A block suspended from a vertical spring is in equilibrium, producing an extension x in the spring. Determine the length of an equivalent simple pendulum (i.e., a pendulum having the same frequency as that of the block).

Step-by-step solution

Let the mass of the block be m and the spring constant be k . In equilibrium, the spring force balances the weight of the block: kx = mg m k = x g The time period of oscillation for the block-spring system is given by: T = 2 m k Substituting the value of m k , we get: T = 2 x g The time period of a simple pendulum of length L is given by: T = 2 L g Since the frequencies (and hence time periods) are equal, we equate the two expressions: 2 x g = 2 L g Squaring both sides, we obtain: L = x