Two wires of same length and radius are joined end to end and loaded. The Young's modulii of the materials of the two wires are $Y_{1}$ and $Y_{2}$. The combination behaves as a single wire then its Young's modulus is :
$Y=\frac{2 Y_{1} Y_{2}}{3\left(Y_{1}+Y_{2}\right)}$
$\mathrm{Y}=\frac{2 \mathrm{Y}_{1} \mathrm{Y}_{2}}{\mathrm{Y}_{1}+\mathrm{Y}_{2}}$
$Y=\frac{Y_{1} Y_{2}}{2\left(Y_{1}+Y_{2}\right)}$
$\mathrm{Y}=\frac{\mathrm{Y}_{1} \mathrm{Y}_{2}}{\mathrm{Y}_{1}+\mathrm{Y}_{2}}$
Correct Option: , 2
In series combination $\Delta \mathrm{l}=\ell_{1}+\ell_{2}$
$\mathrm{Y}=\frac{\mathrm{F} / \mathrm{A}}{\Delta \ell / \ell} \Rightarrow \Delta \ell=\frac{\mathrm{F} \ell}{\mathrm{AY}}$
$\Rightarrow \Delta \ell \propto \frac{\ell}{\mathrm{Y}}$
Equivalent length of rod after joing is $=2 \ell$
As, lengths are same and force is also same in series
$\Delta \ell=\Delta \ell_{1}+\Delta \ell_{2}$
$\frac{\ell_{\mathrm{cq}}}{\mathrm{Y}_{\mathrm{cq}}}=\frac{\ell}{\mathrm{Y}_{1}}+\frac{\ell}{\mathrm{Y}_{2}} \Rightarrow \frac{2 \ell}{\mathrm{Y}}=\frac{\ell}{\mathrm{Y}_{1}}+\frac{\ell}{\mathrm{Y}_{2}}$
$\therefore Y=\frac{2 Y_{1} Y_{2}}{Y_{1}+Y_{2}}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,