A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°C, if the original lengths are at 40.0°C? Is there a ‘thermal stress’ developed at the junction? The ends of the rod are free to expand (Co-efficient of linear expansion of brass = \( 2.0 × 10^{–5} K^{–1}\), steel = \( 1.2 × 10^{–5} K^{–1}\) ).

Asked by Pragya Singh | 1 year ago | 147

Length of the brass rod = length of the steel rod = L_{0} = 50 cm

Diameter of the brass rod = diameter of the steel rod = 3 mm

Initial temperature ( T_{1}) = 40°C

Final temperature ( T_{2}) = 250°C

Therefore, the increase in temperature (∆T)

= 250 – 40 =210°C

Coefficient of linear expansion of brass, ∝ = 2 × 10^{-5} K^{-1}

Coefficient of linear expansion of steel, β = 1.2 × 10^{-5} K^{-1}

Final length of brass, L_{1}

L_{1} = L_{o}(1 + ∝∆T)

= 50( 1 + (2 × 10^{-5} × 210))

=50( 1 + 420 × 10^{-5})

= 50× 1.00420

= 50.21 cm

Increase in length of brass ( ∆L)

= L_{2} – L_{1}

= 50.21 – 50

= 0.21 cm

Final length of steel, L_{2}= L_{o}(1 + β∆T)

= 50( 1 + (1.2 × 10^{-5} × 210))

= 50 × 1.00252

= 50.126 cm

Increase in length ( ∆L’)

= 50.126 cm – 50 cm

= 0.126 cm

Total increase in the length = ∆L + ∆L’

= 0.21 + 0.126

= 0.336 cm

A 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}.