17Thermal expansion

📚
THERMAL EXPANSION
▢ Introduction:
❖ Definition: Increase in length, area or volume of body on heating
❖ General Rule:
  • •
    Usually body expands on heating
  • •
    Usually body contracts on cooling
❖ Cause: Increase in intermolecular separation against intermolecular force
❖ Order: Solids < Liquids < Gases
❖ Reason: Intermolecular force: maximum in solids, less in liquids, least in gases

Table 1: Types of Expansion

Substance
Expansion
Solids
Linear, superficial, cubical
Liquids
Only volume expansion
Gases
Only volume expansion
▢ Expansion of Solids:

Table 1: Linear, Superficial and Cubical Expansion

Feature
Linear expansion
Superficial expansion
Cubical / volume expansion
Meaning
Increase in length
Increase in area
Increase in volume
Dimension
1-D
2-D
3-D
Final value
\(L_2=L_1(1+\alpha\Delta\theta)\)
\(A_2=A_1(1+\beta\Delta\theta)\)
\(V_2=V_1(1+\gamma\Delta\theta)\)
Coefficient
\(\alpha\) = coefficient of linear expansion
\(\beta\) = coefficient of superficial expansion
\(\gamma\) = coefficient of cubical expansion
Change
\(\Delta L=L_1\alpha\Delta\theta\)
\(\Delta A=A_1\beta\Delta\theta\)
\(\Delta V=V_1\gamma\Delta\theta\)
% change
\(\frac{\Delta L}{L_1}\times100=\alpha\Delta\theta\times100\)
\(\frac{\Delta A}{A_1}\times100=\beta\Delta\theta\times100\)
\(\frac{\Delta V}{V_1}\times100=\gamma\Delta\theta\times100\)
❖ Important Points:
  • •
    \(\alpha,\beta,\gamma\) are independent of original dimensions
  • •
    \(\alpha,\beta,\gamma\) depend on nature of material
  • •
    For perfectly rigid body: \(\alpha=\beta=\gamma=0\)
▢ Relation Between Coefficients:

Table 1: Isotropic vs Anisotropic Body

Feature
Isotropic body
Anisotropic body
Linear expansion
\(\alpha_1=\alpha_2=\alpha_3=\alpha\)
\(\alpha_1\neq\alpha_2\neq\alpha_3\)
Cubical expansion
\(\gamma=\alpha+\alpha+\alpha=3\alpha\)
\(\gamma=\alpha_1+\alpha_2+\alpha_3\)
Superficial expansion
\(\beta=\alpha+\alpha=2\alpha\)
\(\beta=\alpha_1+\alpha_2\) or \(\alpha_2+\alpha_3\) or \(\alpha_3+\alpha_1\)
Relation
\(\gamma=\frac{3}{2}\beta\)
Depends on direction
Ratio
\(\alpha:\beta:\gamma=1:2:3\)
No fixed ratio
❖ Special Rod Condition: If two rods of lengths \(l_1,l_2\) and expansivities \(\alpha_1,\alpha_2\) have same difference in length at all temperatures, then \(\alpha_1l_1=\alpha_2l_2\)
▢ Effect of Temperature on Simple Pendulum:
❖ Basic Formula: \(T=2\pi\sqrt{\frac{l}{g}}\)
❖ Relation: \(T\propto\sqrt l\)

Table 1: Temperature Correction in Pendulum

Quantity
Formula
Fractional change in period
\(\frac{\Delta T}{T}=\frac{1}{2}\frac{\Delta l}{l}\)
Using linear expansivity
\(\frac{\Delta T}{T}=\frac{1}{2}\alpha\Delta\theta\)
% change in time period
\(\frac{\Delta T}{T}\times100=\frac{1}{2}\alpha\Delta\theta\times100\)
Change in time period
\(\Delta T=\frac{1}{2}\alpha\Delta\theta\times T\)
❖ Clock Effect:
  • •
    Temperature increases → length increases → time period increases → clock runs slow / loses time
  • •
    Temperature decreases → length decreases → time period decreases → clock runs fast / gains time
  • •
    Summer → pendulum clock runs slower
  • •
    Winter → pendulum clock runs faster
▢ Expansion of Liquids:
❖ Key Point: Liquid is heated in a vessel; vessel also expands
❖ Types of Expansion:

Table 1: Real and Apparent Expansion of Liquid

Type
Meaning
Formula
Real expansion
Actual change in volume of liquid only
\(\Delta V=V_0\gamma_r\Delta\theta\)
Apparent expansion
Observed change considering expansion of vessel
\(\Delta V=V_0\gamma_a\Delta\theta\)
Relation
Real expansion = apparent expansion + expansion of vessel
\(\gamma_r=\gamma_a+\gamma_s\)
❖ Density Change:
◉ Formula: \(\rho=\frac{\rho_0}{1+\gamma\Delta\theta}\approx\rho_0(1-\gamma\Delta\theta)\)
◉ Point: On heating, volume increases and density decreases
▢ Correction of Barometer Reading:
❖ Condition: Mercury column height \(h\) at temperature \(\theta\)
❖ Formula: \(h_0=h[1-(\gamma-\alpha)\theta]\)
❖ Symbols:
  • •
    \(h_0\) = corrected height
  • •
    \(h\) = observed height
  • •
    \(\alpha\) = linear expansivity of scale material
  • •
    \(\gamma\) = cubical expansivity of mercury
▢ Anomalous Expansion of Water:
❖ Definition: Peculiar behaviour of water in which it contracts on heating from 0°C to 4°C and expands beyond 4°C

Table 1: Water Expansion

Temperature range
Behaviour
0°C to 4°C
Water contracts on heating
At 4°C
Volume minimum; density maximum
Above 4°C
Water expands on heating
Density at 4°C
\(1\ g/cc\)
❖ Importance: Aquatic life survives in cold regions because water at bottom of lakes/ponds remains liquid near 4°C even when surface freezes
❖ Coefficient of Cubical Expansion of Water:

Table 1: Cubical Expansivity of Water

Range
\(\gamma\)
Below 4°C
Negative
At 4°C
Zero
Above 4°C
Positive
▢ Bimetallic Strip:
❖ Definition: Two different metal strips joined together to form a single strip
❖ Uses:
  • •
    Thermometer
  • •
    Thermostat
  • •
    Automatic breaking of electric circuits
  • •
    Fire alarm

Table 1: Bimetallic Strip Bending

Condition
Direction of bending
Concave side
Convex side
Heated
Bends towards metal with smaller \(\alpha\)
Smaller \(\alpha\)
Larger \(\alpha\)
Cooled
Bends towards metal with larger \(\alpha\)
Larger \(\alpha\)
Smaller \(\alpha\)
❖ Example: If \(\alpha_A>\alpha_B\), on heating strip bends towards B
▢ Expansion of Gases:
❖ Key Points:
  • •
    Gases have only volume expansion
  • •
    Coefficient of expansion of gases is greater than liquids and solids
  • •
    Volume expansivity is defined at constant pressure
  • •
    Pressure coefficient is defined at constant volume

Table 1: Expansion Coefficients of Gas

Coefficient
Definition
Formula
Value
Volume expansivity
Fractional increase in volume per unit rise of temperature at constant pressure
\(\gamma_v=\left(\frac{\Delta V}{V_0\Delta\theta}\right)_P\)
\(\frac{1}{273}\ ^\circ C^{-1}\)
Pressure coefficient
Fractional increase in pressure per unit rise of temperature at constant volume
\(\gamma_p=\left(\frac{\Delta P}{P_0\Delta\theta}\right)_V\)
\(\frac{1}{273}\ ^\circ C^{-1}\)
Relation
For ideal gas
\(\gamma_v=\gamma_p\)
\(\frac{1}{273}\ ^\circ C^{-1}\)
▢ Thermal Stress in Rigidly Fixed Rod:
❖ Condition: Metallic rod of length \(L\), area \(A\), linear expansivity \(\alpha\), Young's modulus \(Y\), clamped between rigid supports and heated by \(\Delta\theta\)

Table 1: Thermal Stress Formulae

Quantity
Formula
Thermal strain
\(\frac{\Delta L}{L}=\alpha\Delta\theta\)
Thermal stress
\(Y\alpha\Delta\theta\)
Thermal force
\(F=AY\alpha\Delta\theta\)
Thermal strain energy
\(U=\frac{1}{2}YAL\alpha^2\Delta\theta^2\)
Strain energy per unit volume
\(\frac{U}{V}=\frac{1}{2}Y\alpha^2\Delta\theta^2\)
▢ Heating of Holes and Cavities:

Table 1: Expansion of Holes

Case
Result
Metal disc with hole heated
Size of hole increases
Solid ball with concentric spherical cavity heated
Volume of cavity increases
Reason
Hole/cavity expands as if filled with same material
▢ Liquid in Container:
❖ Condition: Liquid with cubical expansivity \(\gamma\) in container of material with linear expansivity \(\alpha\)

Table 1: Level of Liquid on Heating

Condition
Result
\(\gamma>3\alpha\)
Liquid level rises / overflows if completely filled
\(\gamma=3\alpha\)
Liquid level remains stationary
\(\gamma<3\alpha\)
Liquid level falls
❖ Initial Fall Then Rise: If mercury is heated in vessel, liquid column first descends due to vessel expansion and then ascends due to mercury expansion
▢ Apparent Weight with Temperature:

Table 1: Apparent Weight in Liquid

Condition
Formula / Result
Solid cubical expansivity
\(\gamma_s\)
Liquid cubical expansivity
\(\gamma_l\)
Temperature change
\(\Delta\theta=\theta_2-\theta_1\)
Change in apparent weight
\(\Delta W=W_2-W_1=W(\gamma_l-\gamma_s)\Delta\theta\)
\(\gamma_l>\gamma_s\)
\(W_2>W_1\)
\(\gamma_l=\gamma_s\)
\(W_2=W_1\)
\(\gamma_l<\gamma_s\)
\(W_2
❖ Note: Cases above are for heating; converse is true for cooling
❖ General Point: A metallic piece weighed in a liquid whose temperature is raised continuously has increasing apparent weight
▢ Empty Space in Measuring Cylinder:
❖ Condition: Measuring cylinder capacity \(V_0\), cubical expansivity \(\gamma_0\), contains liquid volume \(V\), cubical expansivity \(\gamma\)
❖ Formula: \(\gamma_0V_0=\gamma V\)
❖ Meaning: Condition for empty space to remain unchanged on heating or cooling
▢ Invar:
❖ Property: Very low coefficient of linear expansion
❖ Uses:
  • •
    Metal scales
  • •
    Pendulum clocks
▢ Read and Digest:

Table 1: Important Thermal Expansion Points

Fact
Answer
Water heated from 0°C to 10°C
Volume first decreases then increases
Temperature of liquid increases
Volume increases; density decreases
Copper sphere heated
% increase maximum for volume
Reason
\(\gamma=3\alpha\), \(\beta=2\alpha\)
Expansion on heating
Density decreases
Water density
Maximum at 4°C
Water volume
Minimum at 4°C
Water from 0°C to 4°C
\(C_P
Water at 4°C
\(C_P=C_V\)
Beaker completely filled with water at 4°C
Overflows on heating as well as cooling
Glass jar full of water in freezing mixture
Breaks because water expands from 4°C to 0°C
Two rods same length difference at all temperature
\(L_1\alpha_1=L_2\alpha_2\)
Water bottle at 0°C opened on Moon
Water boils because pressure is nearly zero
Identical hollow and solid spheres heated by equal heat
Hollow sphere expands more due to less mass
Metal disc with hole heated
Hole increases
Spherical cavity heated
Cavity volume increases
Fire alarm
Works on bending of bimetallic strip
Metal floating in mercury heated
Floats at lower level
Same volume of benzene
Weighs less in summer than winter
Same mass of benzene
Occupies more volume in summer than winter
50 g benzene
Weighs same in summer and winter
\(\alpha,\beta,\gamma\)
Depend on unit of temperature
Invar
Used in metal scales and pendulum clocks
▢ High-Yield Recall:

Table 1: Thermal Expansion One-Liners

Fact
Answer
Thermal expansion
Increase in dimensions on heating
Cause
Increase in intermolecular separation
Expansion order
Solid < Liquid < Gas
Solids expansion
Linear, superficial, cubical
Liquids and gases
Only volume expansion
Linear expansion
\(L_2=L_1(1+\alpha\Delta\theta)\)
Superficial expansion
\(A_2=A_1(1+\beta\Delta\theta)\)
Cubical expansion
\(V_2=V_1(1+\gamma\Delta\theta)\)
Isotropic relation
\(\alpha:\beta:\gamma=1:2:3\)
Cubical coefficient
\(\gamma=3\alpha\)
Superficial coefficient
\(\beta=2\alpha\)
Pendulum temperature correction
\(\frac{\Delta T}{T}=\frac{1}{2}\alpha\Delta\theta\)
Summer pendulum clock
Runs slow
Winter pendulum clock
Runs fast
Real expansion of liquid
\(\Delta V=V_0\gamma_r\Delta\theta\)
Apparent expansion of liquid
\(\Delta V=V_0\gamma_a\Delta\theta\)
Real-apparent relation
\(\gamma_r=\gamma_a+\gamma_s\)
Density on heating
\(\rho\approx\rho_0(1-\gamma\Delta\theta)\)
Barometer correction
\(h_0=h[1-(\gamma-\alpha)\theta]\)
Anomalous expansion of water
Contracts from 0°C to 4°C; expands above 4°C
Maximum density of water
At 4°C
Minimum volume of water
At 4°C
Bimetallic strip heated
Bends towards smaller \(\alpha\)
Bimetallic strip cooled
Bends towards larger \(\alpha\)
Gas expansivity
\(\gamma_v=\gamma_p=\frac{1}{273}\ ^\circ C^{-1}\)
Thermal stress
\(Y\alpha\Delta\theta\)
Thermal force
\(AY\alpha\Delta\theta\)
Thermal strain energy
\(\frac{1}{2}YAL\alpha^2\Delta\theta^2\)
Hole in metal disc on heating
Increases
Invar use
Metal scales and pendulum clocks
Q1.
Moment of inertia of body is I and coefficient of linear expansion is α. If temperature increases, new moment of inertia will be
📅CEE MBBS 2025
Q2.
When water is heated from 0 to 4°C:
📅CEE Nursing 2024
Q3.
The cubical expansivity of the sphere is less than that of liquid. With increase of temperature, the apparent weight of sphere will:
📅CEE MBBS 2024B
Q4.
If the coefficient of linear expansion of glass with mass 50 g is given, the mass of liquid it holds at 80°C is:
📅CEE MBBS 2024a
Q5.
A steel tape gives correct measurement at 50°C. A piece of wood is measured with the steel tape at 0°C and the reading is 50 cm. The real length of the piece of wood must be:
📅CEE MBBS 2024a
Q6.
A uniform metal rod of length 'L' and mass 'M' is rotating about an axis passing through one of the ends perpendicular to the rod with angular speed 'ω'. If the temperature increase to 't' degree Celsius then the change in angular velocity is proportional to which of the following? (Coefficient of linear expansion of rod = α)
📅CEE BDS 2024
Q7.
If a material has an initial volume of 500 m³, undergoes a change in volume of 250 cm³, and has a cubical expansivity of 2 × 10⁻⁵, what is the value of change in temperature?
📅CEE BDS 2024
Q8.
A steel metre scale is to be ruled so that millimeter intervals are accurate within about 5×10⁻⁵ mm at a certain temperature. The maximum temperature variation allowable during the ruling is (coefficient of linear expansion of steel = 10×10⁻⁶):
📅CEE BDS 2024
Q9.
A pendulum clock gains 5 seconds per day at 16°C and loses 15 seconds per day at 40°C. Then the clock keeps correct time at:
📅CEE BDS 2021
Q10.
If a bimetallic strip is heated, it will:
📅CEE BDS 2021
Q11.
A liquid is placed in a graduated glass cylinder. If its real volume-expansion coefficient is three times the linear-expansion coefficient of glass, how does the level change relative to the cylinder markings on heating?
📅IOM 2013
Q12.
A brass pendulum clock keeps correct time at \(20^\circ\mathrm C\). How many seconds does it lose per day at \(35^\circ\mathrm C\)? Take \(\alpha_{\mathrm{brass}}=2\times10^{-5}\,\mathrm{K^{-1}}\).
📅IOM 2010
Q13.
Let \(C\) and \(S\) be the apparent volume-expansion coefficients of a liquid in copper and silver vessels, respectively. If the linear-expansion coefficient of copper is \(A\), what is that of silver?
📅IOM 2009
Q14.
A thin copper wire of length \(L\) expands by \(0.1\%\) upon heating. What is the approximate percentage increase in the area of a copper plate initially measuring \(L\times2L\), heated through the same temperature interval?
📅IE 2010
Q15.
The apparent volume-expansion coefficients of a liquid in copper and steel vessels are \(6\times10^{-6}\,\mathrm{K^{-1}}\) and \(24\times10^{-6}\,\mathrm{K^{-1}}\), respectively. If \(\alpha_{\mathrm{Cu}}=18\times10^{-6}\,\mathrm{K^{-1}}\), find \(\alpha_{\mathrm{steel}}\).
📅IE 2013
Q16.
A conductor has resistance \(15\,\Omega\) at \(60^\circ\mathrm C\) and \(20\,\Omega\) at \(100^\circ\mathrm C\). Assuming linear variation, find its resistance at \(10^\circ\mathrm C\).
📅MOE 2012
Q17.
A material has resistance \(1.72\,\Omega\) at \(20^\circ\mathrm C\). Find its resistance at \(100^\circ\mathrm C\), given the temperature coefficient at \(20^\circ\mathrm C\) is \(0.00393\,\mathrm{K^{-1}}\).
📅KU 2010
Q18.
A zinc rod is \(1\,\mathrm m\) long at \(0^\circ\mathrm C\). Find its length at \(50^\circ\mathrm C\) if \(\alpha_{\mathrm{Zn}}=26\times10^{-6}\,\mathrm{K^{-1}}\).
📅MOE 2014
Q19.
Two rods have initial lengths \(L_1,L_2\) and linear-expansion coefficients \(\alpha_1,\alpha_2\). If the difference between their lengths is temperature-independent, which relation must hold?
📅MOE 2014
Q20.
A brass rod and a lead rod, each \(80\,\mathrm{cm}\) long at \(0^\circ\mathrm C\), are fixed together at one end. They are heated to \(100^\circ\mathrm C\). Find the approximate difference in their lengths if \(\alpha_{\mathrm{lead}}-\alpha_{\mathrm{brass}}=1.0\times10^{-5}\,\mathrm{K^{-1}}\).
Q21.
Solid and hollow spheres of the same material have the same initial outer radius. If their temperatures are raised by the same amount, which has the greater increase in outer radius?
Q22.
A solid sphere and a hollow sphere of the same material have the same initial outer radius and temperature. If equal amounts of heat are supplied to both without losses, which has the greater increase in outer radius?
Q23.
A circular hole in a metal disc has diameter \(1.5\,\mathrm{cm}\) at \(20^\circ\mathrm C\). If the disc is heated to \(150^\circ\mathrm C\) and \(\alpha=1.9\times10^{-5}\,\mathrm{K^{-1}}\), what happens to the diameter?
Q24.
A rod is heated from \(0^\circ\mathrm C\) to \(20^\circ\mathrm C\) while its expansion is completely prevented. If \(\alpha=12\times10^{-6}\,\mathrm{K^{-1}}\) and \(Y=10^{11}\,\mathrm{N/m^2}\), find the stored elastic energy per unit volume.
Q25.
The coefficients of cubical expansion of brass and iron are \(54\times10^{-6}\,\mathrm{K^{-1}}\) and \(36\times10^{-6}\,\mathrm{K^{-1}}\), respectively. If the difference in the lengths of their rods stays constant with temperature, find \(L_{\mathrm{brass}}:L_{\mathrm{iron}}\).
Q26.
A liquid in a container has volume-expansion coefficient \(\gamma\), while the container has linear-expansion coefficient \(\alpha\). What happens to the liquid level relative to markings on the container when heated?
Q27.
An iron cube floats in mercury at \(20^\circ\mathrm C\). Assuming mercury has the larger volume-expansion coefficient, how does the cube float after the temperature is increased by \(100^\circ\mathrm C\)?
Q28.
The real volume-expansion coefficient of glycerin is \(0.000597\,\mathrm{K^{-1}}\), and the linear-expansion coefficient of glass is \(0.000009\,\mathrm{K^{-1}}\). Find glycerin’s apparent volume-expansion coefficient in glass.
Q29.
Steel rails of length \(40\,\mathrm m\) are laid at \(-10^\circ\mathrm C\). What gap is required to allow expansion up to \(40^\circ\mathrm C\) if \(\alpha=12\times10^{-6}\,\mathrm{K^{-1}}\)?
Q30.
A steel measuring tape is accurate at \(20^\circ\mathrm C\). At \(0^\circ\mathrm C\), it indicates that a piece of wood is \(25\,\mathrm{cm}\) long. Assuming the wood’s length is effectively unchanged, what is its true length?
Q31.
Mercury has density \(13.6\times10^3\,\mathrm{kg/m^3}\) at \(0^\circ\mathrm C\) and real volume-expansion coefficient \(18.0\times10^{-5}\,\mathrm{K^{-1}}\). Find its approximate density at \(50^\circ\mathrm C\).
Q32.
A thin copper wire elongates by \(1\%\) over a certain temperature interval. What is the approximate percentage increase in the area of a thin copper plate over the same interval?
Q33.
A body has moment of inertia \(I\) about a fixed axis through its centre. If it expands uniformly with linear-expansion coefficient \(\alpha\) over a small temperature rise \(\Delta\theta\), find the approximate change in moment of inertia.
Q34.
An iron tyre is to fit a wooden wheel of diameter \(1.0\,\mathrm m\), but its initial inner diameter is \(6\,\mathrm{mm}\) smaller. If iron’s cubical-expansion coefficient is \(3.6\times10^{-5}\,\mathrm{K^{-1}}\), approximately how much must the tyre’s temperature rise?
Q35.
A material has linear-expansion coefficient \(\alpha\) per degree Celsius. Express its coefficient per degree Fahrenheit.
Q36.
A glass vessel holds \(50\,\mathrm g\) of a liquid when just filled at \(0^\circ\mathrm C\). If \(\alpha_{\mathrm{glass}}=8\times10^{-6}\,\mathrm{K^{-1}}\) and the liquid’s real volume-expansion coefficient is \(5.5\times10^{-4}\,\mathrm{K^{-1}}\), approximately what mass will fill the vessel at \(80^\circ\mathrm C\)?
Q37.
A brass disc fits tightly into a circular hole in a steel plate. Assuming brass has a larger linear-expansion coefficient than steel, how should the assembly be treated to loosen the disc?
Q38.
A metal ball weighs \(W_1\) apparently when submerged in alcohol at \(0^\circ\mathrm C\) and \(W_2\) at \(50^\circ\mathrm C\). If alcohol has a larger cubical-expansion coefficient than the metal, which relation is correct?
Q39.
A fully immersed metallic object is weighed in a liquid while the temperature rises. If the liquid expands more than the metal, its apparent weight generally:
Q40.
A container filled to the brim with a liquid is heated. The liquid has volume-expansion coefficient \(\gamma\), and the container has linear-expansion coefficient \(\alpha\). When will liquid overflow?
Q41.
The apparent loss of weight of a completely submerged solid is \(W_0\) in a liquid at \(0^\circ\mathrm C\). If their cubical-expansion coefficients are \(\gamma_s\) and \(\gamma_l\), respectively, what is the first-order value \(W_t\) at \(t^\circ\mathrm C\)?
Q42.
A glass flask of volume \(1\,\mathrm L\) is completely filled with mercury at \(0^\circ\mathrm C\). If \(\gamma_{\mathrm{Hg}}=1.82\times10^{-4}\,\mathrm{K^{-1}}\) and \(\alpha_{\mathrm{glass}}=10\times10^{-6}\,\mathrm{K^{-1}}\), how much mercury overflows when heated to \(100^\circ\mathrm C\)?
Q43.
A pendulum clock keeps correct time at \(20^\circ\mathrm C\). Its rod has \(\alpha=2\times10^{-5}\,\mathrm{K^{-1}}\). If the temperature falls to \(10^\circ\mathrm C\), what is its approximate daily error?
Q44.
A pendulum clock gains \(5\,\mathrm{s/day}\) at \(16^\circ\mathrm C\) and loses \(15\,\mathrm{s/day}\) at \(40^\circ\mathrm C\). At what temperature will it keep correct time?
Q45.
When a bonded bimetallic strip is heated uniformly, in which direction does it curve?
Q46.
The density of an isotropic solid decreases by \(0.1\%\) for a temperature rise of \(100^\circ\mathrm C\). Find its approximate coefficient of linear expansion.
📅IOM 1997
Q47.
Two rods A and B have the same initial length and temperature. Rod A has \(\alpha_A=12\times10^{-6}\,\mathrm{K^{-1}}\) and rod B has \(\gamma_B=3\times10^{-5}\,\mathrm{K^{-1}}\). After an equal temperature increase, rod A will be:
📅MOE 2058
Q48.
What happens to a fixed mass of water when it is heated above \(4^\circ\mathrm C\)?
📅IE 2006
Q49.
As water at atmospheric pressure is heated continuously from \(0^\circ\mathrm C\) to \(100^\circ\mathrm C\), its volume:
Q50.
A vessel has linear-expansion coefficient \(\alpha\) and is filled to a fixed mark with liquid. If the liquid reaches the same mark after heating, what is its real volume-expansion coefficient?
📅KU 2015