Advertisements
Advertisements
Question
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).
Advertisements
Solution 1
Initial temperature, T1 = 40°C
Final temperature, T2 = 250°C
Change in temperature, ΔT = T2 – T1 = 210°C
Length of the brass rod at T1, l1 = 50 cm
Diameter of the brass rod at T1, d1 = 3.0 mm
Length of the steel rod at T2, l2 = 50 cm
Diameter of the steel rod at T2, d2 = 3.0 mm
Coefficient of linear expansion of brass, α1 = 2.0 × 10–5K–1
Coefficient of linear expansion of steel, α2 = 1.2 × 10–5K–1
For the expansion in the brass rod, we have:
`("Change in length"(trianglel_1))/("Original length"(l_1))" = alpha_1triangleT`
`:. trianglel_1 = 50 xx (2.1xx 10^(-5))xx210`
= 0.2205 cm
For the expansion in the steel rod, we have:
`("Change in length"(trianglel_2))/("Original length"(l_2))" = alpha_2triangleT`
`:.trianglel_2 = 50xx (1.2xx10^(-5))xx210`
= 0.126 cm
Total change in the lengths of brass and steel,
Δl = Δl1 + Δl2
= 0.2205 + 0.126
= 0.346 cm
Total change in the length of the combined rod = 0.346 cm
Since the rod expands freely from both ends, no thermal stress is developed at the junction.
Solution 2
Here `l_"brass" = l_"steel" = 50 cm, d_"brass" = d_"steel" = 3mm = 0.3 cm, trianglel_"brass" = ?, trianglel_"steel" = ?`
`triangleT = 250 - 40 = 210 ""^@C`
`alpha_"brass" = 2xx10^(-5) ""^@C^(-1) and alpha_"steel" = 1.2 xx 10^(-5) ""^@C^(-1)`
Now `trianglel_"brass" = alpha_"brass" xx l_"brass" xx triangleT`
`= 2xx10^(-5) xx 50 xx 210 = 0.21 cm`
Now `trianglel_"steel" = alpha_"steel" xx l_"steel" xx triangleT`
`= 1.2 xx 10^(-5) xx 50 xx 210`
`=0.126 cm = 0.13 cm`
`:. "Total change in length," trianglel = trianglel_"brass" + trianglel_"steel" = 0.21 + 0.13 = 0.34 cm`
Since the rod is not clamped at its ends, no thermal stress developed at the junction.
APPEARS IN
RELATED QUESTIONS
A 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, assuming 50% of power is used up in heating the machine itself or lost to the surroundings Specific heat of aluminium = 0.91 J g–1 K–1
If an automobile engine is overheated, it is cooled by pouring water on it. It is advised that the water should be poured slowly with the engine running. Explain the reason.
Is it possible for two bodies to be in thermal equilibrium if they are not in contact?
A system X is neither in thermal equilibrium with Y nor with Z. The systems Y and Z
The density of water at 0°C is 0.998 g cm–3 and at 4°C is 1.000 g cm–1. Calculate the average coefficient of volume expansion of water in the temperature range of 0 to 4°C.
A steel rod is clamped at its two ends and rests on a fixed horizontal base. The rod is unstrained at 20°C.
Find the longitudinal strain developed in the rod if the temperature rises to 50°C. Coefficient of linear expansion of steel = 1.2 × 10–5 °C–1.
Answer the following question.
Derive the relation between three coefficients of thermal expansion.
Solve the following problem.
A blacksmith fixes iron ring on the rim of the wooden wheel of a bullock cart. The diameter of the wooden rim and the iron ring are 1.5 m and 1.47 m respectively at room temperature of 27 °C. To what temperature the iron ring should be heated so that it can fit the rim of the wheel? (αiron = 1.2 × 10–5K–1).
A metal rod of Young's moduls 'Y' and coefficient of linear expansion 'a' has its temeprature raised by 'Δ θ'. The linear stress to prevent the expansion of rod is ______.
(L and l is original length of rod and expansion respectively)
The volume of a metal block changes by 0.86% when heated through 200 °C then its coefficient of cubical expansion is ______.
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly ______.
An aluminium sphere is dipped into water. Which of the following is true?
The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is α. The sphere is heated a little by a temperature ∆T so that its new temperature is T + ∆T. The increase in the volume of the sphere is approximately ______.
The height of mercury column measured with brass scale at temperature T0 is H0. What height H' will the mercury column have at T = 0°C. Coefficient of volume expansion of mercury is γ. Coefficient of linear expansion of brass is α ______.
A clock with an iron pendulum keeps the correct time at 15°C. If the room temperature is 20°C, the error in seconds per day will be near ______.
(coefficient of linear expansion of iron is 1.2 × 10-5/°C)
When heat is given to a substance, it generally ______.
