Derivation of centripetal acceleration a = rω2 a = v2
___ r
Since v = rω: a = (rω)2
_______ r
Multiplying out the bracket: a = r2ω2
______ r
Cancelling r : a = rω2
Derivation of formulae for centripetal force From F = ma and a = v2
___ r :
F = ma F = m
( v2 ___
F = mv2
_____ r
Using F = ma and a = rω2: F = ma
F = mrω2 QUESTIONS AND ANSWERS
7. A 0.3 kg mass is spun in a circle at a speed of 6 m s–1. The length of string to which it is attached is 1.5 m. Calculate:
(i) the centripetal acceleration of the mass
(ii) the tension in the string. (Ignore gravity.)
Solution (i) a = v2
___ r = (6)2
_____ 1.5 = 24 m s–2
(ii) F = ma = (0.3)(24) = 7.2 N You may now complete Exercise 8C (page 81).
8. Calculate the radius of circle, acceleration and centripetal force involved if a 5 kg particle is travelling at 10 m s–1 in a circle with an angular velocity of 3 rad s–1.
