Advertisements
Advertisements
प्रश्न
Answer the following question.
Derive the relation between three coefficients of thermal expansion.
Advertisements
उत्तर
- Consider a square plate of side l0 at 0° C and lT at T °C.
∴ lT = l0 (1 + αT)
If area of plate at 0° C is A0, A0 = `l_0^2`
If area of plate at T °C is AT,
`"A"_"T" = l_1^2 = l_0^2 (1 +alpha"T")^2`
or `"A"_"T" = "A"_0(1 + alpha"T")^2` ....(1)
Also,
`"A"_"T" = "A"_0(1 + beta"T")` ....(2) `[because beta = ("A"_"r" - "A"_0)/("A"_0("T" - "T"_0))]` - Using Equations (1) and (2),
`"A"_0(1 + alpha"T")^2 = "A"_0(1 + beta"T")`
∴ `1 + 2alpha"T" + alpha^2"T"^2 = 1 + beta"T"` - Since the values of α are very small, the term α2T2 is very small and may be neglected.
∴ β = 2α - The result is general because any solid can be regarded as a collection of small squares.
- Consider a cube of side l0 at 0 °C and lT at T °C.
∴ lT = l0(1 + αT)
If volume of the cube at 0 °C is V0, V0 = `l_0^3`
If volume of the cube at T °C is
`"V"_"T", "V"_"T" = l_"T"^3 = l_0^3 (1 + alpha"T")^3`
`"V"_"T" = "V"_0(1 + alpha"T")^3` ....(1)
Also,
`"V"_"T" = "V"_0(1 + gamma"T")` ....(2) ...`[therefore gamma = ("V"_"T" - "V"_0)/("V"_0("T" - "T"_0))]` - Using Equations (1) and (2),
`"V"_0 (1 + alpha"T")^3 = "V"_0(1 + gamma"T")`
∴ `1 + 3alpha"T" + 3alpha^2"T"^2 + alpha^3"T"^3 = 1 + gamma"T"` - Since the values of α are very small, the terms with higher powers of α may be neglected.
∴ γ = 3α - The result is general because any solid can be regarded as a collection of small cubes.
- Relation between α, β and γ is given by,
`alpha = beta/2 = gamma/3`
where, α = coefficient of linear expansion,
β = coefficient of superficial expansion,
γ = coefficient of cubical expansion.
APPEARS IN
संबंधित प्रश्न
A steel tape 1m long is correctly calibrated for a temperature of 27.0 °C. The length of a steel rod measured by this tape is found to be 63.0 cm on a hot day when the temperature is 45.0 °C. What is the actual length of the steel rod on that day? What is the length of the same steel rod on a day when the temperature is 27.0 °C? Coefficient of linear expansion of steel = 1.20 × 10–5 K–1
If two bodies are in thermal equilibrium in one frame, will they be in thermal equilibrium in all frames?
If mercury and glass had equal coefficients of volume expansion, could we make a mercury thermometer in a glass tube?
For a constant-volume gas thermometer, one should fill the gas at
A gas thermometer measures the temperature from the variation of pressure of a sample of gas. If the pressure measured at the melting point of lead is 2.20 times the pressure measured at the triple point of water, find the melting point of lead.
Show that the moment of inertia of a solid body of any shape changes with temperature as I = I0 (1 + 2αθ), where I0 is the moment of inertia at 0°C and α is the coefficient of linear expansion of the solid.
Answer the following question.
What is thermal stress?
Solve the following problem.
A blacksmith fixes iron ring on the rim of the wooden wheel of a bullock cart. The diameter of the wooden rim and the iron ring are 1.5 m and 1.47 m respectively at room temperature of 27 °C. To what temperature the iron ring should be heated so that it can fit the rim of the wheel? (αiron = 1.2 × 10–5K–1).
A clock pendulum having coefficient of linear expansion. α = 9 × 10-7/°C-1 has a period of 0.5 s at 20°C. If the clock is used in a climate, where the temperature is 30°C, how much time does the clock lose in each oscillation? (g = constant)
A metal rod is heated to t°C. A metal rod has length, area of cross-section, Young's modulus and coefficient of linear expansion as 'L', 'A', 'Y' and 'a' respectively. When the rod is heated, the work performed is ______.
A metal rod of Young's moduls 'Y' and coefficient of linear expansion 'a' has its temeprature raised by 'Δ θ'. The linear stress to prevent the expansion of rod is ______.
(L and l is original length of rod and expansion respectively)
A metal rod of cross-sectional area 3 × 10-6 m2 is suspended vertically from one end has a length 0.4 m at 100°C. Now the rod is cooled upto 0°C, but prevented from contracting by attaching a mass 'm' at the lower end. The value of 'm' is ______.
(Y = 1011 N/m2, coefficient of linear expansion = 10-5/K, g = 10m/s2)
The volume of a metal block changes by 0.86% when heated through 200 °C then its coefficient of cubical expansion is ______.
The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is α. The sphere is heated a little by a temperature ∆T so that its new temperature is T + ∆T. The increase in the volume of the sphere is approximately ______.
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 diameters have been measured at room temperature (27°C). (Given: coefficient of linear thermal expansion of gold αL = 1.4 × 10-5 K-1).
If the length of a cylinder on heating increases by 2%, the area of its base will increase by ______.
The height of mercury column measured with brass scale at temperature T0 is H0. What height H' will the mercury column have at T = 0°C. Coefficient of volume expansion of mercury is γ. Coefficient of linear expansion of brass is α ______.
Length of steel rod so that it is 5 cm longer than the copper rod at all temperatures should be ______ cm.
(α for copper = 1.7 × 10-5/°C and α for steel = 1.1 × 10-5/°C)
A solid metallic cube having a total surface area of 24 m2 is uniformly heated. If its temperature is increased by 10°C, calculate the increase in the volume of the cube.
(Given: α = 5.0 × 10−4°C−1)
A glass flask is filled up to a mark with 50 cc of mercury at 18°C. If the flask and contents are heated to 38°C, how much mercury will be above the mark? (α for glass is 9 × 10-6/°C and coefficient of real expansion of mercury is 180 × 10-6/°C)
When heat is given to a substance, it generally ______.
The increase in the dimensions of a body due to an increase in its temperature is called ______.
Which of the following correctly lists the three types of thermal expansion?
When a solid is heated, its atoms vibrate faster and move farther apart. This happens because ______.
Among solids, liquids, and gases, the thermal expansion on heating is ______.
The volume of a block of metal at 30°C changes by 0.12% when its temperature is increased to 70°C. The coefficient of linear expansion of the metal is ______.
