# of atoms Number of atoms at time to decay 1. Radioactive decay : constant .)(t-to) N () = Nito)e a . the half-life, tip, of a radionuclide is defined as the amount of time needed for half of the atoms in a sample to decay, ie . . N (t) = N(to) 2 when t = to + ti/2. what is the relationship between b. if time is being recorded with units of seconds, what are the units of 2? C. Show that N( t) given above is a solution to the differential equation dNct) = - X Nct). at2. Harmonic Motion of a mass+ Spring. )( (t ) = A cos (J Em t+$) phase displacement shiff maximum of the mass displacement Cindependent of time ) at any time , t oscillation freq. W = \\ 1/m a. How long does it take for the mass to complete one oscillation? b . show that C ( + ) , above , is a Solution to the differential equation equation m d'x = - kx . dt 2 C. velocity is the change in displacement with respect to time, ice , v = dx dt Write an expression for the Velocity of the mass as a function of time . d . consider two systems , huinom, each with Spring constant , K . The mass in the first system is m,, first system is irl,, and the second is mz . algebraically Compare the oscillation frequencies , w, E w 2 ? 3 / 4 for the systems. if m, > m, , which one oscillates at a higher frequency