Class 11th Physics Mechanical Properties of Fluids Physics Surface Tension

A large number of water drops, each of radius r, combine to have a drop of radius R. If the surface tension is T and mechanical equivalent of heat is J, the rise in heat energy per unit volume will be

Option 1 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>T</mi> </mrow> <mrow> <mi>J</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>r</mi> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span></p>

Option 2 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>T</mi> </mrow> <mrow> <mi>r</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </math> </span></p>

Option 3 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>T</mi> </mrow> <mrow> <mi>r</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </math> </span></p>

Option 4 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>T</mi> </mrow> <mrow> <mi>J</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>r</mi> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span></p>

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Payal Gupta | Contributor-Level 10

6 months ago

Correct Option - 1

Detailed Solution:

From volume conservation

$n * \frac{4}{3} π r^{3} = \frac{4}{3} π R^{3} \Rightarrow R^{3} = n r^{3} . . . . . . . . . (1)$

Decrease in surface area = $n * 4 π r^{2} - 4 π R^{2}$

$(Δ A) = 4 π [n r^{2} - R^{2}] = 4 π [\frac{n * r^{3}}{r} - R^{2}] = 4 π [\frac{R^{3}}{r} - R^{2}] = 4 π R^{3} [\frac{1}{r} - \frac{1}{R}]$

Energy released (W) = $T * Δ A = 4 π R^{3} T [\frac{1}{r} - \frac{1}{R}]$

Heat produced (Q) = $\frac{W}{J} = \frac{4 π T R^{3}}{J} [\frac{1}{r} - \frac{1}{R}]$

$N o w, Q = m s Δ θ$

$\Rightarrow \frac{4 π R^{3} T}{J} [\frac{1}{r} - \frac{1}{R}] = (\frac{4}{3} π R^{3}) * | x | * Δ θ = \frac{3 T}{J} [\frac{1}{r} - \frac{1}{R}]$

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Similar Questions for you

n droplets of water each of radius r coalesce to form a big drop of radius R. If the energy released during coalescence goes into heating the drop, then the rise in temperature will be (take T as surface tension of water and J as mechanical equivalent of heat)

$R = n^{1 / 3} r$

$E = T . Δ A = ms Δ T J$

A water drop of diameter 2 cm is broken into 64 equal droplets. The surface tension of water is 0.075N/m. In this process the gain in surface energy will be:

Gain in surface energy, $Δ U = T Δ A$

from volume centenary, $\frac{4}{3} π R^{3} = 64 \times \frac{4}{3} π r^{3}$

$\Rightarrow r = \frac{R}{4}$

Initial surface area, Ai = 4pR²

final surface area, $A_{f} = 64 \times 4 π r^{2}$

$∴ Δ U = 12 π R^{2} . T = 0.075 \times 12 \times 3.14 \times 1 0^{- 4} = 2.8 \times 1 0^{- 4} J$

If temperature of a liquid is increased, choose the correct option regarding change in its surface tension and viscosity.


Fact. Due to decrease in intermolecular forces

When two soap bubbles of radii a and b (b > a) coalesce, the radius of curvature of common surface is:



P? - P? = 4T/a

P? - P? = 4T/b

P? - P? = 4T (1/a - 1/b)

Also, P? - P? = 4T/r

4T/r = 4T (1/a - 1/b)

1/r = (b-a)/ab

r = ab / (b-a)

A water drop of radius 1cm is broken into 729 equal droplets. If surface tension of water is 75 dyne/cm, then the gain in surface energy upto first decimal place will be:

(Given p = 3.14)

$\frac{4}{3} π R^{3} = 729 \frac{4}{3} π r^{3}$

R³= 729r³

$R = (729) \frac{1}{3} (r) (\frac{1}{3})$

R = 9r

$Δ u = T (4 π r^{2}) \times 729 - T \times 4 π R^{2}$

$= T \times 4 π \times 8 R^{2}$

$\begin{array}{l} = 75.39 \times 1 0^{- 5} J \\ Δ u = 7.5 \times 1 0^{- 4} J \end{array}$

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Physics Mechanical Properties of Fluids 2025

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