Swing

A child of mass m on a swing raises her center of mass by a small distance b every time the swing passes the vertical position, and lowers her mass by the same amount at each extremal position. Assuming small oscillations, calculate the work done by the child per period of oscillation. Show that the energy of the swing grows exponentially according to dE/dt = αE and determine the constant α.

SOLUTION

Consider half a period of swinging motion between points 0 → 1 → 2 → 3 → 4 (see Figure 1.1, where the dotted line indicates the position of the center of mass). From 0 to 1, energy is conserved E₀ = E₁.

Figure 1.1

where φ₀ is the initial angle. For small angles, 1- cos φ₀ = φ²₀/2 and

(1)

From 1 to 2, angular momentum is conserved:

(2)

From 2 to 3, again, energy is conserved so we can find the final angle φ_f

(3)

From (1) and (3) we can express mv²₁/2 and obtain the change in the amplitude:

(4)

or using b/l << 1

(5)

and

(6)

The work done by the child is equal to the energy change per period :

(7)

We want to write (7) in the form dE/dt = αE,

(8)

(9)

where