Final answer:
The confusion stems from the units of radians being dimensionless, which means they cancel out, leaving the potential energy of a torsion spring in joules (J). Radians measure the angle of twist but do not contribute a separate dimension to the units. The mechanical energy, a combination of kinetic and potential energy, remains constant in a frictionless system.
Step-by-step explanation:
The confusion arises from misunderstanding the units of angle measurement in the formula for the energy stored in a torsion spring. When calculating the potential energy (U) of a torsion spring, the formula is U = 1/2kφ² where k is the spring constant with units of Nm/rad, and φ is the angle of twist in radians. Since radians are dimensionless (they do not have units of their own, as they represent the ratio of the arc length to the radius of a circle), the radian units in the formula effectively cancel out, leaving the energy in joules (J), which is the correct SI unit for energy.
The potential energy of any spring, including torsion springs, is dependent on Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement or deformation. In the case of a torsional system, this deformation is an angular displacement, and the force is a moment or a torque. The work done to compress or stretch a spring is the same concept used in determining the potential energy stored in the spring, represented as PEs = 1kx² for linear springs, where x is the linear displacement.
Remembering that mechanical energy is the sum of kinetic energy (KE) and potential energy (PE) for a conservative force system, we can see that the total mechanical energy remains constant in a frictionless system. This conservation of energy is foundational to understanding how energy is stored and released in systems like springs, whether they are linear or torsional.