A thin rod having a length of \(1 m\) and area of cross-section \(3 \times 10^{-6} m ^2\) is suspended vertically from one end. The rod is cooled from \(210^{\circ} C\) to \(160^{\circ} C\) After cooling, a mass \(M\) is attached at the lower end of the rod such that the length of rod again becomes \(1m\). Young's modulus and coefficient of linear expansion of the rod are \(2 \times 10^{11} N m ^{-2}\) and \(2 \times 10^{-5} K ^{-1}\), respectively. The value of \(M\) is ____ \(kg\). (Take \(g =10 \ m s ^{-2}\))
Correct Answer: 60
Solution and Explanation
If Δℓ is decease in length of rod due to decease in temperature
\(Δℓ = ℓαΔT\)
\(α = 2×10^{−5} K^{−1}\)
\(ΔT = (210−160) = 50K\)
\(Δℓ =1×2×10^{−5}×50 = 10^{−3} m\)
Young Modulus \(Y = \frac {\frac FA}{\frac {Δℓ}{ℓ}}\) \((A = 3×10^{−6} m^2)\)
\(2×10^{11}=\frac {\frac {mg}{3 \times 10^{-6}}}{\frac {10^{-3}}{1}}\)
\(Mg = 2×10^{11}×3×10^{−9}\)
\(Mg =6×10^{2}\)
\(M \times 10 = 600\)
\(M=60 \ kg\)
So, The correct answer is 60.
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Concepts Used:
Thermal Properties of Matter
Anything that has mass or occupies space in the universe is commonly known as matter. There are five properties of matters namely chemical, mechanical, thermal, dimensional, and physical properties.
Read More: Thermal Properties of Matter
Heat Capacity:
The quantity of heat needed to change the temperature of the matter by 1° is known as the heat capacity of a material. The temperature is indicated in kelvin or Celsius and the amount of heat is shown in calories or joules. Specific heat capacity or molar heat capacity is used to calculate the heat capacity of the matter with the stated dimension.
Linear expansion is the situation when change takes place in one dimension or dimensional.
Thermal Expansion:
When heat is passed through the material, the change in the area, volume, and shape is recognized as the thermal expansion property of the material. The expansion of the railway tracks due to maximal heat which leads to accidents is an example of thermal expansion.
Thermal Conductivity:
This property is interconnected to the conductivity of heat. The amount of heat regulated by the material is directly proportional to the conductivity of the material. Not all objects have the capacity to conduct heat throughout their bodies. Insulators are such objects which do not have the property to conduct heat throughout their body.
Thermal Stress:
The stress due to thermal contraction or expansion of the body is known as thermal stress. The explosion of materials takes place due to thermal stress which is dangerous. The cracks on the truck tyres are caused by an outcome of thermal stress. Trucks at high speed generate heat which is caused by the friction of the truck tyres and the road surface.