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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 Assume Coefficient of Linear Expansion of the Steel to Be Constant Over the Required Temperature Range - CBSE (Science) Class 11 - Physics

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Question

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.

Solution 1

The given temperature, T = 27°C can be written in Kelvin as:

27 + 273 = 300 K

Outer diameter of the steel shaft at Td1 = 8.70 cm

Diameter of the central hole in the wheel at Td2 = 8.69 cm

Coefficient of linear expansion of steel, αsteel = 1.20 × 10–5 K–1

After the shaft is cooled using ‘dry ice’, its temperature becomes T1.

The wheel will slip on the shaft, if the change in diameter, Δd = 8.69 – 8.70

= – 0.01 cm

Temperature T1, can be calculated from the relation:

Δd = d1αsteel (T1 – T)

0.01 = 8.70 × 1.20 × 10–5 (T1 – 300)

(T1 – 300) = 95.78

T1= 204.21 K

= 204.21 – 273.16

= –68.95°C

Therefore, the wheel will slip on the shaft when the temperature of the shaft is –69°C.

Solution 2

Here at temperature `T_1 = 27^@C`, diameter of shaft `D_1 = 8.70 cm`

Let temperature `T_2` the diameter of shaft changes to `D_2`= 8.69 cm for steel

`alpha = 1.20 xx 10^(-5) K^(-1) = 1.20 xx 10^(-5) "" ^@ C^(-1)`

∵ Change in diameter `triangleD  = D_2 - D_1 =  D_1 xx alpha xx (T_2 - T_1)`

`:. 8.69 - 8.70 = 8.70 xx 1.20 xx 10^(-5) xx (T_2 -27)`

`=> T_2  = 27 - 0.01/(8.70xx1.20xx10^(-5)) = 27 - 95.8 = - 68.8 ""^@C` or `-69 ""^@C`

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 NCERT Solution for Physics Textbook for Class 11 (2018 to Current)
Chapter 11: Thermal Properties of Matter
Q: 7 | Page no. 295
Solution 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 Assume Coefficient of Linear Expansion of the Steel to Be Constant Over the Required Temperature Range Concept: Ideal-gas Equation and Absolute Temperature.
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