KVPY 2020-21 SA Exam Previous Papers

Question : 24 of 80

Marks: +1, -0

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 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 = 1 corresponds to completely filled) vs the tipping angle θ

Validate

Solution:

A the point of vertical angular displacement, the gravitational force would pass via point of contact. This would lead to zero torque about point of contact.
h : height of shampoo

h_0

: height of bottle
R : Radius of bottle

\tan \θ={R}/{({h}/{2})}={2 R}/{h}

\⇒\tan \θ={2 R}/{h_{0} f} \⇒\θ=\tan ^{-1}({2 R}/{h_{0} f})

\;\;\;\;(\∵{h}/{h_{0}}=f)

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