Advertisements
Advertisements
Question
A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = 1.70 × 10–5 K–1.
Advertisements
Solution 1
Initial temperature, T1 = 27.0°C
Diameter of the hole at T1, d1 = 4.24 cm
Final temperature, T2 = 227°C
Diameter of the hole at T2 = d2
Co-efficient of linear expansion of copper, αCu= 1.70 × 10–5 K–1
For co-efficient of superficial expansion β,and change in temperature ΔT, we have the relation:
`"Change in area(ΔA)"/"Original area(A)" = betaΔT`
`(((pi (d_2^2))/4 - pi (d_1^2)/4))/((pi d_1^2/4)) = (triangleA)/A`
`:.(triangleA)/A = (d_2^2 - d_1^2)/d_1^2`
But `beta = 2 alpha`
`:.(d_2^2-d_1^2)/d_1^2 = 2alpha triangleT`
`d_2^2/d_1^2 - 1= 2alpha (T_2 - T_1)`
`d_2^2/(4.24)^2 = 2xx 1.7 xx10^(-5)(227- 27) + 1`
`d_2^2 = 17.98 xx 1.0068 = 18.1`
`:. d_2 = 4.2544 cm`
Change in diameter = d2 – d1 = 4.2544 – 4.24 = 0.0144 cm
Hence, the diameter increases by 1.44 × 10–2 cm.
Solution 2
In this superficial expansion of copper sheet will be involved on heating
Here, area of hole at `27^@, A_1 = piD_1^2/4 = pi/4 xx(4.24)^2 cm^2`
if `D_2` cm is the diameter of the hole at `227^@C` then area of the hole at `227^@C`
`A_2 = (piD_2^2)/4 cm^2`
Coefficient of superficial expansion of copper is
`beta = 2 alpha = 2xx1.70 xx 10^(-5) = 3.4 xx 10^(-5)"'^@C^(-1)`
Increase in area = `A_2 - A_1 = betaA_1triangleT or A_2 = A_1 +betaA_1 triangle T = A_1(1 + beta triangleT)`
`(piD_2^2)/4 = pi/4(4.24)^2 [1+3.4xx10^(-5)(228 - 27)]`
`=> D_2^2 = (4.24)^2xx 1.0068` or
`D_2 = 4.2544 cm`
Change4 in diameter = `D_2 - D_1 = 4.2544 - 4.24 = 0.0114 cm`
APPEARS IN
RELATED QUESTIONS
The coefficient of volume expansion of glycerin is 49 × 10–5 K–1. What is the fractional change in its density for a 30 °C rise in temperature?
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?
Solve the following problem.
In olden days, while laying the rails for trains, small gaps used to be left between the rail sections to allow for thermal expansion. Suppose the rails are laid at room temperature 27 °C. If maximum temperature in the region is 45 °C and the length of each rail section is 10 m, what should be the gap left given that α = 1.2 × 10–5K–1 for the material of the rail section?
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 metre scale made of a metal reads accurately at 25 °C. Suppose in an experiment an accuracy of 0.12 mm in 1 m is required, the range of temperature in which the experiment can be performed with this metre scale is ______.(coefficient of linear expansion of the metal is `20 xx 10^-6 / (°"C")`
A metal sphere 10.01 cm in diameter is placed on a brass ring of internal diameter 10 cm and at the same temperature of 12° C. The temperature up to which they should be heated together so that the metal sphere just passes through the ring is `[alpha_"metal"= 12 xx 10^-6//°"C" and alpha_"brass" =18 xx 10^-6//°"C"]` ____________.
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly ______.
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 ______.
A rail track made of steel having length 10 m is clamped on a raillway line at its two ends (figure). On a summer day due to rise in temperature by 20° C, it is deformed as shown in figure. Find x (displacement of the centre) if αsteel = 1.2 × 10–5/°C.
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.
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 ______.
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 metal ball immersed in water weighs w1 at 0°C and w2 at 50°C. The coefficient of cubical expansion of metal is less than that of water. Then ______.
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 light of the above statements.
choose the correct answer from the options given below:
Among solids, liquids, and gases, the thermal expansion on heating is ______.
