Question

In a projectile loom, the energy stored in a torsion rod just before picking is proportional to \( r^n \). If \( r \) is the radius of the torsion rod, then the value of \( n \) (in integer) is ________________.

Hint:

In torsional systems, energy storage is proportional to the fourth power of the radius. This principle is important in mechanical systems like projectile looms, where large forces need to be stored and released efficiently.

Answer 1

Correct Answer: 4

In projectile looms, a torsion rod is typically used to store and release energy needed for the picking mechanism, which helps in transferring the weft yarn through the shed. The energy stored in the torsion rod is a function of its radius. The relation between the stored energy \( E \) and the radius \( r \) of the torsion rod is generally given by: \[ E \propto r^n \] Where:
- \( E \) is the energy stored in the torsion rod,
- \( r \) is the radius of the torsion rod,
- \( n \) is a constant that represents the power to which the radius is raised.
In torsional systems, the energy stored in the rod is proportional to the fourth power of the radius, meaning that \( n = 4 \). This relationship arises from the physics of rotational energy storage in rods and shafts subjected to torsional forces. When a torsion rod is twisted, the strain energy is proportional to the square of the twist angle and the fourth power of the rod's radius. This is because the torque exerted on the rod is related to the radius, and the displacement (which corresponds to energy) is also proportional to the radius raised to the fourth power. This is a well-established result in the study of torsional dynamics. Thus, in the case of this projectile loom, the value of \( n \) is \( \boxed{4} \).