Infinite springs with force constants k , 2 k , 4 k a n d 8 k … . . respectively are connected in series. The effective force constant of the spring will be

Step-by-step solution

Effective force constant is equal to the reciprocal of the sum of individual force constant, hence 1 k e = 1 k 1 + 1 k 2 + 1 k 3 + … Given, k 1 = k , k 2 = 2 k , k 3 = 3 k , … . . ∴ 1 k e = 1 k + 1 2 k + 1 4 k + 1 8 k + … The given series a geometric progression series, hence sum is S ∞ = a 1 - r where a is first term of series and r the common difference. ⇒ 1 k e = 1 k × 1 1 - 1 2 = 2 k ⇒ k e = k 2