Thermometry & Thermal Expansion

The length of mercury column in a mercury-in-glass thermometer is $H_1$ mm at the ice point and $H_2$ mm at the steam point. When the length is $H$ mm, the temperature $T$ is given in $^\circ$C by ($H_1 < H_2$)

If the two temperatures on Fahrenheit scale differ by $54^\circ$F, then the difference in temperature on the Celsius scale is

What is the change in volume of an iron sphere of volume $400$ cm$^3$, when it is heated from $0^\circ\text{C}$ to $50^\circ\text{C}$? ($\alpha_{\text{iron}} = 12 \times 10^{-6}/^\circ\text{C}$)

The temperature of a metallic sphere of radius $R$ is increased by a small amount $\Delta T$. If the linear coefficient of thermal expansion of the metal is $\alpha$, the approximate increase in the volume of the sphere is:

The coefficient of volume expansion of mercury is $18 \times 10^{-5}/^\circ$C. A thermometer bulb has volume $10^{-6}$ m$^3$ and cross-section of stem is $0.004$ cm$^2$. Assuming bulb is filled with mercury at $0^\circ$C then the mercury column at $100^\circ$C is

A celsius and a Fahrenheit thermometer are dipped in boiling water. The temperature of water is lowered until the Farenheit thermometer shows $140^\circ$. What is the fall in temperature on the Celsius thermometer?

An iron tyre is to be fitted on to a wooden wheel $1.0$ m in diameter. The diameter of the tyre is $6$ mm smaller than that of the wheel. The tyre should be heated so that its temperature increases by a minimum of

(coefficient of volume expansion of iron = $3.6 \times 10^{-5}/^\circ$C)

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 ______.

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at $30^\circ\text{C}$. The coefficient of linear expansion of aluminium and steel are $24 \times 10^{-6}/^\circ\text{C}$ and $1.2 \times 10^{-5}/^\circ\text{C}$, respectively. The length of this composite rod when its temperature is raised to $100^\circ\text{C}$, is ______ cm.

A sheet of solid material having area $A$ and coefficient of linear expansion $\alpha$. The mass of sheet is $m$ and specific heat capacity is $s$. If $dQ$ is the amount of heat given to the sheet then change in its area is

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3 =$ constant. The coefficient of volume expansion of the gas is ______.

A metal rod (A) of length $20$ cm expands by $0.075$ cm when its temperature changes from $0^{\circ}$C to $100^{\circ}$ C. Another metal rod (B) of different metal of same length expands by $0.045$ cm for same temperature change. A third rod of same length is composed of 2 parts, one of metal A and another of metal B. For same temperature change, this rod expands by $0.060$ cm. The portion made of metal A has the length

The temperature of a body on Kelvin scale is ' $x$ ' $K$. When it is measured by a Fahrenheit thermometer, it is found to be ' x ' $^\circ\text{F}$. The value of ' $x$ ' is (nearly)

A centigrade and Fahrenheit thermometer are dipped in boiling water. The water temperature is lowered until the Fahrenheit temperature observed is $140^\circ\text{F}$. At that time the temperature registered by the centigrade thermometer is

Which of the following figure represents the relation between Celsius and Fahrenheit temperatures?

The volume of a metal sphere increases by $0.33\%$ when its temperature is raised by $50^\circ\text{C}$. The coefficient of linear expansion of the metal is

A thermometer bulb has volume $10^{-6}\text{ m}^3$ and cross-section of the stem is $0.002\text{ cm}^2$. The bulb is filled with mercury at $0^\circ\text{C}$. If the thermometer reads temperature as $100^\circ\text{C}$, then the length of mercury column is (coefficient of cubical expansion of mercury $= 18 \times 10^{-5}/^\circ\text{C}$)

The length of steel rod is 5 cm longer than the copper rod at all temperatures. The length of the steel and copper rod is respectively (Coefficient of linear expansion for steel and copper is respectively $1.1 \times 10^{-5}/^\circ\text{C}$ and $1.7 \times 10^{-5}/^\circ\text{C}$)

The difference in length between two rods A and B is 60 cm at all temperatures. If $\alpha_{\text{A}} = 18 \times 10^{-6}/^\circ\text{C}$ and $\alpha_{\text{B}} = 27 \times 10^{-6}/^\circ\text{C}$, then the length of $\text{rod A}$ and $\text{rod B}$ at $0^\circ\text{C}$ is respectively

Two rods of different materials have lengths ' $l_1$ ' and ' $l_2$ ' whose coefficient of linear expansions are ' $\alpha_1$ ' and ' $\alpha_2$ ' respectively. If the difference between the two lengths is independent of temperature then

Consider a rectangular sheet of solid material of length $l = 9\text{ cm}$ and width $\text{d} = 4\text{ cm}$. The coefficient of linear expansion is $\alpha = 3.1 \times 10^{-5}\text{ K}^{-1}$ at room temperature and one atmospheric pressure. The mass of sheet $m = 0.1\text{ kg}$ and the specific heat capacity $C_{\text{v}} = 900\text{ J kg}^{-1}\text{ K}^{-1}$. If the amount of heat supplied to the material is $8.1 \times 10^2\text{ J}$ then change in area of the rectangular sheet is:

On Celsius scale the temperature of body increases by $40^\circ\text{C}$. The increase in temperature on Fahrenheit scale is:

A metal rod having coefficient of linear expansion $2 \times 10^{-5}/^\circ\text{C}$ is 0.75 m long at $45^\circ\text{C}$. When the temperature rises to $65^\circ\text{C}$, the increase in length of the rod will be

The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are $8\ \Omega$ and $10\ \Omega$ respectively. After inserting in a hot bath of temperature $400^\circ\text{C}$, the resistance of platinum wire is:

At certain temperature, $\text{rod A}$ and $\text{rod B}$ of different materials have lengths $\text{L}_{\text{A}}$ and $\text{L}_B$ respectively. Their co-efficients of linear expansion are $\alpha_A$ and $\alpha_B$ respectively. It is observed that the difference between their lengths remain constant at all temperatures. The ratio $L_A/L_B$ is given by

Rails of material of steel are laid with gaps to allow for thermal expansion. Each track is 10 m long, when laid at temperature $17^\circ\text{C}$. The maximum temperature that can be reached is $45^\circ\text{C}$. The gap to be kept between the two segments of railway track is

$(\alpha$steel $= 1.3 \times 10^{-5}/^\circ\text{C})$

At certain temperature, $\text{rod A}$ and $\text{rod B}$ of different materials have lengths $\text{L}_{\text{A}}$ and $\text{L}_{\text{B}}$ respectively. Their coefficients of linear expansion are $\alpha_A$ and $\alpha_B$ respectively. It is observed that the difference between their lengths remains constant at all temperatures. The ratio $\text{L}_{\text{A}} : \text{L}_{\text{B}}$ is given by

Railway track is made of steel segments separated by small gaps to allow for linear expansion. The segment of track is 10 m long when laid at temperature $17^\circ\text{C}$. The maximum temperature that can be reached is $45^\circ\text{C}$. Increase in length of the segment of railway track is ' $x$ ' $\times 10^{-5}\text{ m}$. The value of ' $x$ ' is $(\alpha$steel $= 1.2 \times 10^{-5}/^\circ\text{C})$

A sheet of steel is 40 cm long and 5 cm broad at $0^\circ\text{C}$. The surface area of the sheet increases by $1.4\text{ cm}^2$ at $100^\circ\text{C}$. Coefficient of linear expansion of steel is

Two rods, one of copper ( Cu$)$ and the other of iron ( Fe) having initial lengths $L_1$ and $L_2$ respectively are connected together to form a single rod of length $L_1 + L_2$. The coefficient of linear expansion of Cu and Fe are $\alpha_c$ and $\alpha_i$ respectively. If the length of each rod increases by the same amount when their temperatures are raised by $t^{\circ}\text{C}$, then ratio of $\frac{L_1 - L_2}{L_1 + L_2}$ will be

Two rods, one of aluminium and the other of steel, having initial lengths ' $L_1$ ' and ' $L_2$ ' are connected together to form a single rod of length $(L_1 + L_2)$. The coefficients of linear expansion of aluminium and steel are ' $\alpha_1$ ' and ' $\alpha_2$ ' respectively. If the length of each rod increases by the same amount, when their temperatures are raised by $t^{\circ}\text{C}$, then the ratio $\frac{L_1}{L_1 + L_2}$ will be

A metal rod $2\text{ m}$ long increases in length by $1.6\text{ mm}$, when heated from $0^\circ\text{C}$ to $60^\circ\text{C}$. The coefficient of linear expansion of metal rod is

The volume of a metal block increases by $0.225\%$ when its temperature is increased by $30^\circ\text{C}$. Hence coefficient of linear expansion of the material of metal block is

On a temperature scale '$\text{X}$', the boiling point of water is $65^\circ\text{X}$ and the freezing point is $-15^\circ\text{X}$. Assume that the $\text{X}$ scale is linear. The equivalent temperature corresponding to $-95^\circ\text{X}$ on the Fahrenheit scale would be:

The graph between two temperature scales P and Q is shown in the figure. Between upper fixed point and lower fixed point there are 150 equal divisions of scale P and 100 divisions on scale Q. The relationship for conversion between the two scales is given by :-

A faulty thermometer reads $5^\circ\text{C}$ in melting ice and $95^\circ\text{C}$ in steam. The correct temperature on absolute scale will be __________ $\text{K}$ when the faulty thermometer reads $41^\circ\text{C}$.

The difference in length between two rods $\text{A}$ and $\text{B}$ is $60\text{ cm}$ at all temperatures. If $\alpha_{\text{A}} = 18 \times 10^{-6}/^\circ\text{C}$ and $\alpha_{\text{B}} = 27 \times 10^{-6}/^\circ\text{C}$, the lengths of the two rods are

The side of a copper cube is $1\text{ m}$ at $0^\circ\text{C}$. What will be the change in its volume, when it is heated to $100^\circ\text{C}$? $[\alpha$copper $= 18 \times 10^{-6}/^\circ\text{C}]$

Two uniform brass rods $A$ and $B$ of length '$l$' and '$2l$' and their radii '$2r$' and '$r$' respectively are heated to same temperature. The ratio of the increase in the volume of $\text{rod A}$ to that of $\text{rod B}$ is

A hole is drilled in a metal sheet. At $27^\circ\text{C}$, the diameter of hole is 5 cm. When the sheet is heated to $177^\circ\text{C}$, the change in the diameter of hole is $\text{d} \times 10^{-3}$ cm. The value of d will be __________ if coefficient of linear expansion of the metal is $1.6 \times 10^{-5}/^\circ$C.

A steel rod of length $1\text{ m}$ and cross sectional area $10^{-4}\text{ m}^2$ is heated from $0^\circ\text{C}$ to $200^\circ\text{C}$ without being allowed to extend or bend. The compressive tension produced in the rod is ___________ $\times 10^4\text{ N}$. (Given Young's modulus of steel $= 2 \times 10^{11}\text{ N m}^{-2}$, coefficient of linear expansion $= 10^{-5}\text{ K}^{-1}$)

The pressure of the gas in a constant volume gas thermometer is 100 cm of mercury when placed in melting ice at 1 atm. When the bulb is placed in a liquid, the pressure becomes 180 cm of mercury. Temperature of the liquid is:

(Given 0$^\circ$ C = 273 K)

At what temperature a gold ring of diameter 6.230 cm be heated so that it can be fitted on a wooden bangle of diameter 6.241 cm? Both the diameters have been measured at room temperature (27$^\circ$C).

(Given: coefficient of linear thermal expansion of gold $\alpha$L = 1.4 $\times$ 10$-$5 K${}^{-1}$)

A solid metallic cube having total surface area 24 m2 is uniformly heated. If its temperature is increased by 10$^\circ$C, calculate the increase in volume of the cube. (Given $\alpha$ = 5.0 $\times$ 10 $-$4 $^\circ\text{C}^{-}$1).

A unit scale is to be prepared whose length does not change with temperature and remains $20\text{ cm}$, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is $40\text{ cm}$, then length of iron will be __________ $\text{cm} \cdot \left(\alpha\right.$iron $= 1.2 \times 10^{-5}\text{ K}^{-1}$ and $\alpha$brass $= 1.8 \times 10^{-5}\text{ K}^{-1}\Big)$.

Find the value of $-$ 197$^\circ$C temperature in Kelvin.

On an imaginary linear scale of temperature (called 'W' scale) the freezing and boiling points of water are 39$^\circ$ W and 239$^\circ$ W respectively. The temperature on the new scale corresponding to 39 $^\circ$C temperature on Celsius scale will be

A bimetallic strip consists of metals A and B. It is mounted rigidly as shown. The metal A has higher coefficient of expansion compared to that of metal B. When the bimetallic strip is placed in a cold bath, it will:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: When a rod lying freely is heated, no thermal stress is developed in it. Reason R: On heating, the length of the rod increases.

In the light of the above statements, choose the correct answer from the options given below:

Two rods of different metals have coefficients of linear expansion '$\alpha_1$' and '$\alpha_2$' respectvely. Their respective lengths are '$\text{L}_1$' and '$\text{L}_2$'. At all temperatures ($\text{L}_2 - \text{L}_1$) is same. The correct relation is

Let '$\text{W}_1$' be the work done in blowing a soap bubble of radius '$r$' from soap at room temperature. The soap is now heated and second soap bubble of radius '$2r$' is blown from the heated soap solution. If '$W_2$' is the work done in forming this bubble then

Each side of a box made of metal sheet in cubic shape is 'a' at room temperature 'T', the coefficient of linear expansion of the metal sheet is '$\alpha$'. The metal sheet is heated uniformly, by a small temperature $\Delta$T, so that its new temperature is T + $\Delta$T. Calculate the increase in the volume of the metal box.

The area of cross-section of a railway track is 0.01 m2. The temperature variation is 10$^{\circ}$C. Coefficient of linear expansion of material of track is 10$^{-5}/^{\circ}$ C. The energy stored per meter in the track is ____________ J/m.

(Young's modulus of material of track is 1011 Nm$^{-}$2)

A non-isotropic solid metal cube has coefficients of linear expansion as:

5 $\times$ 10-5/oC along the x-axis and 5 $\times$ 10-6/oC along the y and the z-axis. If the coefficient of volume expansion of the solid is C $\times$ 10-6/oC then the value of C is

Two different wires having lengths L1 and L2, and respective temperature coefficient of linear expansion $\alpha$1 and $\alpha$2, are joined end-to-end. Then the effective temperature coefficient of linear expansion is:

When the temperature of a metal wire is increased from 0oC to 10oC, its length increases by 0.02%. The percentage change in its mass density will be closest to:

A bakelite beaker has volume capacity of 500 cc at 30oC. When it is partially filled with Vm volume (at 30oC) of mercury, it is found that the unfilled volume of the beaker remains constant as temperature is varied. If $\gamma$(beaker) = 6 × 10–6oC–1 and $\gamma$(mercury) = 1.5 × 10–4oC–1, where $\gamma$ is the coefficient of volume expansion, then Vm (in cc) is close to ____.

A thermometer graduated according to a linear scale reads a value x0 when in contact with boiling water, and x0/3 when in contact with ice. What is the temperature of an object inoC, if this thermometer in the contact with the object reads x0/2?

A copper rod of 88 cm and an aluminium rod of unknown length have their increase in length independent of increase in temperature. The length of aluminium rod is: ($\alpha$Cu = 1.7 × 10–5 K–1 and $\alpha$Al = 2.2 × 10–5 K–1)

Two rods A and B of identical dimensions are at temperature 30°C. If A is heated upto 180oC and B upto ToC, then the new lengths are the same. If the ratio of the coefficients of linear expansion of A and B is 4: 3, then the value of T is

A rod, of length L at room temperature and uniform area of cross section A, is made of a metal having coefficient of linear expansion $\alpha$/oC. It is observed that an external compressive force F, is applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by $\Delta$T K. Young's modulus, Y, for this metal is:

A clock pendulum having coefficient of linear expansion. $\alpha = 9 \times 10^{-7}/{}^{\circ}\text{C}^{-1}$ has a period of 0.5 s at $20^{\circ}\text{C}$. If the clock is used in a climate, where the temperature is $30^{\circ}\text{C}$, how much time does the clock lose in each oscillation? ( $g =$ constant)

At 40o C, a brass wire of 1 mm radius is hung from the ceiling. A small mass, M is hung from the free end of the wire. When the wire is cooled down from 40oC to 20oC it regains its original length of 0.2 m. The value of M is close to: (Coefficient of linear expansion and Young’s modulus of brass are 10–5 /oC and 1011 N/m 2, respectively; g= 10 ms–2)

A steel rail of length 5 m and area of cross section 40cm2 is prevented from expanding along its length while the temperature rises by 10oC. If coefficient of linear expansion and Young’s modulus of steel are 1.2×10−5 K−1 and 2×1011 Nm−2 respectively, the force developed in the rail is approximately:

A compressive force, F is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta$T. The net change in its length is zero. Let $\ell$ be the length of the rod, A its area of cross-section,Y its Young’s modulus, and $\alpha$ its coefficient of linear expansion. Then, F is equal to:

An external pressure P is applied on a cube at 0oC so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and $\alpha$ is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:

A simple pendulum made of a bob of mass m and a metallic wire of negligible mass has time period 2 s at T=0oC. If the temperature of the wire is increased and the corresponding change in its time period is plotted against its temperature, the resulting graph is a line of slope S. If the coefficient of linear expansion of metal is $\alpha$ then the value of S is:

A pendulum clock loses $12\text{ s}$ a day if the temperature is $40^{\circ}\text{C}$ and gains $4\text{ s}$ a day if the temperature is $20^{\circ}\text{C}.$ The temperature at which the clock will show correct time, and the co-efficient of linear expansion $(\alpha)$ of the metal of the pendulum shaft are respectively:

On a linear temperature scale Y, water freezes at $-160^\circ$ Y and boils at $-50^\circ$ Y. On this Y scale, a temperature of 340 K would be read as: (water freezes at 273 K and boils at 373 K)

The ratio of the coefficient of volume expansion of a glass container to that of a viscous liquid kept inside the container is 1: 4. What fraction of the inner volume of the container should the liquid occupy so that the volume of the remaining vacant space will be same at all temperatures?

A wooden wheel of radius $R$ is made of two semicircular part (see figure). The two parts are held together by a ring made of a metal strip of cross sectional area $S$ and length $L.\ L$ is slightly less than $2\pi R.$ To fit the ring on the wheel, it is heated so that its temperature rises by $\Delta T$ and it just steps over the wheel. As it cools down to surrounding temperature, it process the semicircular parts together. If the coefficient of linear expansion of the metal is $\alpha$, and its Young's modulus is $Y,$ the force that one part of the wheel applies on the other part is:

The coefficients of linear expansions of brass and steel are $\alpha_1$ and $\alpha_2$ respectively. When we take a brass rod of length $l_1$ and a steel rod of length $l_2$ at $0^\circ\text{C}$, then the difference in their lengths $(l_2 - l_1)$ will remain the same at all temperatures if

Mercury thermometer can be used to measure temperature upto