The answer to each of the questions is a single digit integer ranging from 0 to 9.

A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is U(x) = K|x|³, where K is a positive constant. If the amplitude of oscillation is a, then its time period T is proportional to a^–p/q. Find the value of p + q.

3

        U(x) = K|x|³

        \ [K] =

        Now time period may depend on T µ (mass)^x (amplitude)^y (K)^z

        Or [M⁰L⁰T] = [M]^x[L]^y[ML^–1T^–2]^z = [M^x+zL^y–zT^–2z]

        Equating the powers, we get – 2z = 1 or z =

        y – z = 0 or y = z =

        \ T µ (amplitude)–1/2 or T µ (a)–1/2 or T µ .

        On comparing we get, p = 1, q = 2.

        \ p + q = 1 + 2 = 3