A spring of force constant k is cut into lengths of ratio 1 : 3 : 4 . They are connected in series and the new force constant is k' . Then they are connected in parallel and force constant is k'' . Then k' : k'' is

Step-by-step solution

Let the total length of the spring be L . The spring is cut into three parts with lengths l₁ = 1 8 L , l₂ = 3 8 L , and l₃ = 4 8 L . The force constant of a spring is inversely proportional to its length, k 1 l . Thus, k₁ = 8k , k₂ = 8 3 k , and k₃ = 2k . When connected in series, the equivalent force constant k' is given by 1 k' = 1 k₁ + 1 k₂ + 1 k₃ = 1 8k + 3 8k + 1 2k = 1+3+4 8k = 8 8k = 1 k . Therefore, k' = k . When connected in parallel, the equivalent force constant k'' is given by k'' = k₁ + k₂ + k₃ = 8k +