What is a spring pendulum and why does it show energy transfer?

A spring pendulum is a mass hanging from a spring that can both swing side to side like a pendulum and bounce up and down like a spring. The two modes are coupled through the equations of motion. When the spring frequency is about twice the pendulum frequency energy flows dramatically between the two modes in a phenomenon called parametric resonance.

What is the 2 to 1 resonance condition for a spring pendulum?

The 2 to 1 resonance occurs when the spring oscillation period is half the pendulum period. This means omega spring equals 2 times omega pendulum or equivalently k equals 3mg divided by L0. At this ratio energy transfers most efficiently between radial bouncing and angular swinging.

Spring Pendulum — Coupled 2D Oscillations & 2:1 Resonance

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Spring Pendulum (Elastic Pendulum)