Q.The moment of inertia of a hollow sphere about its diameter is (2/3) M R^2. What is its radius of gyration about that axis?
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Rotational Dynamics: The Physics of Spinning Things
Imagine you're trying to open a heavy door. You push near the hinge — it barely moves. Push near the handle — it swings open easily. Same force, different result. That's the first clue: rotation isn't just about how much you push, but where and in what direction.
Now think about a spinning bicycle wheel. Why is it so hard to tilt it sideways when it's spinning fast? And why does a figure skater spin faster when she pulls her arms in? These are the questions rotational dynamics answers.
The Core Idea
Rotational dynamics is the study of why things rotate and how their rotation changes. It's the spinning-world equivalent of Newton's laws for straight-line motion.
In linear motion, you have:
- Force (F) causes acceleration (a)
- Mass (m) resists acceleration
In rotational motion, you have:
- Torque (τ) causes angular acceleration (α)
- Moment of inertia (I) resists angular acceleration
The master equation is:
τnet=Iα
This is the rotational version of F=ma. Every term has a direct parallel.
Breaking It Down
Torque — The Rotational "Push"
Torque isn't just force — it's force multiplied by the distance from the pivot point (the lever arm). That's why the door handle works better than the hinge.
τ=rFsinθ
Where r is the distance from the axis, F is the force, and θ is the angle between them. Maximum torque happens when you push perpendicular to the lever arm (θ=90∘).
Think of torque as "twisting effectiveness." A wrench works because the handle gives you a long lever arm. A short wrench needs more force to do the same job.
Moment of Inertia — The Rotational "Mass"
Mass resists linear acceleration. Moment of inertia resists angular acceleration. But unlike mass, moment of inertia depends on how the mass is distributed relative to the axis of rotation.
For a point mass m at distance r from the axis:
I=mr2
For extended objects, you sum (or integrate) over all mass elements:
I=∑miri2
| Object | Axis | Moment of Inertia |
|--------|------|-------------------|
| Thin hoop | Through center, perpendicular to plane | MR2 |
| Solid disk | Through center, perpendicular to plane | 21MR2 |
| Solid sphere | Through center | 52MR2 |
| Thin rod | Through center, perpendicular to rod | 121ML2 |
Notice: a hoop has more moment of inertia than a disk of the same mass and radius because its mass is farther from the axis. That's why a hoop is harder to start spinning.
Angular Acceleration — How Fast Rotation Changes
Just as acceleration is the rate of change of velocity, angular acceleration α is the rate of change of angular velocity ω:
α=dtdω
And angular velocity is the rate of change of angular displacement θ:
ω=dtdθ
The Complete Picture: Rotational Analogues
| Linear Quantity | Rotational Analogue |
|---|---|
| Displacement x | Angular displacement θ |
| Velocity v | Angular velocity ω |
| Acceleration a | Angular acceleration α |
| Mass m | Moment of inertia I |
| Force F | Torque τ |
| Newton's 2nd law: F=ma | τ=Iα |
| Kinetic energy: 21mv2 | 21Iω2 |
| Momentum: p=mv | Angular momentum: L=Iω |
The Key Insight: Conservation of Angular Momentum
This is where rotational dynamics gets beautiful. Just as linear momentum is conserved when no external force acts, angular momentum is conserved when no external torque acts:
L=Iω=constant …
The radius of gyration is found by equating the given moment of inertia expression to the standard form I = MK^2 and solving for K. …
Comparing I = (2/3) M R^2 with I = M K^2 gives the radius of gyration K = R*sqrt(2/3), about 0.816 times R.
The radius of gyration K of a body about an axis is defined through:
I = M K^2
where I is the moment of inertia of the body about that axis and M is its mass.
We are given, for a hollow sphere about a diameter:
I = (2/3) M R^2
Equating the two expressions for I:
M K^2 = (2/3) M R^2
K^2 = (2/3) R^2
…
Showing the 12 most recent of 25 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Case study: Moment of Inertia of a body about an axis of rotation of the body is defined as the sum of the products of the masses of the constituent particles of the body and square of the distances of the respective particles from the axis of rotation. Its value depends upon the axis of rotation and the distribution of mass of the body. M.I. plays the same role in rotational motion as the force plays in the translation motion. (A) S.I. unit of moment of Inertia is :(a) Kg^2 m^2(b) Kg m^2(c) Kg m^-2(d) Kg m.
›Reveal solutionSolution
Since I = Σ m r², its unit is (unit of mass) × (unit of distance)² = kg·m².
Moment of Inertia is defined as:
I = Σ mi ri²
where mi is the mass of each constituent particle and ri is its perpendicular distance from the axis of rotation.
…
- CBSE 2026Set ANNUAL1 markMCQQ.(B) Moment of Inertia is the rotational analogue of :(a) Torque(b) Mass(c) Force(d) Angular Momentum.
›Reveal solutionSolution
Moment of inertia is the rotational counterpart of mass — both measure a body's inertia (resistance to a change in its state of motion).
In translational motion, Newton's second law is F = ma; mass m measures the body's inertia — its resistance to a change in linear velocity.
In rotational motion, the analogous equation is τ = Iα (torque = moment of inertia × angular acceleration); moment of inertia I measures the body's rotational inertia — its resistance to a change in angular velocity.
…
- CBSE 2026Set sz1 markMCQQ.Radius of gyration is denoted by:(a) R(b) G(c) K(d) I
›Reveal solutionSolution
The radius of gyration is conventionally denoted by K, defined through I = MK^2.
The radius of gyration K of a rigid body about an axis is defined by the relation I = M K^2, where I is the moment of inertia about that axis and M is the total mass. Physically, K is the distance from the axis at which the entire mass of the body could be concentrated without changing its moment of inertia. It …
- CBSE 2026Set ANNUAL1 markMCQQ.On which of the following quantity does the moment of inertia of a body not depend?(a) Angular acceleration(b) Axis of rotation(c) Mass of body(d) Distribution of mass
›Reveal solutionSolution
Moment of inertia is a purely geometric/mass property of a body about a given axis; it does not depend on the body's angular acceleration.
Moment of inertia is defined as I = Σ mi ri², the sum over all mass elements of (mass) × (perpendicular distance from the axis)². This definition shows I depends on:
- (b) Axis of rotation — yes, changing the axis changes every ri, hence I.
- (c) Mass of the body — yes, directly appears in the formula.
- (d) Distribution of mass — yes, mass spread farther from the axis increases I (this is why a hollow cylinder has a larger I than a solid cylinder of the same mass and radius). …
- CBSE 2025Set ANNUAL1 markMCQQ.What will be the moment of inertia of a circular disc about its diameter, if its mass and radius is unity? (A) 1 unit (B) 1/2 unit (C) 1/4 unit (D) 1/8 unit
›Reveal solutionSolution
With unit mass and unit radius, the disc's moment of inertia about a diameter is 1/4 unit.
The moment of inertia of a uniform circular disc of mass M and radius R about an axis through its centre and perpendicular to its plane is Iz=21MR2. By the perpendicular axis theorem, for two perpendicular diameters x and y in the plane of the disc, Iz=Ix+Iy. By sym …
- CBSE 2025Set ANNUAL1 markMCQQ.The unit of moment of inertia of a solid cylinder having mass M and radius R about its geometrical axis is -(a) Nm^2(b) kgm^-1(c) kgm^3(d) kgm^2
›Reveal solutionSolution
Moment of inertia has the SI unit kg m^2, since I is mass multiplied by the square of a distance.
Moment of inertia of a rigid body about an axis is defined as I = summation(m_i r_i^2), where m_i is the mass of each particle and r_i is its perpendicular distance from the axis of rotation. For a solid cylinder of mass M and radius R rotating about its geometrical (central) axis, I = (1/2)MR^2. …
- CBSE 2024Set ANNUAL1 markMCQQ.The mass and radius of a ring are M and r respectively. The moment of inertia of the ring about an axis passing through its centre and perpendicular to its plane is (A) ½Mr^2 (B) Mr^4 (C) M^2r^2 (D) Mr^2
›Reveal solutionSolution
I=Mr2 for a ring about the axis through its centre, perpendicular to its plane.
Moment of inertia is I=∑miri2. In a ring, the entire mass M is distributed at the same perpendicular distance r (the ring's radius) from …
- CBSE 2024Set ANNUAL1 markMCQQ.A solid sphere is rolling on a horizontal plane surface. The ratio of its angular kinetic energy to linear kinetic energy will be equal to (A) 2/5 (B) 5/2 (C) 5/7 (D) 7/5
›Reveal solutionSolution
For a rolling solid sphere, KErot/KEtrans=2/5.
For a solid sphere, I=52mr2. Rolling without slipping means v=rω, so ω=v/r.
KErot=21Iω2=21(52mr2)(rv)2=51mv2.
…
- CBSE 2024Set ANNUAL1 markMCQQ.Rotational kinetic energy is equal to(a) I omega^2(b) I^2 omega^2(c) (3/2) I omega^2(d) (1/2) I omega^2
›Reveal solutionSolution
Rotational kinetic energy = (1/2) I ω^2, the rotational analogue of (1/2) m v^2.
For a rigid body rotating with angular velocity ω, each mass element dm at distance r from the axis has speed v = rω and kinetic energy (1/2) dm v^2 = (1/2) dm r^2 ω^2. Summing (integrating) ove …
- CBSE 2024Set ANNUAL1 markMCQQ.Match the Column given below: Column-I (I) Movement of the tip of the minute hand of a clock (II) A stone falling from a height in straight line path (III) Moving wheel of a sewing machine (IV) Rotational analogue of force Column-II (A) Torque (B) Rational motion (C) Circular motion (D) Linear motion(a) (I)-A (II)-B (III)-C (IV)-D(b) (I)-D (II)-A (III)-B (IV)-C(c) (I)-C (II)-D (III)-B (IV)-A(d) (I)-B (II)-C (III)-D (IV)-A
›Reveal solutionSolution
(I) tip of minute hand -> Circular motion (C); (II) falling stone -> Linear motion (D); (III) sewing-machine wheel -> Rotational motion (B); (IV) rotational analogue of force -> Torque (A). This is option (c).
Matching each item in Column-I to Column-II:
(I) The tip of the minute hand of a clock moves along a fixed circular path of constant radius about the clock's centre -- this is Circular motion -> matches (C).
(II) A stone falling from a height along a straight-line path is a case of one-dimensional, straight-line motion -- this is Linear motion -> matches (D).
(III) The wheel of a sewing machine spins about its own fixed axis -- every point in the wheel (other than the axis) revolves about that axis, which is Rotational motion -> matches (B).
…
- CBSE 2024Set ANNUAL1 markQ.Fill in the blank: The quantity equivalent to mass in rotational motion is ______.
›Reveal solutionSolution
Moment of inertia is the rotational analogue of mass — it measures a body's inertia (resistance to change) in rotational motion.
In translational motion, mass measures a body's inertia — its resistance to a change in linear velocity under an applied force (F = ma). …
- CBSE 2023Set ANNUAL1 markMCQQ.The relation between rotational kinetic energy K, moment of inertia I and angular velocity w is:(a) K = I²w/2(b) K = Iw/2(c) K = I²w²/2(d) K = Iw²/2
›Reveal solutionSolution
The rotational kinetic energy of a rigid body is K = (1/2) I ω², directly analogous to translational KE = (1/2) m v².
A rotating rigid body can be thought of as made of many small mass elements mi, each at distance ri from the axis, moving with speed vi = ri ω. The kinetic energy of each element is (1/2) mi vi² = (1/2) mi ri² ω². Summing over all elements:
K = Σ (1/2) mi ri² ω² = (1/2) ω² Σ mi ri² = (1/2) I ω²
…
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