Can you please show each step preparation, plan, solve and check. Thanks you Pro

Question

Can you please show each step preparation, plan, solve and check. Thanks you
Problem 2 Chapter 13: A light balloon filled with helium of density 0.179 kg/m is tied to a light string of length L"4.0m. The balloon is spherical and has a volume of 2.5 m 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, a) Show that the motion is a simple harmonic oscillator, when 0 is small. b) Determine the period of the motion. Assume that the density of air is 1.29 kg/m

Explanation / Answer

a) Here, Tension, T = mg * sin(theta)

If theta is very small , than sin(theta) =~ theta

=>   Tension, T = ~ mg * theta

Thus, Tension, T is directly proportional to theta .

Thus, motion is simple harmonic oscillator .

b) Period of motion = 2pi * sqrt[4/(1.29 - 0.179) * 2.5]

                               =   7.53 sec

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