Particles of masses $1 \mathrm{~g}, 2 \mathrm{~g}, 3 \mathrm{~g}, \ldots . ., 100 \mathrm{~g}$ are kept at the marks $1 \mathrm{~cm}, 2 \mathrm{~cm}, 3 \mathrm{~cm}, \ldots \ldots$, $100 \mathrm{~cm}$ respectively on a metrescale. Find the moment of inertia of the system of particles about a perpendicular bisector of the metre scale.
The perpendicular bisector passes from $50 \mathrm{~cm}$ mark. So, there will be 49 particles on LHS and 50 particles on RHS.
$\mathrm{I}=\left[49(1)^{2}+51(1)^{2}\right]+\left[48(2)^{2}+52(2)^{2}\right] \ldots \ldots \ldots+\left[1(49)^{2}+99(49)^{2}\right]+100(50)^{2}$
$=(100)\left[1^{2}+2^{2}+\ldots \ldots \ldots+50^{2}\right]$
$=(100)\left[\frac{50 \times(50+1)(2 \times 50+1)}{6}\right] \mathrm{gm}-\mathrm{cm}^{2}$
$\mid \approx 0.43 \mathrm{~kg} / \mathrm{cm}^{2}$
$[\because$ Sum of square of $n$ natural numbers,
$\left.\Delta=\frac{n(n+1)(2 n+1)}{6}\right]$
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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,