Question

Question 11. 7.  A large steel wheel is to befitted 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 x 10-5K-1.______________​

Answer 1

Given temperature,T=27°C=27+273=300 K

Outer diameter of the steel shaft at T,d1=8.70cm

Diameter of the central hole in the wheel at T,d2=8.96cm

Coefficient of linear expansion of steel, steel=1.2.*10-5 K-1

After cooling=T1

Diameter=8.69-8.70= -0.01cm

T1 can be calculated from the relation:
∆d= d1asteel (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℃

Temperature of the shaft = -69℃

Answer 2

Temperature, T=27 C

In kelvin=27+273=300 K

Outer Diameter of thecsteel shaft at T, d1=8.70 cm

Diameter of the central hole in the wheel at T, d2=8.69 cm

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

After cooling=T1

diameter=8.69-8.70 = -0.01

Temperature can be calculated:

0.01=8.70×1.20×10-5(T1-300)

(T1-300)=95.78

1=204.21K

=204.21-273.16

-68.95 degr