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Properties of Matter question

2021 · 31 Aug · Shift 2 · Q43
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Properties of Matter question

2021 · 31 Aug · Shift 2 · Q43

JEE MainPhysicsProperties of MatterMCQ+4 / −1
Four identical hollow cylindrical columns of mild steel support a big structure of mass 50 ×\times× 103 kg. The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local distribution, calculate the compression strain of each column. [Use Y = 2.0 ×\times× 1011 Pa, g = 9.8 m/s2]
  1. A
    3.60 ×\times× 10 −-− 8
  2. B
    2.60 ×\times× 10 −-− 7
  3. C
    1.87 ×\times× 10 −-− 3
  4. D
    7.07 ×\times× 10 −-− 4
View written solutionFree

Correct answer: B

  1. Given data
  • Mass of structure: M=50×103 kgM = 50 \times 10^3\,\text{kg}M=50×103kg
  • Number of identical columns: 444
  • Inner radius of each hollow cylinder: r=50 cm=0.5 mr = 50\,\text{cm} = 0.5\,\text{m}r=50cm=0.5m
  • Outer radius of each hollow cylinder: R=100 cm=1.0 mR = 100\,\text{cm} = 1.0\,\text{m}R=100cm=1.0m
  • Young's modulus of steel: Y=2.0×1011 PaY = 2.0 \times 10^{11}\,\text{Pa}Y=2.0×1011Pa
  • Acceleration due to gravity: g=9.8 m/s2g = 9.8\,\text{m/s}^2g=9.8m/s2

We need the compression strain in each column.


  1. Load supported by each column

Since the load is uniformly distributed among 4 identical columns,

F=Mg4F = \frac{Mg}{4}F=4Mg​

First calculate total weight:

Mg=50×103×9.8=4.9×105 NMg = 50 \times 10^3 \times 9.8 = 4.9 \times 10^5\,\text{N}Mg=50×103×9.8=4.9×105N

So force on each column:

F=4.9×1054=1.225×105 NF = \frac{4.9 \times 10^5}{4} = 1.225 \times 10^5\,\text{N}F=44.9×105​=1.225×105N


  1. Cross-sectional area of each hollow column

For a hollow cylinder,

A=π(R2−r2)A = \pi(R^2 - r^2)A=π(R2−r2)

Substitute values:

A=π(1.02−0.52)=π(1−0.25)=0.75π m2A = \pi(1.0^2 - 0.5^2) = \pi(1 - 0.25) = 0.75\pi\,\text{m}^2A=π(1.02−0.52)=π(1−0.25)=0.75πm2

A≈0.75×3.1416=2.356 m2A \approx 0.75 \times 3.1416 = 2.356\,\text{m}^2A≈0.75×3.1416=2.356m2


  1. Stress in each column

Stress=FA\text{Stress} = \frac{F}{A}Stress=AF​

Stress=1.225×1052.356≈5.20×104 Pa\text{Stress} = \frac{1.225 \times 10^5}{2.356} \approx 5.20 \times 10^4\,\text{Pa}Stress=2.3561.225×105​≈5.20×104Pa


  1. Compression strain

Using

Y=StressStrainY = \frac{\text{Stress}}{\text{Strain}}Y=StrainStress​

So,

Strain=StressY=5.20×1042.0×1011\text{Strain} = \frac{\text{Stress}}{Y} = \frac{5.20 \times 10^4}{2.0 \times 10^{11}}Strain=YStress​=2.0×10115.20×104​

Strain=2.60×10−7\text{Strain} = 2.60 \times 10^{-7}Strain=2.60×10−7


  1. Match with options

The calculated compression strain is

2.60×10−7\boxed{2.60 \times 10^{-7}}2.60×10−7​

So the correct option is B.


  1. Comparison with stored correct answer

Stored correct answer: B

Our derived answer: B

They match.

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