Physics · System Of Particles And Rotational Motion · NEET
No. Mass measures resistance to a change in straight-line (translational) motion, and it is fixed for a body. Moment of inertia measures resistance to a change in rotation, and it is NOT fixed — the same body has different I for different axes. Think of mass as the rotational analogue's cousin: in F = ma, mass is m; in the rotation version τ = Iα, moment of inertia I plays the same role as mass.
Because I = Σmᵢrᵢ² uses rᵢ, the distance of each mass piece from the axis. Move the axis, and every rᵢ changes, so I changes. A rod spun about its centre has less I than the same rod spun about one end, because in the end case the mass sits farther from the axis on average. So always state the axis when you give a moment of inertia.
Each small mass at distance r moves in a circle of radius r. Its speed is v = rω, and its kinetic energy is ½mv² = ½m(rω)² = ½(mr²)ω². The r² comes straight out of squaring the speed. Adding all pieces gives KE = ½(Σmr²)ω² = ½Iω², so I = Σmr². The square is why a mass placed far out matters much more than one placed close in.
No. I depends only on the total mass, how that mass is spread out, and the position and direction of the axis. Angular speed ω does not appear in I = Σmr². A wheel has the same moment of inertia whether it is at rest or spinning fast (as long as it stays rigid).
Inertia is a general idea: a body resists change in its state of motion. For straight-line motion this resistance is measured by mass. For rotation, the same resistance is measured by moment of inertia. So moment of inertia is simply the 'rotational inertia' of a body about a chosen axis.
A light rod of length l carries two masses m₁ and m₂ at its ends. The moment of inertia of the system about an axis perpendicular to the rod through the centre of mass is:
The moment of inertia of a thin rod about an axis through its mid-point and perpendicular to its length is 2400 g·cm². The length of the 400 g rod is nearly:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The SI unit is kilogram metre squared (kg·m²). This follows directly from I = Σmr², where mass is in kg and distance in m, so mass × distance² gives kg·m². It has no special named unit.
For a system of point masses, I = Σmᵢrᵢ² = m₁r₁² + m₂r₂² + ... For a continuous body it becomes an integral, I = ∫r²dm. Standard bodies have ready formulas, for example a thin rod about its centre is ML²/12 and a solid sphere about its diameter is (2/5)MR².
For NEET purposes, moment of inertia about a fixed axis is treated as a scalar (a single positive number in kg·m²). The related vector or tensor idea is studied in higher classes; you do not need it for the exam.
Pulling arms in brings mass closer to the axis, so every r drops and moment of inertia I falls. Since angular momentum L = Iω is conserved (no external torque), a smaller I forces a larger ω, so they spin faster. This is the clearest real-life use of moment of inertia.
It appears everywhere in rotation: kinetic energy KE = ½Iω², torque τ = Iα, and angular momentum L = Iω. Many NEET questions first make you compute I (often with the parallel or perpendicular axis theorem or by removing a hole), then plug it into one of these. Getting I right is usually the hard, scoring step.
