The radius of the moon's orbit about the Earth is about ##3.6 * 10^8## ##m##. The moon's period is ##2.3 * 10^6## seconds. What is the centripetal acceleration of the moon?

I found ##0.00268m/s^2~~0.003m/s^2##

Centripetal acceleration is given by:

##a_c=v^2/r##

Where ##v## is the linear velocity and ##r## the radius of the circular orbit.

We assume uniform velocity and a circular orbit; velocity will be:

##v="distance"/"time"="circumference"/"period"=(2pir)/T##

Our centripetal acceleration will become:

##a_c=(4pi^2r^cancel(2))/(T^2cancel(r))##

##a_c=(4pi^2r)/(T^2)##

Using your data for the radius and period:

##a_c=(4pi^2*3.6*10^8)/((2.3*10^6)^2)##

##a_c=(39.478*3.6*10^8)/(5.29*10^12)##

##a_c=(39.478*3.6*10^8*10^-12)/(5.29)##

##a_c=(39.478*3.6*10^-4)/(5.29)##

##a_c=(142.1208*10^-4)/5.29##

##a_c=26.86.... *10^-4##

##a_c=0.00268m/s^2##

##a_c~~0.003m/s^2##