A large steel wheel is to befitted on to a shaft of the same material. At $27^{o}$ 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 $a_{s t e e l} = 1.20 \times 10^{- 5} K^{- 1}$ .

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Solution

Given $α_{s t e e l} = 1.20 \times 10^{- 5} K^{- 1}$
Temperature $T_{1} = 27^{0} C = 303 K$
$D_{0} = 8.70 c m$ ( outer dia of the shaft)
$D_{H} = 8.69 c m$ (dia of the hole)
The will wheel will slip when change is diameter $△ D = D_{h} - D_{0} = 8.69 - 8.70 = - .01$
From coefficient of linear expansion we get ,
$△ D = D_{0} α_{s t e e l} (T_{2} - T_{1}) (T_{2} - T_{1}) = \frac{△ D}{D_{0} α_{s t e e l}} = \frac{- 0.01 \times 10^{5}}{10.44} = - 95.78 K T_{2} = 303 - 95.78 = 207.21 K T_{2} = - {65.78}^{0} C$

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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. 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: α_steel= 1.20 × 10^–5K^–1.

Q. A large steel wheel is to be fitted on to a shaft of the same material. At

27^{o} C

the outer diameter of the shaft is

8.70 c m

and the diameter of the central hole in the wheel is

8.69 c m

. 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 the constant over the required temperature range :

α_{s t e e l} = 1.20 \times 10^{- 5} K^{- 1}

Q. A large steel wheel is to be fitted on a shaft of the same material. At

27^{\circ} C

, the outer diameter of the shaft is

8.70 cm

and the diameter of the central hole of the wheel is

8.69 cm

. The shaft is cooled using 'dry ice'. At what temperature of the shaft does the wheel slip on it?

[α = 1.20 \times 10^{- 5} /^{\circ} C]

A large steel wheel is to be fitted on a shaft of the same material. At $27^{\circ} C$ , the outer diameter of the shaft is $8.70 cm$ and the diameter of the central hole of the wheel is $8.69 cm$ . The shaft is cooled using 'dry ice'. At what temperature of the shaft does the wheel slip on it?

$[α = 1.20 \times 10^{- 5} /^{\circ} C]$

Q. A wheel is rotating with an angular speed of 750 r.p.m. on a shaft. Another identical wheel is suddenly coupled on the same shaft. The frequency of the system will be ( assuming M.I. of shaft to be neglishible)

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