($\pi = 22/7$)
Work Done Soap Bubble Inflation Calculation
This section provides a step-by-step calculation for the work done in increasing the diameter of a soap bubble, based on surface tension.
Physics Concepts for Work Done
The work done ($W$) to change the surface area of a liquid film is related to surface tension ($T$) and the change in area ($\Delta A$). For a soap bubble, which has two surfaces (inner and outer), the formula is:
$W = 2 \times T \times \Delta A$
The surface area ($A$) of a sphere with radius ($r$) is given by $A = 4 \pi r^2$. Therefore, the change in area is $\Delta A = A_{final} - A_{initial} = 4 \pi r_{final}^2 - 4 \pi r_{initial}^2$.
Calculation Steps
Step 1: Define Given Parameters
- Surface Tension, $T = 0.04 \text{ N/m}$
- Initial Diameter, $d_1 = 7 \text{ cm} = 0.07 \text{ m}$
- Final Diameter, $d_2 = 14 \text{ cm} = 0.14 \text{ m}$
- Work Done Formula: $W = (15000 - x) \text{ }\mu\text{J}$
- Value of Pi, $\pi = 22/7$
Step 2: Determine Initial and Final Radii
- Initial Radius, $r_1 = d_1 / 2 = 0.07 / 2 = 0.035 \text{ m}$
- Final Radius, $r_2 = d_2 / 2 = 0.14 / 2 = 0.07 \text{ m}$
Step 3: Calculate Change in Surface Area ($\Delta A$)
The change in area depends on the squares of the radii:
$\Delta A = 4 \pi (r_2^2 - r_1^2)$
Substituting the values:
$\Delta A = 4 \times \frac{22}{7} \times \left( (0.07)^2 - (0.035)^2 \right) \text{ m}^2$
$\Delta A = \frac{88}{7} \times (0.0049 - 0.001225) \text{ m}^2$
$\Delta A = \frac{88}{7} \times 0.003675 \text{ m}^2$
$\Delta A = 0.0462 \text{ m}^2$
Step 4: Calculate Total Work Done ($W$)
Use the work formula $W = 2 \times T \times \Delta A$:
$W = 2 \times (0.04 \text{ N/m}) \times (0.0462 \text{ m}^2)$
$W = 0.08 \times 0.0462 \text{ J}$
$W = 0.003696 \text{ J}$
Convert the work done to microJoules ($\mu J$):
$W = 0.003696 \times 10^6 \text{ }\mu\text{J} = 3696 \text{ }\mu\text{J}$
Step 5: Determine the value of $x$
Equate the calculated work ($W$) with the given expression for work done:
$15000 - x = 3696$
Solve for $x$:
$x = 15000 - 3696$
$x = 11304$
The value of $x$ is 11304.
Similar Questions
- A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
Suppose there is a uniform circular disc of mass $M \text{ kg}$ and radius $r \text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^2$. The value of $x$ is ________.
- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
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(Given, Young's modulus of steel = $2 \ \times \ 10^{11} \ N/m^2$ and Young's modulus of copper = $1.5 \ \times \ 10^{11} \ N/m^2$)
A fly wheel of mass $10$ kg and radius $50$ cm is rotating at a rate of$360$ rpm. Assuming mass to be concentrated at the rim, the constant retarding torque required to stop the rotation of wheel in $6$ rotations will be x Nm. The value of x is ____________.
A body moving horizontally on a smooth surface with a speed of $20$ $ms^{-1}$ splits into two parts and continue to move in the same direction. The masses of two parts are in the ratio of $1:2$. The smaller part moves with a speed of $30$ $ms^{-1}$. The fractional change is the kinetic energy is given by $\frac{1}{x}$. The value of x is__________
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A thin uniform bar of length L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2v and v respectively. The masses stick to the bar after collision at a distance $\frac{L}{3}$ and $\frac{L}{6}$ respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be :
As shown in the figure, forces of $10^5$ N each are applied in opposite directions, on the upper and lower faces of a cube of side 10 cm, shifting the upper face parallel to itself by 0.5 cm. If the side of another cube of the same material is 20 cm, then under similar conditions as above, the displacement will be :
Important Questions from Mechanics
- A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
Suppose there is a uniform circular disc of mass $M \text{ kg}$ and radius $r \text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^2$. The value of $x$ is ________.
- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
- A body of density '$\rho$' is dropped slowly on the surface of a lake of depth $d$. If the density of the lake water be '$\rho'$' ($\rho' < \rho$) then the time taken by the body to reach the bottom of the lake is
