Question

Determine the relationship between the torque N and the torsion angle $\phi$ for the tube whose wall thickness $\Delta r$ is considerably less than the tube radius.
A. $N = \dfrac{{2\pi {r^3}\Delta r\phi }}{{3l}}G$
B. $N = \dfrac{{3\pi {r^3}\Delta r\phi }}{l}G$
C. $N = \dfrac{{2\pi {r^3}\Delta r\phi }}{l}G$
D. None of the above

Answer 1

To solve this question, we have to know about torque. We know that, A torque angle, otherwise called a dihedral angle, is framed by three successive bonds in an atom and characterized by the point made between the two external bonds. The foundation of a protein has three distinctive twist points.

Complete step by step answer:
We know that, keeping the lower end of the hollow tube fixed, its upper end is twisted by angle $\phi$ by applying a force F. Due to the twist, a shear stress is generated between the lower end and upper end of the tube. Thus, we can say, the point A is displaced to A’ due to the force such that, $AA' = dx$
Now. From sector, AOA’, $AA' = r\phi$
Also we can say, from the sector,$ABA'$, $AA' = 1\theta$
Therefore, $\theta = \dfrac{{r\phi }}{l}$
Tangential stress equal to, force upon area. Which is equal to, $\dfrac{F}{{dx\Delta r}}$
Therefore, shear modulus,
G = \dfrac{{stress}}{\theta } = \dfrac{{\dfrac{F}{{\Delta rdx}}}}{{\dfrac{{r\phi }}{l}}} \\\ \Rightarrow F = \dfrac{{G\phi r}}{l}\Delta rdx \\\
Moment of force,
$dM = Fr = \dfrac{{G\phi {r^2}}}{l}\Delta rdx$
So, we can say, the total restoring torque on the annual surface,
N = \int {dM} \\\ \Rightarrow N= \dfrac{{G{r^2}\phi }}{l}\Delta r\int {dx} \\\ \therefore N = \dfrac{{2\pi G{r^3}\phi }}{l}\Delta r \\\
Hence,option C is correct.

Note: We also have to know that, S.I unit of torque is Newton- meter. We have to keep that in our mind. We have calculated here the torsion angle which is denoted by$\phi$. We know that, a power that produces or will in general create turn or twist a vehicle motor conveys force to the drive shaft likewise: a proportion of the adequacy of such a power that comprises of the result of the power and the opposite separation from the line of activity of the power to the pivot of revolution.