Which of the following represents the polar modulus of the hollow shaft?
[ If Do = External diameter and Di = Internal diameter ]

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RRB JE ME 22 Apr 2025 Shift 2 CBT 2 Official Paper
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  1. [ 16π/Do ] [ \({Do}^4 - {Di}^4\) ]
  2. [ π/16Do ] [ \({Do}^3 - {Di}^3\) ]
  3. [ π/16Do ] [ \({Do}^4 - {Di}^4\) ]
  4. [ 16/πDo ] [ \({Do}^4 - {Di}^4\) ]

Answer (Detailed Solution Below)

Option 3 : [ π/16Do ] [ \({Do}^4 - {Di}^4\) ]
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Detailed Solution

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Explanation:

The polar modulus is a geometric property of the cross section of a shaft that is used to calculate the strength of the shaft in torsion. For a hollow shaft, the polar modulus is derived from the external and internal diameters.

Polar Modulus of a Hollow Shaft:

The polar modulus (Zp) for a hollow circular shaft is given by the formula:

Zp = J / R

where J is the polar moment of inertia, and R is the outer radius of the shaft.

The polar moment of inertia (J) for a hollow circular shaft is:

J = \(\frac{\pi}{32}\) × \({Do}^4 - {Di}^4\)

where Do is the external diameter and Di is the internal diameter.

The outer radius (R) is half of the external diameter:

R = Do / 2

Now, substituting J and R into the formula for the polar modulus, we get:

∴ Zp = \(\frac{π}{16}\)Do × [ \({Do}^4 - {Di}^4\)

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