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Question: A rod is found to be 200cm long at \(40^\circ C\)and 200.24 at\(100^\circ C\). The coefficient of cu...

A rod is found to be 200cm long at 40∘C40^\circ Cand 200.24 at100∘C100^\circ C. The coefficient of cubical expansion of the material is:

  1. 2×10−5/∘C2 \times {10^{ - 5}}/^\circ C
  2. 6×10−5/∘C6 \times {10^{ - 5}}/^\circ C
  3. 3×10−5/∘C3 \times {10^{ - 5}}/^\circ C
  4. 4×10−5/∘C4 \times {10^{ - 5}}/^\circ C
Explanation

Solution

Hint:- Coefficient of cubical expansion is defined as an increment in the unit volume of solid for a unit increase in temperature at pressure that is constant. There are many coefficients of expansions like; coefficient of volumetric expansion; coefficient of thermal expansion, expansion coefficient, etc.
Formula used: The formula for coefficient of cubical expansion of the material is:
β=ΔVV1ΔT\beta = \dfrac{{\Delta V}}{{{V_1}\Delta T}}
Where:
β\beta = coefficient of cubical expansion;
ΔV\Delta V= Difference in volume;
V1{V_1}= Volume of the increased rod.
ΔT\Delta T= Difference in temperature.

Complete step-by-step solution
The formula for coefficient of cubical expansion of the material is:
β=ΔVV1ΔT\beta = \dfrac{{\Delta V}}{{{V_1}\Delta T}};
Calculate the difference in volume and temperature:
The difference in volume is:
ΔV=v1−v\Delta V = {v_1} - v;
The volume of the cylinder is given as:
vcylinder=πr2h{v_{cylinder}} = \pi {r^2}h;
Put the value of height “h”.
v=r2×π×200v = {r^2} \times \pi \times 200;
Volume for the increased rod is given by:
v1=r2×π×200.24{v_1} = {r^2} \times \pi \times 200.24;
Put the above two values in the equationΔV=v1−v\Delta V = {v_1} - v.
ΔV=πr2(200−200.24)\Delta V = \pi {r^2}(200 - 200.24);
Solve mathematically, no need to put the value or “r” or ”pi”
ΔV=πr2(0.24)\Delta V = \pi {r^2}(0.24);
Now, calculate the temperature difference:
ΔT=T−T1\Delta T = T - {T_1};
Put the values of the given temperature in the above equation:
ΔT=40−100\Delta T = 40 - 100;
ΔT=60∘C\Delta T = 60^\circ C;
Now we have the needed values. Put the required in the equation for coefficient for cubical expansion.
β=πr2×0.24πr2×200.24×60\beta = \dfrac{{\pi {r^2} \times 0.24}}{{\pi {r^2} \times 200.24 \times 60}};
Solve mathematically,
β=0.24200.24×60\beta = \dfrac{{0.24}}{{200.24 \times 60}};
β=1.99×105\beta = 1.99 \times {10^5};
We can approximate the above value to:
β≈2×105\beta \approx 2 \times {10^5};
Final Answer: Option”1” is correct. The coefficient of cubical expansion of the material is2×1052 \times {10^5}.

Note:- Here go step by step first find the volume of the rod, there is no need for actual calculation of the volume. In the later part the common factors will cancel each other out. Find out the difference in the volume of rod and difference in temperature. Then put the value in the equation for the coefficient of cubical expansion and solve.