A body has centripetal acceleration with magnitude a such that
a = v ² / R
where v is the body's tangential speed and R is the radius of the circular path the body takes.
Convert the child's angular speed ω into linear/tangential speed. Assume angular speed is measured in rad/s and tangential speed in m/s. For every 2π rad that he revolves around his mother, the child travels a distance of 2πR m, so that
ω = (ω rad/s) • (2πR/(2π) m/rad) = Rω = v
Then the child's acceleration is
a = (Rω)² / R = Rω ²
When the mother pulls her arms in, the distance R gets halved and changes to R/2, so that the child's new acceleration is
a = (R/2 • ω)² / (R/2) = (1/4 • (Rω)²) / (1/2 • R) = 1/2 Rω ²
so the child's centripetal acceleration decreases by a factor of 2.
