In this paper, a new fifth-order finite volume central weighted essentially non-oscillatory (CWENO) scheme is proposed for solving hyperbolic conservation laws on staggered meshes. The high-order spatial reconstruction procedure using a convex combination of a fourth degree polynomial with two linear polynomials (in one dimension) or four linear polynomials (in two dimensions) in a traditional WENO fashion and a time discretization method using the natural continuous extension (NCE) of the Runge-Kutta method are applied to design this new fifth-order CWENO scheme. This new finite volume CWENO scheme uses the information defined on the same largest spatial stencil as that of the same order classical CWENO schemes [37, 46] with the application of smaller number of unequal-sized spatial stencils. Since the new nonlinear weights are adopted, the new finite volume CWENO scheme could obtain the same order of accuracy and get smaller truncation errors in $L^1$ and $L^∞$ norms in smooth regions, and control the spurious oscillations near strong shocks or contact discontinuities. The new CWENO scheme has advantages over the classical CWENO schemes [37, 46] on staggered meshes in its simplicity and easy extension to multi-dimensions. Some one-dimensional and two-dimensional benchmark numerical examples are provided to illustrate the good performance of this new fifth-order finite volume CWENO scheme.