Four identical particles of equal masses $1 \mathrm{~kg}$ made to move along the circumference of a circle of radius $1 \mathrm{~m}$ under the action of their own mutual gravitational attraction. The speed of each particle will be :
$\sqrt{\frac{G}{2}(1+2 \sqrt{2})}$
$\sqrt{G(1+2 \sqrt{2})}$
$\sqrt{\frac{G}{2}(2 \sqrt{2}-1)}$
$\sqrt{\frac{(1+2 \sqrt{2}) G}{2}}$
Correct Option: , 4
$\mathrm{F}_{1}=\frac{\mathrm{Gmm}}{(2 \mathrm{R})^{2}}=\frac{\mathrm{Gm}^{2}}{4 \mathrm{R}^{2}}$
$\mathrm{~F}_{2}=\frac{\mathrm{Gmm}}{(\sqrt{2} \mathrm{R})^{2}}=\frac{\mathrm{Gm}^{2}}{2 \mathrm{R}^{2}}$
$\mathrm{~F}_{3}=\frac{\mathrm{Gmm}}{(\sqrt{2} \mathrm{R})^{2}}=\frac{\mathrm{Gm}^{2}}{2 \mathrm{R}^{2}}$
$\Rightarrow \mathrm{F}_{\text {net }}=\mathrm{F}_{1}+\mathrm{F}_{2} \cos 45^{\circ}+\mathrm{F}_{3} \cos 45^{\circ}$
$=\frac{\mathrm{Gm}^{2}}{4 \mathrm{R}^{2}}+\frac{\mathrm{Gm}^{2}}{2 \mathrm{R}^{2}} \frac{1}{\sqrt{2}}+\frac{\mathrm{Gm}^{2}}{2 \mathrm{R}^{2}} \frac{1}{\sqrt{2}}$
$=\frac{\mathrm{Gm}^{2}}{\mathrm{R}^{2}}\left(\frac{1}{4}+\frac{1}{2 \sqrt{2}}+\frac{1}{2 \sqrt{2}}\right)$
$=\frac{\mathrm{Gm}^{2}}{\mathrm{R}^{2}}\left(\frac{1}{4}+\frac{1}{\sqrt{2}}\right)=\frac{\mathrm{Gm}^{2}}{4 \mathrm{R}^{2}}(1+2 \sqrt{2})$
$\mathrm{F}_{\mathrm{net}}=\frac{\mathrm{Gm}^{2}}{4 \mathrm{R}^{2}}(1+2 \sqrt{2})=\frac{\mathrm{mv}^{2}}{\mathrm{R}}$
$\Rightarrow \mathrm{v}=\frac{\sqrt{\mathrm{G}(1+2 \sqrt{2})}}{2}$
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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,