A 21-mm-diameter steel bar is to be used as a torsion spring. If the torsional stress in the bar is not to exceed 109 MPa when one end is twisted through an angle of 15°, what must be the length of the bar?

2m

Explanation:

The modulus of rigidity was omitted in the question and I'm using the modulus of rigidity of structural steel (G=79 GPa)

For the shaft, the maximum moment

T=π*rho*d³/16

Where π=3.14

Rho = torsional stress=109MPa

d=diameter of of shaft = 21mm

Inserting values of π, rho and d

T = 198,230.54Nmm

Polar moment of the shaft is given by

J = π D⁴ / 32

Also torsional moment (in radian) of the shaft is given as

α = L * T / (J * G)

α = 32 * L*T / (G * π * D⁴)

And in degrees,

αdegrees = 584 * L*T / (G*D⁴)

Where L=length of shaft and G is modulus of rigidity

L=α*d⁴*G/584*T

Where α=15⁰, d=0.021m, G=79*10⁹Pa, T=198.23Nm

Substituting values for α, d, G and T

L=1.99m approximately 2m