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A uniform bar of circular cross section (radius r, length L, modulus of rigidity eta) is clamped at its upper end. What is the moment of the couple required to twist its lower (free) end through an angle of 90 degrees (pi/2 radians)?

1. pi²*eta*r⁴/(4L)
2. pi*eta*r⁴/(2L)
3. pi²*eta*r⁴/(2L)
4. pi*eta*r⁴/(4L)

Correct answer: pi²*eta*r⁴/(4L)

Solution

For a cylinder of radius r, length L and rigidity modulus eta, the couple needed to produce a twist phi (in radians) is C = (pi*eta*r⁴)/(2L) * phi. This comes from integrating the shear couple over concentric shells. For a twist of 90 deg, phi = pi/2 rad, so C = (pi*eta*r⁴)/(2L) * (pi/2) = pi²*eta*r⁴/(4L).

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