Please show all steps. ***2.5m^3 not square!! Thank you A light balloon filled w

Question

Please show all steps. ***2.5m^3 not square!! Thank you A light balloon filled with helium of density 0.179 kg/m^3 is tied to a light string of length L = 4.0m. The balloon is spherical and has a volume of 2.5 m^2. The string is tied to the ground, forming an "inverted" simple pendulum, see the figure. If the balloon is displaced slightly from equilibrium, as shown in the figure, Show that the motion is a simple harmonic oscillator, when theta is small. Determine the period of the motion. Assume that the density of air is 1.29 kg/m^3.

Explanation / Answer

Givendata
density =0.179 kg/m^3
length L=4m
volume = 2.5 m^2
density of air = 1.29 kg/m^3
(a) One would certainly
expect the balloon to reach an equal displacement on the other side,
and for the velocity to be a maximum at the equilibrium position,
so it will certainly LOOK like SHM.

(b) Let's say the balloon has volume V.
mass m = V = 0.179kg/m³ * V
buoyant force F = (' - )Vg = (1.29 - 0.179)kg/m³ * V * 9.8m/s²
F = 10.88N/m³ * V
acceleration a = F / m = 10.88N/m³ * V / (0.179kg/m³ * V) = 60.82 m/s²

period T = 2(L/a) = 2(4m / 60.82m/s²) = 1.61 seconds

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