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 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, assuming 50% of power is used up in heating the machine itself or lost to the surroundings Specific heat of aluminium = 0.91 J g–1 K–1
If an automobile engine is overheated, it is cooled by pouring water on it. It is advised that the water should be poured slowly with the engine running. Explain the reason.
Is it possible for two bodies to be in thermal equilibrium if they are not in contact?
The density of water at 0°C is 0.998 g cm–3 and at 4°C is 1.000 g cm–1. Calculate the average coefficient of volume expansion of water in the temperature range of 0 to 4°C.
A steel rod is clamped at its two ends and rests on a fixed horizontal base. The rod is unstrained at 20°C.
Find the longitudinal strain developed in the rod if the temperature rises to 50°C. Coefficient of linear expansion of steel = 1.2 × 10–5 °C–1.
Answer the following question.
What is thermal stress?
A hot body at a temperature 'T' is kept in a surrounding of temperature 'T0'. It takes time 't1' to cool from 'T' to 'T2', time t2 to cool from 'T2' to 'T3' and time 't3' to cool from 'T3' to 'T4'. If (T - T2) = (T2 - T3) = (T3 - T4), then ______.
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 ______.
A bimetallic strip is made of aluminium and steel (αAl > αsteel) . On heating, the strip will ______.
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly ______.
An aluminium sphere is dipped into water. Which of the following is true?
As the temperature is increased, the time period of a pendulum ______.
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 ______.
Find out the increase in moment of inertia I of a uniform rod (coefficient of linear expansion α) about its perpendicular bisector when its temperature is slightly increased by ∆T.
Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of 57°C is drunk. You can take body (tooth) temperature to be 37°C and α = 1.7 × 10–5/°C, bulk modulus for copper = 140 × 109 N/m2.
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 'α'. The metal sheet is heated uniformly, by a small temperature ΔT, so that its new temeprature is T + ΔT. Calculate the increase in the volume of the metal box.
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 ______.
An anisotropic material has coefficient of linear thermal expansion α1, α2 and α3 along x, y and z-axis respectively. Coefficient of cubical expansion of its material will be equal to ______.
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:
