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Properties of Matter question

2021 · 18 Mar · Shift 1 · Q65
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Properties of Matter question

2021 · 18 Mar · Shift 1 · Q65

JEE MainPhysicsProperties of MatterNumerical+4 / −1
Two separate wires A and B are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire B is 4 times that of the radius of wire A. The length of the wires A and B are in the ratio of a : b. Then a/b can be expressed as 1/x where x is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 32

  1. Use the formula for extension of a wire

For a wire under tension,

ΔL=FLAY\Delta L = \frac{FL}{AY}ΔL=AYFL​

where:

  • ΔL\Delta LΔL = extension,
  • FFF = applied force,
  • LLL = original length,
  • AAA = cross-sectional area,
  • YYY = Young's modulus.

Since both wires are made of the same material, YYY is same for both. Also, both are subjected to the same force F=2 NF=2\,\text{N}F=2N.

So,

ΔL∝LA\Delta L \propto \frac{L}{A}ΔL∝AL​
  1. Write the ratio of extensions

Given:

  • Wire A stretches by 2 mm2\,\text{mm}2mm
  • Wire B stretches by 4 mm4\,\text{mm}4mm

Hence,

ΔLAΔLB=24=12\frac{\Delta L_A}{\Delta L_B} = \frac{2}{4} = \frac{1}{2}ΔLB​ΔLA​​=42​=21​

Now,

ΔLAΔLB=LA/AALB/AB=LALB⋅ABAA\frac{\Delta L_A}{\Delta L_B} = \frac{L_A/A_A}{L_B/A_B} = \frac{L_A}{L_B}\cdot \frac{A_B}{A_A}ΔLB​ΔLA​​=LB​/AB​LA​/AA​​=LB​LA​​⋅AA​AB​​
  1. Use the radius information

Radius of wire B is 4 times radius of wire A:

rB=4rAr_B = 4r_ArB​=4rA​

Area is proportional to square of radius:

A∝r2A \propto r^2A∝r2

Therefore,

ABAA=rB2rA2=42=16\frac{A_B}{A_A} = \frac{r_B^2}{r_A^2} = 4^2 = 16AA​AB​​=rA2​rB2​​=42=16
  1. Substitute into the extension ratio
12=LALB⋅16\frac{1}{2} = \frac{L_A}{L_B}\cdot 1621​=LB​LA​​⋅16

So,

LALB=132\frac{L_A}{L_B} = \frac{1}{32}LB​LA​​=321​

Given LA:LB=a:bL_A:L_B = a:bLA​:LB​=a:b, therefore

ab=132\frac{a}{b} = \frac{1}{32}ba​=321​

This is of the form

ab=1x\frac{a}{b} = \frac{1}{x}ba​=x1​

so,

x=32x = 32x=32
  1. Comparison with stored answer

Derived answer: 323232

Stored correct answer: 323232

They match.

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