Two shafts in torsion will have equal strength if

General
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two shafts in torsion will have equal strength if

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Two shafts in torsion will have equal strength if

Answer: In the context of shafts under torsion, the concept of “equal strength” pertains to their ability to withstand the same amount of torque (twisting force) without failing. Two shafts in torsion will have equal strength under the following conditions:

  1. Same Material Properties:

    • They must be made from materials that have the same shear strength and elastic modulus.
    • Shear strength is the maximum shear stress the material can withstand before yielding.
    • The elastic modulus, specifically the shear modulus (often denoted as ( G )), indicates the material’s response to shear stress.

  2. Equal Polar Moment of Inertia (Polar Second Moment of Area):

    • The shafts must have the same polar moment of inertia, ( J ).
    • ( J ) is a geometric property that indicates how the cross-sectional area is distributed around the axis of rotation.
    • For a circular shaft, the polar moment of inertia is given by:
      J = \frac{\pi d^4}{32}
      where ( d ) is the diameter of the shaft.

  3. Equal Cross-sectional Area and Shape:

    • The shafts should have the same cross-sectional shape and size.
    • This ensures that the stress distribution across the shafts’ cross-sections is the same.

  4. Same Length (if considering stiffness):

    • If we consider the deflection or angle of twist as a criterion for strength, the shafts must also be of equal length ( L ).
    • The angle of twist ( \theta ) for a shaft of length ( L ) under a torque ( T ) is given by:
      \theta = \frac{TL}{GJ}
      This implies that to have the same angle of twist, ( L ) and ( G ) should be equal, complementing the conditions above.

Deriving the Condition for Equal Strength:

To further elucidate, let’s look at the maximum shear stress ( \tau_{\text{max}} ) induced in each shaft:

\tau_{\text{max}} = \frac{T \cdot r}{J}

where:

  • ( T ) is the applied torque,
  • ( r ) is the radius of the shaft,
  • ( J ) is the polar moment of inertia.

For equal strengths, ( \tau_{\text{max}} ) should be the same for both shafts. Given the relationship above, this will be true if the following conditions are maintained:

  1. The material properties (shear strength and modulus) are the same.
  2. The polar moments of inertia are the same.

Conclusion:

Therefore, to ensure two shafts have equal strength in torsion, they should be made from the same material (with equal shear strength and modulus), have the same polar moment of inertia, and have equal cross-sectional properties. When all these conditions are met, both shafts will sustain the same amount of torque without yielding or failing, hence having equal strength.