Figure below shows a shampoo bottle in a perfect cylindrical shape

In a simple experiment, the stability of the bottle filled with different amount of shampoo volume is observed. The bottle is tilted from one side and then released. Let the angle `theta` depicts the critical angular displacement resulting in the bottle losing its stability and tripping over. Choose the graph correctly depicting the fraction f of shampoo filled (f= 1 corresponds to completely filled) vs the tripping angle `theta`

Options

A.
B.
C.
D.

Answer

D

Explanation

Mass of bottle = `m_0`
Length of bottle = L
base Area = A = `pir^2`
density of shampoo = p
mass of shampoo = pfAL

Center of mass of system
`y=(m_(0)L/(2)+(pfAL)((fL)/2))/(m_(0)+pfAL)`
for critical angular displacement, mg will pass through tilted side.

From the diagram `tantheta=r/y`
`tantheta=(r(m_0+pALf))/(L/2(m_0+pALf^2))`
at f = 0 & f = 1, tipping angle ‘`theta`’ will be same. for very small values of ‘f’, we can neglect `f^2` terms
`rArrtantheta=r/(L/2)((m_0+pALf)/(m_0)`
`theta=tan^(-1)(r/(L/2)((m_0+pALf))/(m_(0)))`
So if f increases `theta` will increase.