Leaky funnel

Level pending

An inverted funnel of base radius r = 8 c m r = 8 cm and height H = 15 c m H = 15 cm is placed on a smooth horizontal table and is being filled with a liquid of density 0.96 g m / c m 3 0.96 gm/cm^{3} . It is seen that the liquid starts leaking out from the bottom of the funnel when the height of the liquid is 3 4 \frac{3}{4} times the height of the conical part of the funnel i.e. h = 3 H 4 h= \frac{3H}{4} .

If the mass of the funnel is m (in kg), then find m \lfloor m \rfloor .

( x \lfloor x \rfloor denotes the greatest integer less than or equal to x x )


The answer is 1.

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1 solution

Shreyam Natani
Dec 28, 2013

The normal force acting on the base of the funnel due to the water will be

N = h ρ g π r 2 N = h\rho g \pi r^2

This normal force will balance the water's weight as well as the weight of the funnel.

N = m w g + m f g N = m_{w}g + m_{f}g .

As m w = [ π r 2 H 3 m_{w} = [\frac{ \pi r^2 H}{3} - π r 2 H 4 16 3 ] ρ \frac{\pi r^2 H}{4*16*3}] \rho

m w = 21 π r 2 H 64 ρ m_{w} = \frac{ 21 \pi r^2 H}{64} \rho

Therefore m f = 3 H ρ π r 2 4 21 H ρ π r 2 64 m_{f} = \frac{ 3H \rho \pi r^2}{4} - \frac{ 21H\rho \pi r^2 }{64} = 27 H ρ π r 2 64 = \frac{ 27H \rho \pi r^2}{64}

Putting in the values we get m f = 1.22 k g m_{f} = 1.22 kg

Therefore m = 1 \lfloor{m} \rfloor = \boxed{1}

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