Half of a Spring.? (a) Suppose you cut a massless ideal spring in half. If the full spring had a force constant ?k?, what is the force constant of each half, in terms of k?? (?Hint:? Think of the original spring as two equal halves, each producing the same force as the entire spring. Do you see why the forces must be equal?) (b) If you cut the spring into three equal segments instead, what is the force constant of each one, in terms of ?k??
Solution 44E Step 1: The spring constant is an inherent property of a spring. Constant by meaning itself is an invariant. But when you cut a spring into several parts, it is not the same spring any more. As, the mass of the spring also gets changed regardless of the invariance of mass. Step 2: When the spring is divided into 2 equal parts, the spring constant changes as follows, Hooke’s law states that, F = kx---------------------(1) When the spring gets divided into 2 equal parts, it can be treated as 2 springs in a series connection of equal spring constants each. So, the formula says that, 1 1 1 K = K + K -------------------(2) old 1 2 But the spring constants of the two new springs must be same. So, 1 1 1 1 1 2 K = K + K = K + K = K old 1 2 1 1 1 K 1 K 0ld= 2 K = 2K . 1 old Where k i1 the spring constant of the two new springs. It gets doubled to produce the same force.