A child, standing in a swing, bends her knees and then straightens them twice per swing period, making the distance from the swings support

A child, standing in a swing, bends her knees and then straightens them twice per swing period, making the distance ℓ from the swing’s support to her center of mass oscillate as

is the swing’s mean angular frequency.

(a) Show that the swing’s angular displacement from vertical, θ, obeys the equation of motion

where ω₁ = √gℓ₁, and θ is assumed to be small, θ «1.

(b) Write θ = X₁ cos ω₀t + X₂ sin ω₀t .Assuming that ℓ₁ « ℓ₀ so ω₁ « ω₀, show that the child’s knee bending (her “pumping” the swing) squeezes X₁ and amplifies X₂ (parametric amplification):

(c) Explain how this squeezing is related to the child’s conscious manipulation of the swing (i.e., to her strategy for increasing the swing’s amplitude when she starts up, and her strategy for reducing the amplitude when she wants to quit swinging).

l= lo + l sin 2wot. Here wo = glo: