Advertisements
Advertisements
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).
Advertisements
उत्तर १
Initial temperature, T1 = 40°C
Final temperature, T2 = 250°C
Change in temperature, ΔT = T2 – T1 = 210°C
Length of the brass rod at T1, l1 = 50 cm
Diameter of the brass rod at T1, d1 = 3.0 mm
Length of the steel rod at T2, l2 = 50 cm
Diameter of the steel rod at T2, d2 = 3.0 mm
Coefficient of linear expansion of brass, α1 = 2.0 × 10–5K–1
Coefficient of linear expansion of steel, α2 = 1.2 × 10–5K–1
For the expansion in the brass rod, we have:
`("Change in length"(trianglel_1))/("Original length"(l_1))" = alpha_1triangleT`
`:. trianglel_1 = 50 xx (2.1xx 10^(-5))xx210`
= 0.2205 cm
For the expansion in the steel rod, we have:
`("Change in length"(trianglel_2))/("Original length"(l_2))" = alpha_2triangleT`
`:.trianglel_2 = 50xx (1.2xx10^(-5))xx210`
= 0.126 cm
Total change in the lengths of brass and steel,
Δl = Δl1 + Δl2
= 0.2205 + 0.126
= 0.346 cm
Total change in the length of the combined rod = 0.346 cm
Since the rod expands freely from both ends, no thermal stress is developed at the junction.
उत्तर २
Here `l_"brass" = l_"steel" = 50 cm, d_"brass" = d_"steel" = 3mm = 0.3 cm, trianglel_"brass" = ?, trianglel_"steel" = ?`
`triangleT = 250 - 40 = 210 ""^@C`
`alpha_"brass" = 2xx10^(-5) ""^@C^(-1) and alpha_"steel" = 1.2 xx 10^(-5) ""^@C^(-1)`
Now `trianglel_"brass" = alpha_"brass" xx l_"brass" xx triangleT`
`= 2xx10^(-5) xx 50 xx 210 = 0.21 cm`
Now `trianglel_"steel" = alpha_"steel" xx l_"steel" xx triangleT`
`= 1.2 xx 10^(-5) xx 50 xx 210`
`=0.126 cm = 0.13 cm`
`:. "Total change in length," trianglel = trianglel_"brass" + trianglel_"steel" = 0.21 + 0.13 = 0.34 cm`
Since the rod is not clamped at its ends, no thermal stress developed at the junction.
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
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.
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?
Is it possible for two bodies to be in thermal equilibrium if they are not in contact?
A system X is neither in thermal equilibrium with Y nor with Z. The systems Y and Z
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.
Answer the following question.
State applications of thermal expansion.
Answer the following question.
Give an example of the disadvantages of thermal stress in practical use?
A glass flask has a volume 1 × 10−4 m3. It is filled with a liquid at 30°C. If the temperature of the system is raised to 100°C, how much of the liquid will overflow? (Coefficient of volume expansion of glass is 1.2 × 10−5 (°C)−1 while that of the liquid is 75 × 10−5 (°C)−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 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)
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 ______.
A student records the initial length l, change in temperature ∆T and change in length ∆l of a rod as follows:
| S.No. | l(m) | ∆T (C) | ∆l (m) |
| 1. | 2 | 10 | `4 xx 10^-4` |
| 2. | 1 | 10 | `4 xx 10^-4` |
| 3. | 2 | 20 | `2 xx 10^-4` |
| 4. | 3 | 10 | `6 xx 10^-4` |
If the first observation is correct, what can you say about observations 2, 3 and 4.
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 ______.
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)
