Advertisements
Advertisements
प्रश्न
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
उत्तर १
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.
उत्तर २
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
संबंधित प्रश्न
A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250 °C, if the original lengths are at 40.0 °C? Is there a ‘thermal stress’ developed at the junction? The ends of the rod are free to expand (Co-efficient of linear expansion of brass = 2.0 × 10–5 K–1, steel = 1.2 × 10–5 K–1).
If two bodies are in thermal equilibrium in one frame, will they be in thermal equilibrium in all frames?
Answer the following question.
Derive the relation between three coefficients of thermal expansion.
Answer the following question.
Give an example of the disadvantages of thermal stress in practical use?
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 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 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 length Land cross-sectional area A is heated through T °C. What is the force required to prevent the expansion of the rod lengthwise?
(Y = Young's modulus of material of the rod, α = coefficient of linear expansion of the rod.)
An aluminium sphere is dipped into water. Which of the following is true?
As the temperature is increased, the time period of a pendulum ______.
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.
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:
If the temperature of the sun were to increase from T to 2T and its radius from R to 2R, then the ratio of the radiant energy received on earth to what it was previously will be ______.
A clock with an iron pendulum keeps the correct time at 15°C. If the room temperature is 20°C, the error in seconds per day will be near ______.
(coefficient of linear expansion of iron is 1.2 × 10-5/°C)
When heat is given to a substance, it generally ______.
Which of the following correctly lists the three types of thermal expansion?
