.

Key Points:
- Power transmitted by a shaft is directly proportional to the torque and the angular speed.
- Torque is directly proportional to the force applied and the radius of the shaft.
- Therefore, for a given power and angular speed, the torque is the same for both shafts.
- As torque is directly proportional to the shaft radius, the shaft with a greater diameter will have a greater radius and vice versa.

Calculating the Shaft Diameter:
Since both shafts transmit the same power, we can equate the power formulas for shafts A and B:

Power (A) = Power (B)

Torque (A) x Angular Speed (A) = Torque (B) x Angular Speed (B)

Since the torque is the same for both shafts, we can write:

Angular Speed (A) / Angular Speed (B) = Diameter (B) / Diameter (A)

Given that Angular Speed (A) = 250 rpm and Angular Speed (B) = 300 rpm, we can rearrange the equation to solve for the ratio of diameters:

Diameter (B) / Diameter (A) = Angular Speed (A) / Angular Speed (B)

Diameter (B) / Diameter (A) = 250 rpm / 300 rpm

Diameter (B) / Diameter (A) = 5/6

Therefore, the diameter of shaft B is 5/6 times the diameter of shaft A. Since shaft A has a greater diameter, the correct answer is option 'A'.

Conclusion:
In shaft design, the power transmitted is directly proportional to the torque and angular speed. The torque is directly proportional to the radius of the shaft, which is half of its diameter. Therefore, for a given power and angular speed, the shaft with a greater diameter will have a greater radius and vice versa. In this question, since both shafts transmit the same power, the one with a higher angular speed will have a smaller diameter.