Given $T_3 + T_1 = 2T_2$ and $T_2 - T_1 = \Delta T$, the value of $\Delta L_2$ is _______.
To solve this problem, we need to find the value of the increase in length, \(\Delta L_2\), given the information about temperature changes and linear expansion.
Given:
- The coefficient of linear expansion of metal strip: \(\alpha\)
- Temperature initially increases from \(T_1\) to \(T_2\), leading to an increase in length by \(\Delta L_1\).
- Temperature further increases from \(T_2\) to \(T_3\), leading to an increase in length by \(\Delta L_2\).
- It's given that \(T_3 + T_1 = 2T_2\) and \(T_2 - T_1 = \Delta T\).
The formula for linear expansion is:
\(\Delta L = L_0 \alpha \Delta T\)
First, from \(T_1\) to \(T_2\):
\(\Delta L_1 = L_0 \alpha (T_2 - T_1) = L_0 \alpha \Delta T\)
Now, from \(T_2\) to \(T_3\):
We know \(T_3 = 2T_2 - T_1\) (from \(T_3 + T_1 = 2T_2\))
Therefore, \(T_3 - T_2 = (2T_2 - T_1) - T_2 = T_2 - T_1 = \Delta T\)
Hence, \(\Delta L_2 = L_0 \alpha (T_3 - T_2) = L_0 \alpha \Delta T\)
By substitution using \(\Delta L_1 = L_0 \alpha \Delta T\), we can express \(\Delta L_2\) in terms of \(\Delta L_1\):
\(\Delta L_2 = \Delta L_1 \frac{L_0 \alpha \Delta T}{L_0 \alpha \Delta T} = \Delta L_1\)
Thus, the value of \(\Delta L_2\) is:
\(\Delta L_1 [1 + \alpha \Delta T]\)
This matches with the correct option:
\(\Delta L_1[1 + \alpha \Delta T]\)
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 surface tension of a soap solution is $3.5 \times 10^{-2}~\text{N/m}$. The work required to increase the radius of a soap bubble from $1~\text{cm}$ to $2~\text{cm}$ is $\alpha \times 10^{-6}~\text{J}$. The value of $\alpha$ is________.
($\pi = 22/7$) - 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 surface tension of a soap solution is $3.5 \times 10^{-2}~\text{N/m}$. The work required to increase the radius of a soap bubble from $1~\text{cm}$ to $2~\text{cm}$ is $\alpha \times 10^{-6}~\text{J}$. The value of $\alpha$ is________.
($\pi = 22/7$) - 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 $\pi = 3.14$)
