The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a.
Let the breadth, height and strength of the beam be $b, h$ and $S$, respectively.
$a^{2}=\frac{h^{2}+b^{2}}{4}$
$\Rightarrow 4 a^{2}-b^{2}=h^{2}$ ....(1)
Here,
Strength of beam, $S=K b h^{2}$
$\Rightarrow S=k b\left(4 R^{2}-b^{2}\right)$ [Where $K$ is some constant]
$\Rightarrow S=k\left(b 4 a^{2}-b^{3}\right)$ $[$ From eq. (1) $]$
$\Rightarrow \frac{d S}{d b}=k\left(4 a^{2}-3 b^{2}\right)$
For maximum or minimum values of $S$, we must have
$\frac{d S}{d b}=0$
$\Rightarrow k\left(4 a^{2}-3 b^{2}\right)=0$
$\Rightarrow 4 a^{2}-3 b^{2}=0$
$\Rightarrow 4 a^{2}=3 b^{2}$
$\Rightarrow b=\frac{2 a}{\sqrt{3}}$
Substituting the value of $b$ in eq. (1), we get
$\Rightarrow 4 a^{2}-\left(\frac{2 a}{\sqrt{3}}\right)^{2}=h^{2}$
$\Rightarrow \frac{12 a^{2}-4 a^{2}}{3}=h^{2}$
$\Rightarrow h=\frac{2 \sqrt{2}}{\sqrt{3}} a$
Now,
$\frac{d^{2} S}{d b^{2}}=-6 K b$
$\Rightarrow \frac{d^{2} S}{d b^{2}}=-6 K \frac{2 a}{\sqrt{3}}$
$\Rightarrow \frac{d^{2} S}{d b^{2}}=\frac{-12 K a}{\sqrt{3}}<0$
So, the strength of beam is maximum when $b=\frac{2 a}{\sqrt{3}}$ and $h=\frac{2 \sqrt{2}}{\sqrt{3}} a$.
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- 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
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- Chemical Effects of Electric Current,