Q6 of 26 Page 294

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.

Given,

Length of the steel tape at temperature T = 27°C = 27 + 273.15 K = 300.15 K, L = 1 m


Coefficient of linear expansion of steel, α = 1.20 × 10-5 K-1


At temperature T1 = 45°C = 45 + 273.15 K = 318.15 K, the length of the steel rod,


L1 = 63 cm = 0.63m


Let L2 be the actual length of the steel rod and L2’ be the length measured by the steel tape at 45°C.


We know,


L2’ = L + αL(T1 - T)


∴ L2’ = 1 m + (1.20 × 10-5 K-1)(1 m)(318.15 K – 300.15 K)


⇒ L2’ = 1 + 0.000216


⇒ L2’ = 1.000216 m = 100.0216 cm


Hence, the actual length of the steel rod at 45°C is given by


L2 = (L2’/L)×L1


⇒ L2 = [(100.0216 cm)/(100 cm)](63 cm) = 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.


More from this chapter

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4

Answer the following:

(a) The triple-point of water is a standard fixed point in modern thermometry.


Why? What is wrong in taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale)?


(b) There were two fixed points in the original Celsius scale as mentioned above which were assigned the number 0°C and 100°C respectively. On the absolute scale, one of the fixed points is the triple-point of water, which on the Kelvin absolute scale is assigned the number 273.16 K. What is the other fixed point on this (Kelvin) scale?


(c) The absolute temperature (Kelvin scale) T is related to the temperature tc on the Celsius scale by


tc = T – 273.15


Why do we have 273.15 in this relation, and not 273.16?


(d) What is the temperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of the Fahrenheit scale?

 5

Two ideal gas thermometers A and B use oxygen and hydrogen respectively. The following observations are made:


















Temperature



Pressure thermometer A



Pressure


thermometer B



Triple-point of water



1.250 × 105 Pa



0.200 × 105 Pa



Normal melting point of sulphur



1.797 × 105 Pa



0.287 × 105 Pa



(a) What is the absolute temperature of normal melting point of sulphur as read by thermometers A and B ?


(b) What do you think is the reason behind the slight difference in answers of thermometers A and B? (The thermometers are not faulty). What further procedure is needed in the experiment to reduce the discrepancy between the two readings?

 7

A large steel wheel is to be fitted on to a shaft of the same material. At 27°C, the outer diameter of the shaft is 8.70 cm and the diameter of the central hole in the wheel is 8.69 cm. The shaft is cooled using ‘dry ice’. At what temperature of the shaft does the wheel slip on the shaft? Assume coefficient of linear expansion of the steel to be constant over the required temperature range: αsteel = 1.20 × 10–5 K–1.

 8

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.