The correct option is (a) \(\frac{\rho A}{3}\)

The explanation is: Using the Timoshenko beam theory applied to natural vibrations, mode shape is approximated using the cubic Hermite polynomials \(\psi_i^e\) and \(\psi_j^e\). The first element of a mass matrix is \(M_{ij}^{11} = \int_{x_a}^{x_b} \rho A \psi_i^e \psi_j^e\) dx, where x is the length of the element. For the 1^st element, using appropriate values of \(\psi_i^e\) and \(\psi_j^e\), the term \(M_{ij}^{11}\) reduces to \(\frac{\rho A}{3}\), where ρ is the density of the beam material, and A is the cross-section area of the beam.