A particle of mass m is executing oscillation about origin on x - axis. Its potential energy is U(x)=K|X|n. If
the time period T is function of its mass, amplitude(a) and k; find the value of n for Tαa−12.
[K]=[U][X]n=[ML2T−2][Ln]=[ML2−nT−2]
T∝(mass)X(amp)Y(K)Z
[M0L0T]=[M]X[L]Y[ML2−nT−2]Z
= [MX+ZLY+2Z−nZT−2Z]
So, -2Z = 1 or Z=−12 ................................(i)
and, X + Z = 0 or X=−Z=12 ........................(ii)
and, Y + 2Z-nZ = 0 orY=1−n2[from(i) and (ii)]
So, T∝(mass)12(amp)1−n2(K)−12
Given T∝(amp)−12
So 1−n2=−12 ⇒n=3