compare the weight of equal lenghts of hollow and solid shafts to transmitt a torque T for the same maximum shear stress. For hollow shaft, the inner and outer diameters have relationship Di=2/3 and Do=2/3DH.

When comparing the weight of hollow and solid shafts designed to transmit the same torque while maintaining the same maximum shear stress, we need to delve into the mechanics of materials and the geometry of the shafts. The relationship between the inner diameter (Di) and outer diameter (Do) of the hollow shaft, as given, is crucial for our calculations.

Understanding Shear Stress and Torque

Shear stress (\( \tau \)) in a shaft is defined as the torque (\( T \)) divided by the polar moment of inertia (\( J \)) and the radius (\( r \)) of the shaft. The formula for shear stress is:

τ = T / J * r

Polar Moment of Inertia

The polar moment of inertia is a measure of an object's resistance to torsion and is calculated differently for solid and hollow shafts:

• For a solid shaft: J_s = (π/32) * D^4
• For a hollow shaft: J_h = (π/32) * (D_o^4 - D_i^4)

Calculating Dimensions for the Hollow Shaft

Given the relationships for the hollow shaft, where \( D_i = \frac{2}{3}D_h \) and \( D_o = \frac{2}{3}D_h \), we can express the outer diameter in terms of the maximum diameter \( D_h \). This means:

• Let \( D_h = D \) (the maximum diameter).
• Then, \( D_i = \frac{2}{3}D \) and \( D_o = \frac{2}{3}D \).

Weight Considerations

The weight of the shafts is proportional to their volume and density. The volume of a solid shaft is:

V_s = (π/4) * D^2 * L

For the hollow shaft, the volume can be calculated as:

V_h = (π/4) * (D_o^2 - D_i^2) * L

Substituting the values of \( D_o \) and \( D_i \):

V_h = (π/4) * \left( \left(\frac{2}{3}D\right)^2 - \left(\frac{2}{3}D\right)^2 \right) * L

This simplifies to:

V_h = (π/4) * \left( \frac{4}{9}D^2 - \frac{4}{27}D^2 \right) * L

Calculating this gives us the volume of the hollow shaft in terms of \( D \) and \( L \).

Comparative Analysis of Weights

To compare the weights, we can express the weight of both shafts as:

• Weight of solid shaft: W_s = ρ * V_s
• Weight of hollow shaft: W_h = ρ * V_h

Where \( ρ \) is the density of the material. Since the volume of the hollow shaft is less than that of the solid shaft, we can conclude that:

W_h < W_s

Conclusion

In summary, for shafts of equal length transmitting the same torque and designed to maintain the same maximum shear stress, the hollow shaft will generally weigh less than the solid shaft. This is due to the reduced volume of material in the hollow shaft while still providing adequate strength and rigidity for torque transmission. This principle is particularly beneficial in engineering applications where weight savings are critical, such as in automotive or aerospace designs.