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Properties of Matter question

2022 · 26 Jul · Shift 2 · Q57
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  5. /2022 · 26 Jul · Shift 2 · Q57

Properties of Matter question

2022 · 26 Jul · Shift 2 · Q57

JEE MainPhysicsProperties of MatterMCQ+4 / −1
The area of cross section of the rope used to lift a load by a crane is 2.5×10−4 m22.5 \times 10^{-4} \mathrm{~m}^{2}2.5×10−4 m2. The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, the required area of cross section of the rope should be : (take g=10 ms−2g=10 \,m s^{-2}g=10ms−2 )
  1. A
    6.25×10−4 m26.25\times 10^{-4} \mathrm{~m}^{2}6.25×10−4 m2
  2. B
    10×10−4 m210\times 10^{-4} \mathrm{~m}^{2}10×10−4 m2
  3. C
    1×10−4 m21\times 10^{-4} \mathrm{~m}^{2}1×10−4 m2
  4. D
    1.67×10−4 m21.67\times 10^{-4} \mathrm{~m}^{2}1.67×10−4 m2
View written solutionFree

Correct answer: A

  1. Use the idea of stress limit

For a rope lifting a load safely, the maximum allowable stress is assumed constant:

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

where FFF is the load force and AAA is the cross-sectional area.

So, for the same material,

F1A1=F2A2\frac{F_1}{A_1} = \frac{F_2}{A_2}A1​F1​​=A2​F2​​

Since F=mgF = mgF=mg, we get

m1gA1=m2gA2\frac{m_1 g}{A_1} = \frac{m_2 g}{A_2}A1​m1​g​=A2​m2​g​

ggg cancels out, so

m1A1=m2A2\frac{m_1}{A_1} = \frac{m_2}{A_2}A1​m1​​=A2​m2​​

Thus,

A2=A1m2m1A_2 = A_1\frac{m_2}{m_1}A2​=A1​m1​m2​​
  1. Substitute the given values

Initial area:

A1=2.5×10−4 m2A_1 = 2.5 \times 10^{-4}\,\text{m}^2A1​=2.5×10−4m2

Initial lifting capacity:

m1=10 metric tonsm_1 = 10\,\text{metric tons}m1​=10metric tons

Required lifting capacity:

m2=25 metric tonsm_2 = 25\,\text{metric tons}m2​=25metric tons

Therefore,

A2=2.5×10−4×2510A_2 = 2.5 \times 10^{-4} \times \frac{25}{10}A2​=2.5×10−4×1025​ A2=2.5×10−4×2.5A_2 = 2.5 \times 10^{-4} \times 2.5A2​=2.5×10−4×2.5 A2=6.25×10−4 m2A_2 = 6.25 \times 10^{-4}\,\text{m}^2A2​=6.25×10−4m2
  1. Match with the options

The required area is

6.25×10−4 m26.25 \times 10^{-4}\,\text{m}^26.25×10−4m2

So the correct option is A.

  1. Comparison with stored answer

Stored correct answer: A

My derived answer: A

They match.

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