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Rotation formulas

Master Rotation through 32 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Rotation, every formula

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MOI of a particle
$I = m r^{2}$
MOI of a system
$I = \sum m_{i} r_{i}^{2}$
Q1MCQMOI system
The moment of inertia of a system of particles is:
- A
  $\sum m_i r_i^{2}$
- B
  $\sum m_i r_i$
- C
  $\sum m_i$
- D
  $mr$
Perpendicular axis theorem
$I_{z} = I_{x} + I_{y}$
plane lamina
Q1MCQPerpendicular axis
The perpendicular axis theorem states (for a lamina):
- A
  $I_z=I_x+I_y$
- B
  $I_z=I_x-I_y$
- C
  $I_z=I_x I_y$
- D
  $I_z=\tfrac12(I_x+I_y)$
Parallel axis theorem
$I = I_{c m} + M d^{2}$
Q1MCQParallel axis
The parallel axis theorem states:
- A
  $I=I_{cm}+Md^{2}$
- B
  $I=I_{cm}-Md^{2}$
- C
  $I=I_{cm}+Md$
- D
  $I=Md^{2}$
Radius of gyration
$I = M K^{2}$
Solid sphere
$I = \frac{2}{5} M R^{2}$
Q1MCQSolid sphere
The moment of inertia of a solid sphere about a diameter is:
- A
  $\tfrac25 MR^{2}$
- B
  $\tfrac23 MR^{2}$
- C
  $MR^{2}$
- D
  $\tfrac12 MR^{2}$
Hollow sphere
$I = \frac{2}{3} M R^{2}$
Q1MCQHollow sphere
The moment of inertia of a hollow sphere about a diameter is:
- A
  $\tfrac23 MR^{2}$
- B
  $\tfrac25 MR^{2}$
- C
  $MR^{2}$
- D
  $\tfrac12 MR^{2}$
Ring about its axis
$I = M R^{2}$
Q1MCQRing
The moment of inertia of a ring about its axis is:
- A
  $MR^{2}$
- B
  $\tfrac12 MR^{2}$
- C
  $\tfrac25 MR^{2}$
- D
  $\tfrac23 MR^{2}$
Disc about its axis
$I = \frac{1}{2} M R^{2}$
Q1MCQDisc
The moment of inertia of a disc about its central axis is:
- A
  $\tfrac12 MR^{2}$
- B
  $MR^{2}$
- C
  $\tfrac25 MR^{2}$
- D
  $\tfrac14 MR^{2}$
Solid cylinder
$I = \frac{1}{2} M R^{2}$
Q1MCQSolid cylinder
The moment of inertia of a solid cylinder about its axis is:
- A
  $\tfrac12 MR^{2}$
- B
  $MR^{2}$
- C
  $\tfrac25 MR^{2}$
- D
  $\tfrac13 MR^{2}$
Rod about its centre
$I = \frac{M L ^{2}}{12}$
Q1MCQRod centre
The moment of inertia of a rod about its centre is:
- A
  $\dfrac{ML^{2}}{12}$
- B
  $\dfrac{ML^{2}}{3}$
- C
  $ML^{2}$
- D
  $\dfrac{ML^{2}}{6}$
Rod about its end
$I = \frac{M L ^{2}}{3}$
Q1MCQRod end
The moment of inertia of a rod about one end is:
- A
  $\dfrac{ML^{2}}{3}$
- B
  $\dfrac{ML^{2}}{12}$
- C
  $ML^{2}$
- D
  $\dfrac{ML^{2}}{6}$
Rectangular plate
$I = \frac{M ( a ^{2} + b ^{2} )}{12}$
Q1MCQRectangular plate
The moment of inertia of a rectangular plate ($a\times b$) about its central perpendicular axis is:
- A
  $\dfrac{M(a^{2}+b^{2})}{12}$
- B
  $\dfrac{Ma^{2}}{12}$
- C
  $\dfrac{M(a^{2}+b^{2})}{6}$
- D
  $M(a^{2}+b^{2})$
Torque
$τ = r \times F, τ = r F sin θ$
Rotational Newton's law
$τ = I α$
Angular momentum
$L = r \times p = I ω$
Torque–angular momentum
$τ = \frac{d L}{d t}$
Q1MCQτ=dL/dt
Torque equals:
- A
  $\dfrac{dL}{dt}$
- B
  $\dfrac{dI}{dt}$
- C
  $I\omega$
- D
  $L t$
Conservation of angular momentum
$I_{1} ω_{1} = I_{2} ω_{2}$
τ_ext=0
Q1MCQAngular momentum conservation
When no external torque acts, a spinning skater pulling in her arms:
- A
  spins faster ($I$ down, $\omega$ up)
- B
  spins slower
- C
  stops
- D
  is unchanged
Rotational kinetic energy
$K = \frac{1}{2} I ω^{2}$
Work by a torque
$W = τ θ$
Rotational power
$P = τ ω$
Angular impulse
$J = \int τ d t = Δ L$
Q1MCQAngular impulse
Angular impulse equals:
- A
  $\Delta L$
- B
  $\Delta\omega$
- C
  $I\alpha$
- D
  $\tau$
Pure rolling condition
$v = R ω$
Q1MCQPure rolling
For pure rolling, the speed of the centre relates to $\omega$ by:
- A
  $v=R\omega$
- B
  $v=\dfrac{\omega}{R}$
- C
  $v=R^{2}\omega$
- D
  $v=\omega$
KE of a rolling body
$E = \frac{1}{2} m v^{2} (1 + \frac{K ^{2}}{R ^{2}})$
Q1MCQRolling KE
The total KE of a rolling body is:
- A
  $\tfrac12 mv^{2}\left(1+\dfrac{K^{2}}{R^{2}}\right)$
- B
  $\tfrac12 mv^{2}$
- C
  $\tfrac12 I\omega^{2}$
- D
  $mv^{2}$
Acceleration down an incline
$a = \frac{g sin θ}{1 + \frac{K ^{2}}{R ^{2}}}$
Q1MCQIncline acceleration
The acceleration of a body rolling down an incline is:
- A
  $\dfrac{g\sin\theta}{1+\tfrac{K^{2}}{R^{2}}}$
- B
  $g\sin\theta$
- C
  $\dfrac{g\sin\theta}{2}$
- D
  $g$
Speed at the bottom of an incline
$v = \frac{2 g h}{1 + \frac{K ^{2}}{R ^{2}}}$
Q1MCQBottom speed
The speed of a body at the bottom of an incline of height $h$ is:
- A
  $\sqrt{\dfrac{2gh}{1+\tfrac{K^{2}}{R^{2}}}}$
- B
  $\sqrt{2gh}$
- C
  $\sqrt{gh}$
- D
  $2gh$
Rolling race order
$v_{s p h er e} > v_{d i sc} > v_{h o l l o w} > v_{r in g}$
Q1MCQRolling race
Rolling down the same incline, which reaches the bottom first?
- A
  solid sphere
- B
  ring
- C
  hollow sphere
- D
  disc
Speed of the top of a rolling wheel
$v_{t o p} = 2 v$
Q1MCQTop of wheel
The speed of the topmost point of a rolling wheel (centre speed $v$) is:
- A
  $2v$
- B
  $v$
- C
  $0$
- D
  $\dfrac{v}{2}$
Speed of a rim point
$v = 2 v_{c m} cos \frac{θ}{2}$
Q1MCQContact point
The speed of the contact point of a purely rolling wheel is:
- A
  $0$
- B
  $v$
- C
  $2v$
- D
  $\dfrac{v}{2}$
Friction for rolling on an incline
$f = \frac{m g sin θ}{1 + \frac{R ^{2}}{K ^{2}}}$
Q1MCQRolling friction
The friction on a body rolling down an incline is:
- A
  $\dfrac{mg\sin\theta}{1+\tfrac{R^{2}}{K^{2}}}$
- B
  $mg\sin\theta$
- C
  $0$
- D
  $\mu mg\cos\theta$
Combined kinetic energy
$E = \frac{1}{2} M v_{c m}^{2} + \frac{1}{2} I_{c m} ω^{2}$
Q1MCQCombined KE
The kinetic energy of a rolling rigid body is:
- A
  $\tfrac12 Mv_{cm}^{2}+\tfrac12 I_{cm}\omega^{2}$
- B
  $\tfrac12 Mv_{cm}^{2}$
- C
  $\tfrac12 I_{cm}\omega^{2}$
- D
  $Mv_{cm}^{2}$
Rotational equilibrium
$\sum τ = 0$
Q1MCQRotational equilibrium
The condition for rotational equilibrium is:
- A
  $\sum\vec\tau=0$
- B
  $\sum\vec F=0$
- C
  $\omega=0$
- D
  $I=0$

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Rotation formulas

Rotation, every formula

MOI of a particle

MOI of a system

Perpendicular axis theorem

Parallel axis theorem

Radius of gyration

Solid sphere

Hollow sphere

Ring about its axis

Disc about its axis

Solid cylinder

Rod about its centre

Rod about its end

Rectangular plate

Torque

Rotational Newton's law

Angular momentum

Torque–angular momentum

Conservation of angular momentum

Rotational kinetic energy

Work by a torque

Rotational power

Angular impulse

Pure rolling condition

KE of a rolling body

Acceleration down an incline

Speed at the bottom of an incline

Rolling race order

Speed of the top of a rolling wheel

Speed of a rim point

Friction for rolling on an incline

Combined kinetic energy

Rotational equilibrium

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