Precariously balanced rocks provide constraints on the level of ground motion that could have occurred during the time the rocks have been in their current positions. Field measurements of the quasi-static toppling accelerations of precariously balanced rocks provide information for computer modeling of the response of the rocks to given ground-motion histories. The quasi-static toppling acceleration is determined by the ratio of the quasi-static force through the center of mass whose moment about the rocking point counterbalances that of gravity and the mass of the rock. For an estimate of dynamic toppling peak ground accelerations, we use a finite-difference numerical code to model the dynamic response of rocks to arbitrarily complex acceleration time histories. We test our computational methodology and results using the University of Nevada shake table. The shake-table tests of both idealized rectangular shapes and actual rocks confirm the accuracy of our methodology for estimating dynamic toppling accelerations of precarious rocks for different waveforms representing earthquake ground-motion time histories. A study of statistical variation of the dynamic toppling acceleration for a suite of seismograms provides constraints on the peak ground acceleration on hard-rock sites in Mojave Desert, about 15 km from the San Andreas fault, of about 0.5g for synthetic seismograms with relatively high frequency content, or about 0.4g for seismograms with spectra similar to the recent Izmit, Turkey, earthquake. A major source of uncertainty is the uncertainty in the spectrum of ground motion for large earthquakes. Waveforms with relatively more low frequencies can topple the rocks with lower peak ground accelerations. We describe methods for reducing the uncertainties with further study and more data.