Two light spring of force constants `k_(1) and k_(2)` and a block of mass m are in the line AB on a smooth horizontal table such that one end of each spring is fixed on right supports and the other end is free as shown in figure

The distance CD between the free ends of the spring is `60 cm`. If the block `(k_(1) = 1.8 N//m, k_(2) = 3.2 N//m and m = 200g)`. Is the motion simple harmonic?

Explanation

When the block touches `D`, it will compress the spring and its `KE` will be converted into elastic energy of the spring. The compressed spring will push the block to `D` with same speed so time taken by the block to move from D towards B and back to D will be
`t_(1) = (T_(2))/(2) = pi sqrt((m)/(k_(1))) = pi sqrt((0.2)/(3.2)) = (pi)/(4) s`
Moreover during complete oscllation between `A and b` block moves the distance `CD` twice with uniform velocity `v` once from `C to D` and again from D to C. So
`T_(1) = (2 L)/(v) = (2 xx 0.6)/(1.2) = 1 s`
`T = t_(1) + t_(2) + t_(3) = pi ((1)/(3) + (1)/(4)) + 1 = 2.82 s`
Now a motion is simple harmonic only and only if throughout the motion `F = - k x`. Here between `C and D, F = 0 (asv = consrtant)`, the motion is not simple harmonic but oscillatory.