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Question

A hydrometer has a bulb of volume 10.5 cm3 and the area of the cross-section of its stem is 0.3 cm2. The hydrometer weighs 13.5 gf. Calculate the density of the liquid if only 4 cm of the length of the stem gets immersed in the liquid.


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Solution

Step 1: Given data

Volume of the bulb of the hydrometer, Vb=10.5cm3

Area of the cross-section of the stem of the hydrometer, A=0.3cm2

Weight of hydrometer, W=13.5gf

Weight of liquid displaced =Wl

Immersed length of the stem of the hydrometer, Li=4cm

Submerged volume of the stem =Vs

Density of liquid, ρl=?

Step 2: Formula used

ρl=mlvl ………………………………(a)

Where ml is the mass of the liquid displaced and vl is the volume of the liquid displaced

Step 3: Calculation of the density of the liquid

From the basic crux of the law of floatation, it follows that the volume of the liquid displaced will be equal to the summation of the volume of the bulb of the hydrometer and the submerged volume of the stem of the hydrometer.

Vl=Vb+Vs ………………………(b)

The submerged volume of the stem of the hydrometer will be equal to the product of the area of the cross-section of the stem and the immersed length.

Vs=As×LiVs=0.3cm2×4cm

Vs=1.2cm3 …………………….(c)

Substituting the value obtained from the equation (c) and the given values in equation (b), we get

Vl=10.5cm3+1.2cm3

Vl=11.7cm3 ……………………(d)

From the basic crux of the law of floatation, it follows that the weight of the liquid displaced will be equal to the weight of the hydrometer.

Wl=WWl=13.5gf

Converting the weight of the liquid displaced into the mass of the liquid displaced, we get

ml=Wlg

ml=13.5g ……………………….(e)

Substituting the values obtained from the equations (d) and (e) in equation (a), we get

ρl=13.5g11.7cm3ρl=1.153g/cm3

Hence, the density of the liquid is equal to 1.153 g/cm3.


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