Characteristic precessions of spherical orbit around a rotating braneworld black hole

We study the orbital dynamics and relativistic precession effects in the spacetime of rotating braneworld black holes within the Randall-Sundrum framework. For test particles on spherical orbits, we analyze three conserved quantities-energy, angular momentum, and Carter constant-and examine how the innermost stable spherical orbit depends on the tidal charge and orbital inclination. Compared to Kerr black holes, braneworld corrections significantly modify both nodal and periastron precession frequencies: positive tidal charges suppress precession rates, while negative charges enhance them. For stationary gyroscopes, we calculate the Lense-Thirring precession frequency and demonstrate its sensitivity to the tidal charge, black hole spin, and gyroscope orientation. Our results show that a positive tidal charge weakens frame-dragging effects even as it enhances gravitational attraction-offering a distinctive signature of extra-dimensional gravity. These results have important implications for astrophysical observations, including accretion disk behavior, stellar orbital dynamics, and gravitational wave detection. The modified orbital and gyroscopic precession provide new ways to test braneworld gravity in strong-field regimes.