Persnickety with Pendulums

Students often worry about small inaccuracies when we use approximate models to solve things in physics. Of course, ignoring some things is useful when trying to understand the world. This is because they don$$t significantly impact the result of a problem but they do significantly impact the investment required to get results. Choosing what to ignore while still getting results is a high$-$level skill. This problem will show you some things we have ignored and why.

A pendulum swings in a rhythmic fashion. It consists of a ball on the end of a massless string that is $30$ cm long.

Part I: Making the usual assumptions

What is the period of the pendulum?

Part II: Correcting for buoyant force

Suppose the ball on the end of the string is a very thin lead shell about the size of a

baseball $($radius of $3\mathrm{.}7$ cm$)$ filled with helium gas. What is the buoyant force on this ball?

Is this significant? Explain how you gauge $$significance$?$

How much lead $($in grams$)$ could be used to make the shell before the period of the

resulting pendulum would be infinite?

How thin would the lead shell be $($for simplicity, be approximate$)?$ Would it significantly

affect the volume of the ball if we took this more seriously into account?

Part II: Correcting for large amplitudes $($treat the ball as a point mass again$)$

What is the tangential acceleration of the ball as it swings through its lowest point $($i$.$e$.$ the string is vertical$)?$

What is the tangential acceleration of the ball at a moment it is released from rest with the string horizontal?

What is the tangential acceleration of the ball as a function of the angle of the string from a vertical line?

At what angular amplitude $($in degrees$)$ is the error in the tangential acceleration due to the small angle approximation $()10\%$ or less? $($You may have to be a little clever to find a number, but you can do it$.$ Just show your process$)$