($\pi = 22/7$)
Surface Tension Work Calculation for Soap Bubble
The work required to increase the surface area of a bubble is given by the formula:
$W = S \times \Delta A$
where $S$ is the surface tension and $\Delta A$ is the change in the total surface area. A soap bubble has two surfaces (inner and outer). The surface area of a sphere is $4 \pi r^2$. Therefore, the total surface area of a soap bubble is $A = 2 \times (4 \pi r^2) = 8 \pi r^2$.
The change in total surface area is:
$\Delta A = A_2 - A_1 = 8 \pi r_2^2 - 8 \pi r_1^2 = 8 \pi (r_2^2 - r_1^2)$
So, the work done is:
$W = S \times 8 \pi (r_2^2 - r_1^2)$
Given Values
- Surface tension, $S = 3.5 \times 10^{-2}~\text{N/m}$
- Initial radius, $r_1 = 1~\text{cm} = 0.01~\text{m}$
- Final radius, $r_2 = 2~\text{cm} = 0.02~\text{m}$
- $\pi = 22/7$
Calculation
Substitute the values into the work formula:
$W = (3.5 \times 10^{-2}~\text{N/m}) \times 8 \times (\frac{22}{7}) \times ((0.02~\text{m})^2 - (0.01~\text{m})^2)$
First, calculate the difference in squares of radii:
$(0.02)^2 - (0.01)^2 = 0.0004~\text{m}^2 - 0.0001~\text{m}^2 = 0.0003~\text{m}^2$
Now, substitute this back into the equation for W:
$W = (3.5 \times 10^{-2}) \times 8 \times (\frac{22}{7}) \times (0.0003)$
Simplify the calculation:
$W = (\frac{3.5}{7}) \times 8 \times 22 \times 10^{-2} \times 0.0003$
$W = 0.5 \times 8 \times 22 \times 10^{-2} \times 3 \times 10^{-4}$
$W = 4 \times 22 \times 3 \times 10^{-6}$
$W = 88 \times 3 \times 10^{-6}$
$W = 264 \times 10^{-6}~\text{J}$
Result
The work required is given as $\alpha \times 10^{-6}~\text{J}$. Comparing this with the calculated value:
$\alpha \times 10^{-6}~\text{J} = 264 \times 10^{-6}~\text{J}$
Therefore, the value of $\alpha$ is 264.
Similar Questions
The strain-stress plot for materials $A, B, C$ and $D$ is shown in the figure. Which material has the largest Young's modulus ?
- Eight mercury drops, each of radius $r$, coalesce to form a bigger drop. The surface energy released in this process is ___________. ($\text{S}$ is the surface tension of mercury).
- An ideal gas at pressure $P$ and temperature $T$ is expanding such that $P T^{3} = \text{constant}$. The coefficient of volume expansion of the gas is ___________.
- The temperature of a metal strip having coefficient of linear expansion $\alpha$ is increased from $T_1$ to $T_2$ resulting in increase of its length by $\Delta L_1$. The temperature is further increased from $T_2$ to $T_3$ such that the increase in its length is $\Delta L_2$.
Given $T_3 + T_1 = 2T_2$ and $T_2 - T_1 = \Delta T$, the value of $\Delta L_2$ is _______. - A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4 \times 10^8 \text{ N/m}^2$ and its radius is 4 mm, then maximum acceleration the lift can take is _________ $\text{m/s}^2$.
(take $g = 10 \text{ m/s}^2$ and $\pi = 3.14$) - Given below are two statements:
Statement I: Pressure of a fluid is exerted only on a solid surface in contact as the fluid-pressure does not exist everywhere in a still fluid.
Statement II: Excess potential energy of the molecules on the surface of a liquid, when compared to interior, results in surface tension.
In the light of the above statements, choose the correct answer from the options given below A water drop of radius $1$ cm is broken into eight equal droplets. Surface tension of water is $0.075$ N $m^{-1}$. The gain in surface energy is __________$\times10^{-7}$ J.(Take $\pi = 3.14$)
A gas is kept in a container having walls which are thermally non-conducting. Initially the gas has a volume of $800 \ cm^3$ and temperature $27^{\circ}C$. The change in temperature when the gas is adiabatically compressed to $200 \ cm^3$ is:
(Take $\gamma = 1.5$; $\gamma$ is the ratio of specific heats at constant pressure and at constant volume)
- Pressure of an ideal gas, contained in a closed vessel, is increased by $0.4\%$ when heated by $1^\circ C$. Its initial temperature must be:
A wire of length 10 cm and diameter 0.5 mm is used in a bulb. The temperature of the wire is 1727°C and power radiated by the wire is 94.2 W. Its emissivity is $\frac{x}{8}$ where x =……
(Given $\sigma = 6.0 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$, $\pi = 3.14$ and assume that the emissivity of wire material is same at all wavelength.)
Important Questions from Physical and Thermal Properties of Bulk Matter
The strain-stress plot for materials $A, B, C$ and $D$ is shown in the figure. Which material has the largest Young's modulus ?
- Eight mercury drops, each of radius $r$, coalesce to form a bigger drop. The surface energy released in this process is ___________. ($\text{S}$ is the surface tension of mercury).
- An ideal gas at pressure $P$ and temperature $T$ is expanding such that $P T^{3} = \text{constant}$. The coefficient of volume expansion of the gas is ___________.
- The temperature of a metal strip having coefficient of linear expansion $\alpha$ is increased from $T_1$ to $T_2$ resulting in increase of its length by $\Delta L_1$. The temperature is further increased from $T_2$ to $T_3$ such that the increase in its length is $\Delta L_2$.
Given $T_3 + T_1 = 2T_2$ and $T_2 - T_1 = \Delta T$, the value of $\Delta L_2$ is _______. - A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4 \times 10^8 \text{ N/m}^2$ and its radius is 4 mm, then maximum acceleration the lift can take is _________ $\text{m/s}^2$.
(take $g = 10 \text{ m/s}^2$ and $\pi = 3.14$)
