Thermal Expansion Volume Increase Calculation
This problem asks for the approximate increase in the volume of a metallic sphere when its temperature increases.
Key Concepts
- The initial volume ($V_0$) of a sphere with radius $R$ is given by the formula $V_0 = \frac{4}{3}\pi R^3$.
- The linear coefficient of thermal expansion is $\alpha$.
- For isotropic materials like metals, the volume expansion coefficient ($\beta$) is approximately three times the linear expansion coefficient: $\beta \approx 3\alpha$.
- The change in volume ($\Delta V$) due to a temperature change ($\Delta T$) is calculated as $\Delta V = V_0 \beta \Delta T$.
Volume Increase Calculation
- Initial Volume: Calculate the initial volume of the sphere:
$ V_0 = \frac{4}{3}\pi R^3 $
- Volume Expansion Coefficient: Determine the volume expansion coefficient:
$ \beta \approx 3\alpha $
- Change in Volume: Apply the formula for volume expansion:
$ \Delta V = V_0 \beta \Delta T $
Substitute the expressions for $V_0$ and $\beta$:$ \Delta V = \left(\frac{4}{3}\pi R^3\right) (3\alpha) (\Delta T) $
Simplify the expression:$ \Delta V = 4\pi R^3 \alpha \Delta T $
Conclusion
The approximate increase in the volume of the sphere is $4\pi R^3 \alpha \Delta T$. This corresponds to Option B.
Similar Questions
What will be the approximate terminal velocity of a rain drop of diameter $ 1.8\times {10}^{-3}. \mathrm{m}$, when density of rain water $ \approx {10}^{3}{\mathrm{kgm}}^{-3}$ and the coefficient of viscosity of air $ \approx 1.8\times {10}^{-5}. \mathrm{N}-\mathrm{sm}$$ {}^{-2} ?$ (Neglect buoyancy of air)
- A submarine is designed to withstand an absolute pressure of $100 \text{ atm}$. How deep can it go below the water surface?
(Consider the density of water = $1000 \text{ kg m}^{-3}$, $1 \text{ atm} = 1 \times 10^5 \text{ Pa}$ and gravitational acceleration $g = 10 \text{ m/s}^2$) Match List I with List II :
List I List II A. Young's Modulus I. $\frac{\Delta d}{\Delta L} ( \frac{L}{d})$ B. Compressibility II. $\frac{FL}{A(\Delta L)}$ C. Bulk Modulus III. $-\frac{1}{\Delta P} (\frac{\Delta V}{V})$ D. Poisson's Ratio IV. $-P(\frac{V}{\Delta V})$
Choose the correct answer from the options given below :- In the measurement of viscosity of liquids using terminal velocity experiment, spherical balls of same radius but having different densities are used. The variation of the terminal velocity ($v$) with the ratio of density of spherical ball ($\sigma$) to density of the liquid ($\rho$), is best represented by :
Water flows in a streamline motion through a horizontal pipe of circular cross-section as shown in the figure. The pressure difference of water between P and Q is $15\text{ Nm}^{-2}$. The area of cross-section at P and Q are $40\text{ cm}^2$ and $20\text{ cm}^2$, respectively. The rate of flow of water through the pipe, in $\text{cm}^3\text{s}^{-1}$, is :
[Take density of water $= 1000\text{ kg m}^{-3}$]
Important Questions from Physical and Thermal Properties of Bulk Matter
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.)
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 $\pi = 3.14$)
- Two water drops of radius 'r' each coalesce to form a single large drop. If 'T' is the surface tension, then the surface energy released in this process is:
- 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 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)
