Question

A thin square plate of side 5cm is suspended vertically from a balance so that the lower side just dips into water with side to surface. When the plate is clean (θ = 0°), it appears to weigh 0.044 N. But when the plate is greasy (θ = 180°), it appears to weigh 0.03 N. The surface tension of water is?
$
  {\text{A}}{\text{. 3}}{\text{.5 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}{\text{ N/m}} \\
  {\text{B}}{\text{. 7}}{\text{.0 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}{\text{ N/m}} \\
  {\text{C}}{\text{. 14}}{\text{.0 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}{\text{ N/m}} \\
  {\text{D}}{\text{. 1}}{\text{.08 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}{\text{ N/m}} \\
$

Answer 1

- Hint: In order to find the surface tension, we observe that the apparent weight of the plate in each case is not only the weight of the plate but also the surface tension of water. The surface tension of the water alters the weight of the plate in each case.

Formula Used,
Surface tension of water is given by ${\sigma _{{\text{water}}}} = \dfrac{{\text{F}}}{{\text{l}}}$, where F is the force caused acting along the length l.

Complete step-by-step solution -

In the above scenario, depending on the angle in which the plate is placed along the water. Due to the angle 180° the force of the surface tension is opposite to the actual weight of the body in the second case.
Apparent weight is the net weight felt by the body die to the force of surface tension acting on it.
Let us assume the apparent weight of the plate in case 1 (θ = 0°) and case 2 (θ = 180°) to be ${{\text{W}}_1}$ and ${{\text{W}}_2}$ respectively, where the original weight of the plate is taken as W. Let the force acting on the plate due to surface tension be F.

In case - 1,
The force of surface tension acts along the weight of the body, so the apparent weight is more than the actual weight of the plate and is given by
${{\text{W}}_1} = {\text{W + F}}$. --- (1)

Case – 2,
The force of surface tension acts in the direction opposite to the weight of the body, so the apparent weight is less than the actual weight of the plate and is given by
${{\text{W}}_2} = {\text{W - F}}$. -- (2)

Subtracting equation (1) and (2), we get
The force of surface tension acts along the weight of the body, so the apparent weight is more than the actual weight of the plate and is given by
${{\text{W}}_1} - {\text{ }}{{\text{W}}_2} = {\text{ 2F}}$. -- (3)

We know the force caused by the surface tension is defined as the product of surface tension of the liquid $\left( \sigma \right)$ and the length (l) along which it acts. It is given by
${\text{F = }}\sigma {\text{ }} \times {\text{ l}}$
Substituting this in equation (3) we get
${{\text{W}}_1} - {\text{ }}{{\text{W}}_2} = {\text{ 2}}\left( {\sigma \times {\text{l}}} \right)$ --- (4)

Given data,
${{\text{W}}_1} = 0.044{\text{N}}$
${{\text{W}}_2} = 0.03{\text{N}}$
$\sigma = {\sigma _{{\text{water}}}}$
L = 5cm = $\dfrac{5}{{100}}$m = 0.05m (∵1m = 100cm)
But here we consider two times the given length of the plate because the plate is wetted on both sides,
Hence our l becomes l × 2 = 0.05 × 2 = 0.1 m

Substituting all these values in equation (4), we get
$
   \Rightarrow {{\text{W}}_1} - {\text{ }}{{\text{W}}_2} = {\text{ 2}}\left( {\sigma \times {\text{l}}} \right) \\
   \Rightarrow 0.044{\text{ }} - {\text{ 0}}{\text{.03 = 2}}\left( {{\sigma _{{\text{water }}}} \times {\text{ 0}}{\text{.1}}} \right) \\
   \Rightarrow {\sigma _{{\text{water}}}} = \dfrac{{0.014}}{{0.2}} = 0.07{\text{ N/m}} \\
$
Hence the surface tension of water is ${\text{7}}{\text{.0 }} \times {\text{ 1}}{{\text{0}}^{ - 2}}{\text{ N/m}}$. Option B is the correct answer.

Note – In order to answer this type of questions the key step is to know that the square plate is exposed to water on both sides and hence the length along which the force of surface tension acts becomes twice the length of the side of a square plate.
Also the formula and definition of the concept of surface tension is to be known.
While substituting the values in a formula, we have to make sure all the values are converted into the S.I system of units.
Other examples of the phenomena of surface tension are insects walking on water and a paper sheet floating on water.