Question

A homogeneous shaft PQR with fixed supports at both ends is subjected to a torsional moment \( T \) at point \( Q \), as shown in the figure. The polar moments of inertia of the portions \( PQ \) and \( QR \) of the shaft with circular cross-sections are \( J_1 \) and \( J_2 \), respectively. The torsional moment reactions at the supports \( P \) and \( R \) are \( T_P \) and \( T_R \), respectively. \includegraphics[width=0.5\linewidth]{34image.png} \text{ If \( T_P / T_R = 4 \) and \( J_1 / J_2 = 2 \), the ratio of the lengths \( L_1 / L_2 \) is:}

• A. \( 0.50 \)
• B. \( 0.25 \)
• C. \( 4.00 \)
• D. \( 2.00 \)

Hint:

When dealing with torsional moments in beams or shafts, it's crucial to understand the inverse relationship between the polar moment of inertia and the length of the segment with respect to the torsional stiffness and moment distribution.

Answer 1

The Correct Option is A

Step 1: Establish the relationship between torsional moments and shaft properties. Given that \( T_P / T_R = 4 \) and \( J_1 / J_2 = 2 \), we know from the theory of torsional moments that: \[ \frac{T_P}{T_R} = \frac{J_1 \cdot L_2}{J_2 \cdot L_1}. \] Plugging in the given ratios: \[ 4 = \frac{2 \cdot L_2}{L_1}. \] Step 2: Solve for \( L_1 / L_2 \). Rearrange the equation to solve for \( L_1 / L_2 \): \[ 4L_1 = 2L_2 \quad \Rightarrow \quad \frac{L_1}{L_2} = \frac{1}{2}. \]