Centripetal acceleration, Ac = V^2 / R.
V = tangential speed
R = radius of the orbit
V = 2πR / T
R = 240,000 miles
T = 28 days [ i am correcting what I think is an error, because the moon orbits the earth every 28 days]
V = 2π(240,000miles)*(1,609m/miles)/(28days*24h/day*3600s/h)
V = 1,002.9m/s
Ac = V^2 / R = 0.0026 m/s^2
The force that cauces this acceleration is the gravity of our planet, as stated Lord Isaac Newton.