  We present exact results for the classical version of the Out-of-Time-Order
Commutator (OTOC) for a family of power-law models consisting of $N$ particles
in one dimension and confined by an external harmonic potential. These
particles are interacting via power-law interaction of the form $\propto
\sum_{\substack{i, j=1 (i\neq j)}}^N|x_i-x_j|^{-k}$ $\forall$ $k>1$ where $x_i$
is the position of the $i^\text{th}$ particle. We present numerical results for
the OTOC for finite $N$ at low temperatures and short enough times so that the
system is well approximated by the linearized dynamics around the many body
ground state. In the large-$N$ limit, we compute the ground-state dispersion
relation in the absence of external harmonic potential exactly and use it to
arrive at analytical results for OTOC. We find excellent agreement between our
analytical results and the numerics. We further obtain analytical results in
the limit where only linear and leading nonlinear (in momentum) terms in the
dispersion relation are included. The resulting OTOC is in agreement with
numerics in the vicinity of the edge of the "light cone". We find remarkably
distinct features in OTOC below and above $k=3$ in terms of going from non-Airy
behaviour ($1<k<3$) to an Airy universality class ($k>3$). We present certain
additional rich features for the case $k=2$ that stem from the underlying
integrability of the Calogero-Moser model. We present a field theory approach
that also assists in understanding certain aspects of OTOC such as the sound
speed. Our findings are a step forward towards a more general understanding of
the spatio-temporal spread of perturbations in long-range interacting systems.
