Question:
A symmetric rigid body has moment of inertia \(I_1, I_2, I_3\) about its principal axes 1, 2, and 3, respectively, with \(I_1 = I_3 = I_\perp\) and \(I_2 \neq I_\perp\). It is rotating in space with no torque on it so that its angular momentum \(\vec{L}\) is constant. Let \(\omega_1, \omega_2, \omega_3\) be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?
A symmetric rigid body has moment of inertia \(I_1, I_2, I_3\) about its principal axes 1, 2, and 3, respectively, with \(I_1 = I_3 = I_\perp\) and \(I_2 \neq I_\perp\). It is rotating in space with no torque on it so that its angular momentum \(\vec{L}\) is constant. Let \(\omega_1, \omega_2, \omega_3\) be the components of its angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following quantities is/are constant during the motion of this rigid body?
Show Hint
Since \(I_1=I_3\), Euler's equation for \(\omega_2\) has a zero coefficient, so \(\omega_2\) stays fixed.
That in turn fixes \(\omega_1^2+\omega_3^2\) (via the conserved \(L^2\) and kinetic energy) and fixes the angle between axis 2 and \(\vec{L}\), while \(\omega_1+\omega_3\) alone keeps oscillating.
That in turn fixes \(\omega_1^2+\omega_3^2\) (via the conserved \(L^2\) and kinetic energy) and fixes the angle between axis 2 and \(\vec{L}\), while \(\omega_1+\omega_3\) alone keeps oscillating.
Updated On: Aug 17, 2026
- \(\omega_1 + \omega_3\)
- \(\omega_1^2 + \omega_3^2\)
- Angle between axis 2 and \(\vec{L}\)
- \(\omega_2\)
Show Solution
Verified By Collegedunia
The Correct Option is B, C, D
Solution and Explanation
Step 1: Write Euler's equations for the torque-free body.
For a rigid body with no external torque, Euler's equations in the body frame read
\[ I_1 \dot{\omega}_1 = (I_2 - I_3)\, \omega_2 \omega_3 \]
\[ I_2 \dot{\omega}_2 = (I_3 - I_1)\, \omega_3 \omega_1 \]
\[ I_3 \dot{\omega}_3 = (I_1 - I_2)\, \omega_1 \omega_2 \]
Step 2: Use \(I_1 = I_3 = I_\perp\) to get \(\omega_2\).
The second equation carries the factor \((I_3 - I_1)\), which is zero since \(I_1 = I_3\). So \(I_2 \dot{\omega}_2 = 0\), giving \(\dot{\omega}_2 = 0\). The component \(\omega_2\) along the odd-one-out axis never changes with time, so statement (D) is TRUE.
Step 3: Reduce the remaining two equations.
With \(I_1 = I_3 = I_\perp\), the first and third equations become \(I_\perp \dot{\omega}_1 = (I_2 - I_\perp)\, \omega_2 \omega_3\) and \(I_\perp \dot{\omega}_3 = (I_\perp - I_2)\, \omega_1 \omega_2\). Since \(\omega_2\) is constant, define the constant \(\Omega = \dfrac{(I_2 - I_\perp)\omega_2}{I_\perp}\). The equations become \(\dot{\omega}_1 = \Omega \omega_3\) and \(\dot{\omega}_3 = -\Omega \omega_1\), the equations of uniform circular motion in the \(\omega_1\)-\(\omega_3\) plane.
Step 4: Check statement (B).
Differentiating \(\omega_1^2 + \omega_3^2\) gives \(2\omega_1\dot{\omega}_1 + 2\omega_3\dot{\omega}_3 = 2\omega_1(\Omega\omega_3) + 2\omega_3(-\Omega\omega_1) = 0\). So \(\omega_1^2 + \omega_3^2\) does not change with time, and statement (B) is TRUE.
Step 5: Check statement (A).
The circular-motion solution gives \(\omega_1(t) = R\cos(\Omega t + \phi)\) and \(\omega_3(t) = R\sin(\Omega t + \phi)\) for constants \(R, \phi\). The sum \(\omega_1 + \omega_3\) keeps oscillating with time rather than staying fixed, so statement (A) is FALSE.
Step 6: Check statement (C).
The angle between the symmetry axis (axis 2) and the space-fixed vector \(\vec{L}\) satisfies \(\cos\theta = \dfrac{\vec{L}\cdot \hat{e}_2}{|\vec{L}|} = \dfrac{I_2\omega_2}{|\vec{L}|}\). Since \(\vec{L}\) is fixed and conserved, \(|\vec{L}|\) is constant; \(I_2\) is a fixed body property; and \(\omega_2\) is constant from Step 2. So \(\cos\theta\) is built entirely from constants, meaning \(\theta\) itself is constant. Statement (C) is TRUE, matching the well-known picture of a symmetric top's axis tracing a fixed cone around \(\vec{L}\).
Final Answer:
The symmetry of the body (\(I_1 = I_3\)) freezes \(\omega_2\), which in turn keeps \(\omega_1^2+\omega_3^2\) and the axis-2/\(\vec{L}\) angle both fixed; only the plain sum \(\omega_1+\omega_3\) keeps changing. \[ \boxed{\text{B, C, D}} \]
For a rigid body with no external torque, Euler's equations in the body frame read
\[ I_1 \dot{\omega}_1 = (I_2 - I_3)\, \omega_2 \omega_3 \]
\[ I_2 \dot{\omega}_2 = (I_3 - I_1)\, \omega_3 \omega_1 \]
\[ I_3 \dot{\omega}_3 = (I_1 - I_2)\, \omega_1 \omega_2 \]
Step 2: Use \(I_1 = I_3 = I_\perp\) to get \(\omega_2\).
The second equation carries the factor \((I_3 - I_1)\), which is zero since \(I_1 = I_3\). So \(I_2 \dot{\omega}_2 = 0\), giving \(\dot{\omega}_2 = 0\). The component \(\omega_2\) along the odd-one-out axis never changes with time, so statement (D) is TRUE.
Step 3: Reduce the remaining two equations.
With \(I_1 = I_3 = I_\perp\), the first and third equations become \(I_\perp \dot{\omega}_1 = (I_2 - I_\perp)\, \omega_2 \omega_3\) and \(I_\perp \dot{\omega}_3 = (I_\perp - I_2)\, \omega_1 \omega_2\). Since \(\omega_2\) is constant, define the constant \(\Omega = \dfrac{(I_2 - I_\perp)\omega_2}{I_\perp}\). The equations become \(\dot{\omega}_1 = \Omega \omega_3\) and \(\dot{\omega}_3 = -\Omega \omega_1\), the equations of uniform circular motion in the \(\omega_1\)-\(\omega_3\) plane.
Step 4: Check statement (B).
Differentiating \(\omega_1^2 + \omega_3^2\) gives \(2\omega_1\dot{\omega}_1 + 2\omega_3\dot{\omega}_3 = 2\omega_1(\Omega\omega_3) + 2\omega_3(-\Omega\omega_1) = 0\). So \(\omega_1^2 + \omega_3^2\) does not change with time, and statement (B) is TRUE.
Step 5: Check statement (A).
The circular-motion solution gives \(\omega_1(t) = R\cos(\Omega t + \phi)\) and \(\omega_3(t) = R\sin(\Omega t + \phi)\) for constants \(R, \phi\). The sum \(\omega_1 + \omega_3\) keeps oscillating with time rather than staying fixed, so statement (A) is FALSE.
Step 6: Check statement (C).
The angle between the symmetry axis (axis 2) and the space-fixed vector \(\vec{L}\) satisfies \(\cos\theta = \dfrac{\vec{L}\cdot \hat{e}_2}{|\vec{L}|} = \dfrac{I_2\omega_2}{|\vec{L}|}\). Since \(\vec{L}\) is fixed and conserved, \(|\vec{L}|\) is constant; \(I_2\) is a fixed body property; and \(\omega_2\) is constant from Step 2. So \(\cos\theta\) is built entirely from constants, meaning \(\theta\) itself is constant. Statement (C) is TRUE, matching the well-known picture of a symmetric top's axis tracing a fixed cone around \(\vec{L}\).
Final Answer:
The symmetry of the body (\(I_1 = I_3\)) freezes \(\omega_2\), which in turn keeps \(\omega_1^2+\omega_3^2\) and the axis-2/\(\vec{L}\) angle both fixed; only the plain sum \(\omega_1+\omega_3\) keeps changing. \[ \boxed{\text{B, C, D}} \]
Was this answer helpful?
0
0
Top GATE PH Physics Questions
- An infinite one-dimensional lattice extends along the x-axis. At each lattice site, there exists an ion with spin 1/2. The spin can point either in the +z or -z direction only. Let \( S_P \), \( S_F \), and \( S_A \) denote the entropies of paramagnetic, ferromagnetic, and antiferromagnetic configurations, respectively. Which of the following relations is/are true?
- Two point charges of charge \( +q \) each are placed a distance \( 2d \) apart. A grounded solid conducting sphere of radius \( a \) is placed midway between them. Assume \( a^2 \ll d^2 \). Which of the following statements is/are true?
- A particle of mass $m$ is moving in the potential
\[ V(x) = \begin{cases} V_0 + \frac{1}{2} m \omega_{0P}^2 x^2 & \quad \text{if } x > 0 \\ \infty & \quad \text{if } x \leq 0 \end{cases} \]
Figures P, Q, R, and S show different combinations of the values of $ \omega_0 $ and $ V_0 $. - Decays of mesons and baryons can be categorized as weak, strong and electromagnetic decays depending upon the interactions involved in the processes. Which of the following option is/are true?
- The temperature \( T \) dependence of magnetic susceptibility \( \chi \) (Column I) of certain magnetic materials (Column II) are given below. Which of the following options is/are correct?
Column I Column II (1) 
(P) Diamagnetic (2) 
(Q) Paramagnetic (3) 
(R) Ferromagnetic (4) 
(S) Antiferromagnetic
View More Questions
Top GATE PH Classical Mechanics Questions
- On a horizontal plane, a projectile of mass \(m\) is launched from the ground with speed \(v_0\) at an angle \(\theta_0\) with the horizontal. In addition to the gravitational force (\(mg\)), it also experiences a drag force \(\vec{F}_{drag} = -\gamma \vec{v}\), where \(\vec{v}\) is its velocity and \(\gamma\) is a constant. It hits the ground at a distance \(R\) from the point of launch with its velocity making an angle \(\theta\) with the horizontal, as shown schematically in the figure below.
Then which of the following options is correct? - A rocket of length 18.0 m is moving at speed \(0.9c\) (where \(c\) is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is
- The Lagrangian \(L_0 = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}m\omega^2q^2\) with the generalized coordinate \(q\) is transformed to \(L = L_0 + \alpha \frac{df(q)}{dt}\). Consider the following statements:
(i) Expression for the canonical momentum does not change.
(ii) The equation of motion does not change.
Which of the following options is correct for the above statements? - Two frames \(S\) (solid lines) and \(S'\) (dashed lines) with common origin are shown in the figure below. Frame \(S\) is inertial while \(S'\) is rotating about the common \(z\)-axis. There is a point mass fixed at \(P\) on the \(x\)-axis of the \(S\) frame. The magnitude of the centrifugal force and the Coriolis force experienced by the mass in the \(S'\) frame is \(F_{cen}\) and \(F_{cor}\), respectively. Which of the following options is correct for these forces?
- Two 1 kg blocks are connected to two massless springs of spring constants \(8\text{ N.m}^{-1}\) and \(4\text{ N.m}^{-1}\). The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the x-axis with the normal mode frequencies \(\omega_H\) and \(\omega_L\) (\(\omega_H > \omega_L\)). The ratio \(\dfrac{\omega_H}{\omega_L}\) (rounded off to two decimal places) is ______
View More Questions
Top GATE PH Questions
- If \( F_1(Q, q) = Qq \) is the generating function of a canonical transformation from \((p, q)\) to \((P, Q)\), then which one of the following relations is correct?
- When an unpolarized plane electromagnetic wave in a dielectric medium with refractive index \( n_1 \) is incident on a plane interface separating it from another dielectric medium with refractive index \( n_2 \) (where \( n_2 > n_1 \)), and the angle of incidence is \(\tan^{-1} \left( \frac{n_2}{n_1} \right) \), the following statement is true:
- The wavefunction of a particle in an infinite one-dimensional potential well at time \( t \) is
\[ \Psi(x, t) = \sqrt{\frac{2}{3}} e^{-iE_1 t/\hbar}\psi_1(x) + \frac{1}{\sqrt{6}} e^{i\pi/6} e^{-iE_2 t/\hbar} \psi_2(x) + \frac{1}{\sqrt{6}} e^{i\pi/4} e^{-iE_3 t/\hbar} \psi_3(x) \]where \(\psi_1\), \(\psi_2\), and \(\psi_3\) are the normalized ground state, the normalized first excited state, and the normalized second excited state, respectively. \(E_1\), \(E_2\), and \(E_3\) are the eigen-energies corresponding to \(\psi_1\), \(\psi_2\), and \(\psi_3\), respectively. The expectation value of energy of the particle in state \(\Psi(x,t)\) is - If a thermodynamical system is adiabatically isolated and experiences a change in volume under an externally applied constant pressure, then the thermodynamical potential minimized at equilibrium is the
- The mean distance between the two atoms of an HD molecule is \( r \), where H and D denote hydrogen and deuterium, respectively. The mass of the hydrogen atom is \( m_H \). The energy difference between the two lowest lying rotational states of HD in multiples of \(\frac{h^2}{m_H r^2}\) is:
View More Questions