A semi-circular wire has a length $\mathrm{L}$ and mass $\mathrm{M}$. A particle of mass $\mathrm{m}$ is placed at the centre of the circle.
Find the gravitational attraction on the particle due to the wire.
Consider two small elements at an angle $\theta$ from horizontal length of the element $=\mathrm{Rd} \theta$
Mass of each element $\mathrm{dm}=\frac{M}{L}(\mathrm{Rd} \theta)$
Force by elemental mass on particle
$\mathrm{dF}=\frac{\mathrm{Gm}(\mathrm{dm})}{R^{2}}$
$\mathrm{dF}=\frac{\mathrm{GmM}(\mathrm{Rd \theta})}{R^{2} L}$
$\mathrm{dF}=\frac{\mathrm{GmM}}{R L} d \theta$
Resultant of the both elemental forces $=2 \mathrm{dF} \sin \theta$ Net Force
$\mathrm{F}_{\mathrm{N}}=\int 2 \mathrm{dF} \sin \theta$
$=\int_{0}^{\frac{\pi}{2}} 2\left(\frac{\mathrm{Gm} M}{R L}\right) d \theta \sin \theta$
$=2 \frac{\mathrm{Gm} M}{R L}[-\cos \theta]_{0}^{\pi / 2}$
$=2 \frac{\mathrm{Gm} M}{R L}$
$\mathrm{F}_{\mathrm{N}=} \frac{2 \pi \mathrm{GmM}}{L^{2}}($ because $\pi R=L)$
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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,