 # Torsion Equation Derivation

Torsion equation or torsion constant is defined as the geometrical property of a bar’s cross-section that is involved in the axis of the bar that has a relationship between the angle of twist and applied torque whose SI unit is m4. The torsion equation is given as follows:

$$\frac{T}{J}=\frac{\tau}{r}=\frac{G\Theta}{L}$$

## Torsion equation derivation

Following are the assumptions made for the derivation of torsion equation:

• The material is homogeneous (elastic property throughout)
• The material should follow Hook’s law
• The material should have shear stress proportional to shear strain
• The cross-sectional area should be plane
• The circular section should be circular
• Every diameter of the material should rotate through the same angle
• The stress of the material should not exceed the elastic limit

Consider a solid circular shaft with radius R that is subjected to a torque T at one end and the other end under the same torque.

Angle in radius = $$\frac{arc}{Radius}$$

Arc AB = RӨ = Lγ

$$\gamma = \frac{R\Theta }{L}$$

Where,

A and B: two fixed points on the circular shaft

γ: angle subtended by AB

$$G=\frac{\tau }{\gamma }$$ (Modulus of rigidity)

Where,

𝞃: shear stress

γ: shear strain

$$\frac{\tau }{G}=\Gamma$$ $$∴ \frac{R\Theta }{L}=\frac{\tau }{G}$$

Consider a small strip of radius with thickness dr that is subjected to shear stress.

$${\Gamma }’*2\pi rdr$$

Where,

dr: thickness of the strip

γ: shear stress

$$2\pi {\tau }’r^{2}dr$$ (torque at the center of the shaft)

$$T=\int_{0}^{R}2\pi {\tau }’r^{2}dr$$ $$T=\int_{0}^{R}2\pi \frac{G\Theta }{L}r^{3}dr$$ (substituting for 𝛕’ )

$$T=\frac{2\pi G\Theta }{L}\int_{0}^{R}r^{3}dr$$ $$=\frac{G\Theta }{L}\left [ \frac{\pi d^{4}}{32} \right ]$$ (after integrating and substituting for R )

$$\frac{G\Theta }{L}J$$ (substituting for the polar moment of inertia)

$$∴ \frac{T}{J}=\frac{\tau }{r}=\frac{G\Theta }{L}$$

Above are the steps of Doppler effect derivation. To know more, stay tuned with BYJU’S.

Related Physics articles:

Test your knowledge on Torsion equation derivation