Q44 / 55 · KVPY 2020 (SA) · Physics

Figure below shows a shampoo bottle in a perfect cylindrical shape

Figure

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 θ depicts the critical angular displacement resulting in the bottle losing its stability and tipping over. Choose the graph correctly depicting the fraction f of shampoo filled (f=l corresponds to completely filled) vs the tipping angle θ.

  • A
    Figure
  • B
    Figure
  • C
    Figure
  • Correct answer
    D
    Figure

Solution

Mass of bottle =m0

Length of bottle =L

base Area =A=πr2

density of shampoo =ρ

mass of shampoo =ρfAL

Figure

Center of mass of system

y=m0L2+ρfALfL2m0+ρfAL

for critical angular displacement, mg will pass through tilted side.

Figure

From the diagram tanθ=ry

tanθ=rm0+ρALfL2m0+ρALf2

at f=0 & f=1, tipping angle θ will be same.

for very small values of f, we can neglect f2 terms

⇒tanθ=rL2m0+ρALfm0

θ=tan-1rL2m0+ρALfm0

So if f increases θ will increase

Hence, the correct answer is option: 'D'.

More Center of Mass Momentum and Collision Questions