Past exam of the physics B course of the University of Cambridge / 2018 / Paper 1 / Section C / C12 / Torsional waves / Solution

The torque difference. Let $\varphi (x,t)$ now be the local angular displacement, and take $G$ and the section constant. The local twist per unit length is ${\partial }_x\varphi$, so the transmitted torque is $C(x,t)=G{J}_p{\partial }_x\varphi$. Across a short slice,Thusto first order in slice length.

The slice's angular acceleration. The slice's axial moment of inertia is $\Delta I=\rho J_p\Delta x$. Its net torque is the difference of the end torques, soCanceling the common geometry gives the torsional waves in a solid circular rod,For the given steel, $c_t\simeq 3.16\times 10^3\mathrm{m}{\mathrm{s}}^{-1}$. The radius cancels because both torsional stiffness and slice inertia contain the same $J_p$. This ideal one-dimensional wave model neglects dissipation and additional elastic modes.