Final answer:
The situation described deals with the application of torque to a disc which results in angular deformation or twist, subject to the laws of elasticity and torsion. The twist angle θ(r) can be calculated based on the modulus of rigidity, the applied torque, and the geometry of the disc and rod through the elasticity theory of torsion.
Step-by-step explanation:
The scenario described involves the physical concept of torque and involves a principle in solid mechanics called twist, which is particularly related to the elasticity theory. When a torque is applied to the rod attached to a disc, the result is a twisting of the disc around the axis of the rod. This creates a twisting angle θ(r) that varies with the radial distance from the axis, r. The expectation, based on the given boundary condition, is that θ(r=r0)=0, which means at the edge of the disk where it is fixed, there is no twist.
The angular displacement of points in the disc as a function of radius can be calculated using the theory of elastic torsion. The calculations would include the modulus of rigidity (also known as the shear modulus) of the material from which the disc is made. The formula to calculate the twist per unit length (angle of twist per unit length of the rod) is θ'=T/(JG) where T is the applied torque, J is the polar moment of inertia of the section about the axis of twist, and G is the modulus of rigidity of the material.
However, this calculation assumes a uniform torque applied along the length of the rod, and the resulting twist is linear from the point of application of the torque to the fixed point. For a disk, calculating θ(r) as a function of radial distance would involve considering the distribution of shear stress across the disk's cross-section, which can be complex. Detailed calculations might rely on numerical methods or further simplifying assumptions based on the disk geometry and material.