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 the coefficient of linear expansion of the steel to be constant over the required temperature range: αsteel = \( 1.20 × 10^{–5} K^{–1}.\)

Asked by Pragya Singh | 1 year ago | 114

Temperature, T= 27°C

The outer diameter of the shaft at 27°C is d_{1} = 8.70 cm

Diameter of the central hole in the wheel at 27°C is d_{2} = 8.69 cm

Coefficient of linear expansion of steel, α_{steel}=1.2×10^{–5} K^{−1}

Temperature at which the wheel will slip on the shaft = T_{1}_{}

Change due to cooling

d_{2}= d_{1}(1+αΔT)

d_{2} = d_{1}[1+α(T_{1} – T)]

d_{2} – d_{1} = d_{1} α(T_{1} – T)

8.69 – 8.70 = 8.70 x 1.2×10^{–5}x (T_{1} – 27)

-0.01 = 10.44 ×10^{–5 }x (T_{1} – 27)

\( \dfrac{-0.01}{(10.44 ×10^{– 5} )}\) = T_{1} – 27

T_{1} = 27 – [\( \dfrac{-0.01}{(10.44 ×10^{– 5})}\)]

T_{1} = 27 – 95.7 = -68.7

Therefore, the wheel will slip on the shaft when the temperature of the shaft is -68.7°C.

Answered by Pragya Singh | 1 year agoA hot ball cools from 90°C to 10°C in 5 minutes. If the surrounding temperature is 20°C, what is the time taken to cool from 60°C to 30°C?

Answer the following questions based on the P-T phase diagram of carbon dioxide:

**(a)** At what temperature and pressure can the solid, liquid and vapour phases of \( CO_2\) co-exist in equilibrium?

**(b)** What is the effect of the decrease of pressure on the fusion and boiling point of \( CO_2\)?

**(c) **What are the critical temperature and pressure for \( CO_2\)? What is its significance?

**(d)** Is \( CO_2\) solid, liquid or gas at

**(a) **–70°C under 1 atm,

**(b)** –60°C under 10 atm,

**(c) **15°C under 56 atm?

A body cools from 80°C to 50°C in 5 minutes. Calculate the time it takes to cool from 60°C to 30°C. The temperature of the surroundings is 20°C.

Explain why :

**(a)** a body with large reflectivity is a poor emitter

**(b)** a brass tumbler feels much colder than a wooden tray on a chilly day

**(c)** an optical pyrometer (for measuring high temperatures) calibrated for an ideal black body radiation gives too low a value for the temperature of a red hot iron piece in the open, but gives a correct value for the temperature when the same piece is in the furnace

**(d) **the earth without its atmosphere would be inhospitably cold

**(e)** heating systems based on the circulation of steam are more efficient in warming a building than those based on the circulation of hot water

A brass boiler has a base area of 0.15 m^{2} and thickness 1.0 cm. It boils water at the rate of 6.0 kg/min when placed on a gas stove. Estimate the temperature of the part of the flame in contact with the boiler. Thermal conductivity of brass = 109 J s^{–1} m^{–1 }K^{–1}; Heat of vaporisation of water = 2256 × 10^{3} J kg^{–1}.