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The dimensional formula of effective torsional constant of spring is__________
A) ${M^1}{L^2}{T^{ - 3}}$
B) ${M^1}{L^2}{T^{ - 2}}{A^{ - 2}}$
C) ${M^1}{L^2}{T^{ - 2}}$
D) ${M^0}{L^0}{T^0}$

Last updated date: 14th Jun 2024
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Hint: We know that in linear motion, friction exists between the two surfaces which is a contact. Similarly, in rotational motion, torsion plays the same role as friction.

Complete step by step solution:
A torsional constant is a physical property of a material. It is mostly used to describe metal beams and is denoted by the variable "C." When a torque is applied to a metal beam, it will twist a certain angle. The angle that the beam twists is dependent on the rigidity, length, and torsional constant. A torsional constant of a beam depends on not only the beam material but also the beam shape.
The torsional constant is a ratio of restoring couples per unit twist.
That is, ${\tau _r} = C\theta $
Where $C$ is the torsion constant of spring
$\theta $ is the deflection in the coil or angle suspended.
${\tau _r}$is the restoring torque.
Then torsion constant of spring is given by $C = \dfrac{{{\tau _r}}}{\theta }$
Apply dimensional formula,
$\theta $ is dimensionless quantity therefore, ${\tau _r} = \overrightarrow r \times \overline F $

$\therefore$ The dimensional formula for torsion constant is, $\left[ {M{L^2}{T^{ - 2}}} \right]$. The correct answer is option (C).

Additional information:
The turning effect of a force about an axis of rotation is called torque or moment of force and it is a measure of the rotational effect of the force.
A pair of equal and opposite forces with different lines of action acting on a body is known as a couple. A couple produces rotation without translational motion. For example, our fingers apply a couple to the lid while opening the lid of a bottle. Also, we apply a couple using our fingers to twist wire or thread.

(i) The torsional constant is a ratio of restoring couples per unit twist.
(ii) In linear motion we have force, and in the case of rotational motion, torque is applied.