A block of mass M is attached to the free end of a uniform spring of mass mₛ and spring constant k. The other end of the spring is fixed. During the fundamental longitudinal oscillation, the speed of the spring material varies approximately linearly from zero at the fixed end to the block speed at the free end. What is the approximate time period?
Options

Question

A block of mass M is attached to the free end of a uniform spring of mass mₛ and spring constant k. The other end of the spring is fixed. During the fundamental longitudinal oscillation, the speed of the spring material varies approximately linearly from zero at the fixed end to the block speed at the free end. What is the approximate time period?

Options

G1 Physics · Accepted Answer

Let the free-end speed be v. For a point of the spring at distance x from the fixed end, v(x) ≈ vx/L. Integrating the kinetic energy of the spring gives Kₛ = ½(mₛ/3)v². Thus the spring contributes an effective mass mₛ/3. Total effective mass: Mₑ = M+mₛ/3. Therefore, T = 2π√[Mₑ/k] = 2π√[(M+mₛ/3)/k]. Correct option: C.