Question

Angle of twist of a shaft of diameter 'd' is inversely proportional to

• A. d
• B. $d^2$
• C. $d^3$
• D. $d^4$

Hint:

Remember that torsional stiffness is proportional to $J \propto d^4$, meaning doubling the diameter reduces the angle of twist to $\frac{1}{16}\text{th}$ of its original value.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question asks for the relationship between the angle of twist of a circular shaft under torsion and its diameter ($d$).

Step 2: Key Formula or Approach:
From the torsion equation:
\[ \frac{T}{J} = \frac{G \theta}{L} \]
where:
$T$ is the applied torque,
$J$ is the polar moment of inertia,
$G$ is the shear modulus (modulus of rigidity),
$\theta$ is the angle of twist, and
$L$ is the length of the shaft.

Step 3: Detailed Explanation:

• Rearranging the torsion equation to solve for the angle of twist ($\theta$):
\[ \theta = \frac{T L}{G J} \]

• For a solid circular shaft of diameter $d$, the polar moment of inertia ($J$) is given by:
\[ J = \frac{\pi d^4}{32} \]

• Substituting $J$ into the angle of twist equation:
\[ \theta = \frac{32 T L}{\pi G d^4} \]

• Keeping $T$, $L$, and $G$ constant, we observe that the angle of twist $\theta$ is inversely proportional to the fourth power of the diameter:
\[ \theta \propto \frac{1}{d^4} \]

Step 4: Final Answer:
The angle of twist of a shaft is inversely proportional to $d^4$.