Question

Two shafts A and B are made up of the same material. The diameter of the shaft A is twice as that of shaft B. The power transmitted by the shaft A will be 'X' times of shaft B. What is the value of 'X'?

• A. 2
• B. 4
• C. 8
• D. 16

Hint:

Torsional strength and power capacity of a solid shaft scale with the cube of the diameter ($d^3$).
Therefore, doubling the diameter increases the power transmission capacity by a factor of $2^3 = 8$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This problem compares the power transmission capacity of two solid circular shafts made of the same material under torsion, where one shaft has twice the diameter of the other.

Step 2: Key Formula or Approach:
The power ($P$) transmitted by a shaft rotating at speed $N$ is directly proportional to the torque ($T$):
\[ P = \frac{2\pi \cdot N \cdot T}{60} \]
From the torsion equation, the maximum torque capacity of a solid circular shaft is limited by its maximum allowable shear stress ($\tau_{\text{max}}$):
\[ T = \frac{\pi}{16} \cdot \tau_{\text{max}} \cdot d^3 \]
where $d$ is the shaft diameter.

Step 3: Detailed Explanation:

• Since both shafts are made of the same material, they have the same allowable shear stress ($\tau_{\text{max}}$).

• Assuming both shafts rotate at the same speed ($N$), the power transmission capacity is directly proportional to the torque carrying capacity:
\[ P \propto T \propto d^3 \]

• Let $d_A$ and $d_B$ be the diameters of shafts A and B, respectively. We are given:
\[ d_A = 2 \cdot d_B \]

• Take the ratio of the power transmitted by shaft A to shaft B:
\[ \frac{P_A}{P_B} = \left(\frac{d_A}{d_B}\right)^3 \]

• Substitute the diameter relationship:
\[ \frac{P_A}{P_B} = (2)^3 = 8 \]

• This means $P_A = 8 \cdot P_B$, so the value of the multiplier 'X' is $8$.

Step 4: Final Answer:
The power transmitted by shaft A will be 8 times that of shaft B.