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

Hint: 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.