Estimating accurately the natural frequencies of electrically actuated carbon nanotubes (CNTs) has been the center of research attention over the past few years. Despite this important topic, a robust knowledge is still missing to understand the role of various physical parameters that affect the natural frequencies, such as the stretching of doubly-clamped CNTs, the DC electrostatic force, and the curvature of CNTs with slacks. In this investigation, we use a 2D nonlinear curved beam model (arch) to simulate the coupled in-plane and the out-of-plane motions of a CNT with curvature (slack). We calculate the variation of its natural frequencies and mode-shapes with the level of slack and the DC electrostatic load. Toward this, we derive a reduced-order model using a multi-mode Galerkin procedure based on the mode shapes of the straight unactuated CNT. We calculate the natural frequencies of the slacked CNT for a given voltage by substituting the static solution into the Jacobian of the reduced-order-model and then finding the corresponding eigenvalues. We show various scenarios of mode crossing and mode veering as the levels of slack and DC load are varied.