Question

A mother spins her son in a circle of radius R at angular

Answer 1

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.