A particle is said to be executing simple harmonic motion, if the acceleration ‘a’ produced in it satisfies the following two conditions :
(i) ‘a’ is directly proportional to the displacement (say y) from the mean position. i.e. a ∝ y.
(ii) ‘a’is directed towards mean position i.e. acts opposite to the direction in which y increases.
i.e. Mathematically a=–ω2y ...(i)
where ω= angular frequency.
(a) a=0.7x does not satisfy eqn. (i), so it does not represent simple harmonic motion.
(b) a=–200x2 , does not satisfy equation (i) hence it does not represent simple harmonic motion.
(c) a = –10x, here x = displacement. It satisfies equation (i), so it represents simple harmonic motion.
(d) a=100x3 does not satisfy equation (i), hence it does not represent simple harmonic motion.