Advertisements
Advertisements
Question
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
Advertisements
Solution 1
On a day when the temperature is 27 °C, the length of 1 cm division on the steel tape is exactly 1 cm, because the tape has been calibrated for 27 °C.When the temperature rises to 45 °C (that is, ΔT = 45 – 27 = 18 °C), the increase in the length of 1 cm division is Δl = αlΔT = (1.2 x 10-5C-1) x 1 cm x 18 °C = 0.000216 cm Therefore, the length of 1 cm division on the tape becomes 1.000216 cm at 45 °C. As the length of the steel rod is read to be 63.0 cm on the steel tape at 45 °C, the actual length of the rod at 45 °C is 63.0 x 1.000216 cm = 63.0136 cm The length of the same rod at 27 °C is 63.0 cm, because 1 cm mark on the steel tape is exactly 1 cm at 27 °C.
Solution 2
Length of the steel tape at temperature T = 27°C, l = 1 m = 100 cm
At temperature T1 = 45°C, the length of the steel rod, l1 = 63 cm
Coefficient of linear expansion of steel, α = 1.20 × 10–5 K–1
Let l2 be the actual length of the steel rod and l' be the length of the steel tape at 45°C.
`l' = l + al(T-1- T)`
`:. l' = 100 + 1.20 xx 10^(-5) xx 100(45- 27)`
= 100.0216 cm
Hence, the actual length of the steel rod measured by the steel tape at 45°C can be calculated as:
`l_2= 100.0216/100 xx 63 = 63.0136 cm`
Therefore, the actual length of the rod at 45.0°C is 63.0136 cm. Its length at 27.0°C is 63.0 cm.
RELATED QUESTIONS
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.
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?
Answer the following question.
State applications of thermal expansion.
Answer the following question.
What is thermal stress?
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 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)
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly ______.
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.
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.
If the length of a cylinder on heating increases by 2%, the area of its base will increase by ______.
A disc is rotating freely about its axis. The percentage change in angular velocity of a disc if temperature decreases by 20°C is ______.
(coefficient of linear expansion of material of disc is 5 × 10-4/°C)
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 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)
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 a solid is heated, its atoms vibrate faster and move farther apart. This happens because ______.
