Question $7\mathrm{.}2$

A balloon filled with a gas that is less dense than air is attached to a $5$metre uniform string. The mass of the entire string is $100$ grams. The balloon is spherical in shape with a radius of $40$ cm $.$ When the balloon is released, a length $h$ of the string is lifted while the rest of the string is lying on the floor as shown in the figure. The envelope of the balloon has a mass of $250$ grams. Given that the density of air at standard temperature and pressure is $1292\frac{g}{m^3},$

$($a$)(10$ marks$)$ Calculate the buoyant force acting on the balloon by air.

$($b$)(10$ marks$)$ If the balloon and the string are stationary when $h=361$ cm $,$ what is the density of the gas used to inflate the balloon?