Many modern-day applications involve transport of objects suspended through cables such as in overhead cranes or landing of rovers on the Martian surface. Any undesired oscillation of the payload has the potential risk of instability, and the problem of damping such oscillation and stabilizing the payload at the desired length is the control objective of this article. The system is modeled as a variable length pendulum (VLP), which comprises a payload suspended via a string wrapped around a pulley. The length of the pendulum is varied using clockwise/counterclockwise rotation of the pulley through torque applied by a motor. For a known payload mass, a nonlinear control design is first presented that guarantees asymptotic stability of the desired equilibrium with limited state measurements. The design is then modified for it to handle significant uncertainty in payload mass. The effectiveness of both designs is validated in simulations.