Centripetal force is not a new force. To move on a circle of radius r at speed v a body needs a net force m v^2 / r pointing at the centre. Something must supply it: tyre friction on a flat road, a component of the road's normal force on a banked road, a component of the string tension for the pendulum.
Banked road. Take bank angle t, tyre grip mu, horizontal radius R. Balancing normal force N and friction f along the slope gives N = m (g cos t + (v^2/R) sin t) and f = m ((v^2/R) cos t - g sin t) (down-slope positive). Grip holds while |f| <= mu N, which gives v_max^2 = R g (sin t + mu cos t)/(cos t - mu sin t) and v_min^2 = R g (sin t - mu cos t)/(cos t + mu sin t) (zero when mu >= tan t). With no friction the ideal speed is sqrt(R g tan t). On a flat road this reduces to v = sqrt(mu g R); the mass cancels everywhere. If mu >= cot t the denominator is not positive and any speed is held (in this simple model).
How the simulator works. The car is a point mass on the cone-shaped road surface in cylindrical coordinates; the constraint force is eliminated exactly, giving N = m (g cos t + rho phi'^2 sin t). The driver asks the tyres for the force that would keep the radius fixed (plus a gentle correction toward the lane centre and a cruise control for speed); the force is capped on the friction circle sqrt(F_slope^2 + F_heading^2) <= mu N. Beyond the cap the car slides, leaves skid marks and drifts off the lane. Integration is 4th-order Runge-Kutta at 240 Hz.
Conical pendulum. For a string of length L at angle t from the vertical, T cos t = m g and T sin t = m v^2/(L sin t), so tan t = v^2 / (g L sin t) and the period is 2 pi sqrt(L cos t / g): independent of the mass. The simulator integrates the full spherical pendulum t'' = phi'^2 sin t cos t - (g/L) sin t, phi'' = -2 t' phi' cos t / sin t, tension T = m (g cos t + L t'^2 + L phi'^2 sin^2 t); launched with other spin it is no longer a circle.
What is simplified. Real tyres have slip curves, load transfer, suspension, aerodynamic downforce and drag; here friction is one coefficient with a circular limit, the car is a point, and rolling resistance and drag are ignored. Friction values are typical, not measured. The string is massless and inextensible with no air drag; tension below zero is flagged because the model is then invalid. The Moon and Mars gravity options change g only.
Checks on this page. The skid and slide-down speeds are found by running the simulator and are compared with the closed forms (the closed forms come from force balance, the simulator from time integration of the constrained equations of motion); the pendulum period is measured from the integrated azimuth and compared with the formula, energy is monitored for a non-circular orbit, and tension is compared with mg / cos t.