Rigid-body Dynamics - Classical Mechanics - Physics Previous Year Questions

1

2007 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2007

Consider a vector \(\vec{p} = 2 \hat{i} + 3 \hat{j} + 2 \hat{k}\) in the coordinate system \((\hat{i}, \hat{j}, \hat{k})\). The axes are rotated anti-clockwise about the Y axis by an angle of \(60^0\). The vector \(\vec{p}\) in the rotated coordinate system \((\hat{i}', \hat{j}', \hat{k}')\) is

A

\((1 - \sqrt{3}) \hat{i}' + 3 \hat{j}' + (1 + \sqrt{3}) \hat{k}'\)

B

\((1 + \sqrt{3}) \hat{i}' + 3 \hat{j}' + (1 - \sqrt{3}) \hat{k}'\)

C

\((1 - \sqrt{3}) \hat{i}' + (3 + \sqrt{3}) \hat{j}' + 2 \hat{k}'\)

D

\((1 - \sqrt{3}) \hat{i}' + (3 - \sqrt{3}) \hat{j}' + 2 \hat{k}'\)

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2

2007 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2007

The moment of inertia of a uniform sphere of radius \( r \) about an axis passing through its centre is given by \( \frac{2}{5} \left( \frac{4 \pi}{3} r^3 \rho \right) \). A rigid sphere of uniform mass density \( \rho \) and radius \( R \) has two smaller spheres of radius \( R/2 \) hollowed out of it, as shown in the figure. The moment of inertia of the resulting body about the Y axis is

A

\(\frac{\pi \rho R^5}{4} \)

B

\(\frac{5 \pi \rho R^5}{12} \)

C

\(\frac{7 \pi \rho R^5}{12} \)

D

\(\frac{3 \pi \rho R^5}{4} \)

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3

2008 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2008

A rigid body is rotating about its centre of mass, fixed at the origin, with an angular velocity \(\vec{\omega}\) and angular acceleration \(\vec{\alpha}\). If the torque acting on it is \(\vec{\tau}\) and its angular momentum is \(\vec{L}\), the rate of change of its kinetic energy is

A

\(\frac{1}{2} \vec{\tau} \cdot \vec{\omega}\)

B

\(\frac{1}{2} \vec{L} \cdot \vec{\omega}\)

C

\(\frac{1}{2} (\vec{\tau} \cdot \vec{\omega} + \vec{L} \cdot \vec{\alpha})\)

D

\(\frac{1}{2} \vec{L} \cdot \vec{\alpha}\)

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4

2008 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2008

The moment of inertia tensor of a rigid body is given by \(I = \begin{pmatrix} 8 & 0 & -4 \\ 0 & 4 & 0 \\ -4 & 0 & 8 \end{pmatrix}\). The magnitude of the moment of inertia about an axis \(\hat{n} = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}, 0\right)\) is

A

6

B

5

C

2

D

8/3

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5

2008 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2008

A hoop of radius R is pivoted at a point on the circumference. The period of small oscillations in the plane of the hoop is

A

\(2\pi \sqrt{\frac{2R}{g}}\)

B

\(2\pi \sqrt{\frac{R}{4g}}\)

C

\(2\pi \sqrt{\frac{R}{g}}\)

D

\(2\pi \sqrt{\frac{9R}{7g}}\)

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6

2011 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2011

A heavy symmetrical top is rotating about its own axis of symmetry (the z-axis). If I₁, I₂ and I₃ are the principal moments of inertia along x, y and z axes respectively, then

A

I₂ = I₃; I₁ ≠ I₂

B

I₁ = I₃; I₁ ≠ I₂

C

I₁ = I₂; I₁ ≠ I₃

D

I₁ ≠ I₂ ≠ I₃

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7

2011 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2011

A heavy symmetrical top is rotating about its own axis of symmetry (the z-axis). If \( I_1, I_2 \) and \( I_3 \) are the principal moments of inertia along x, y and z axes respectively, then

A

\(I_2 = I_3; \ I_1 \neq I_2 \)

B

\(I_1 = I_3; \ I_1 \neq I_2 \)

C

\(I_1 = I_2; \ I_3 \neq I_1 \)

D

\(I_1 \neq I_2 \neq I_3 \)

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8

2012 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2012

Two uniform thin rods of equal length, $L$, and masses $M_1$ and $M_2$ are joined together along the length. The moment of inertia of the combined rod of length $2L$ about an axis passing through the mid-point and perpendicular to the length of the rod is,

A

$(M_1 + M_2)\frac{L^2}{12}$

B

$(M_1 + M_2)\frac{L^2}{6}$

C

$(M_1 + M_2)\frac{L^2}{3}$

D

$(M_1 + M_2)\frac{L^2}{2}$

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9

2013 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2013

A uniform circular disk of radius \( R \) and mass \( M \) is rotating with angular speed \( \omega \) about an axis, passing through its center and inclined at an angle 60 degrees with respect to its symmetry axis. The magnitude of the angular momentum of the disk is,

A

\(\frac{\sqrt{3}}{4} \omega M R^2 \)

B

\(\frac{\sqrt{3}}{8} \omega M R^2 \)

C

\(\frac{\sqrt{7}}{8} \omega M R^2 \)

D

\(\frac{\sqrt{7}}{4} \omega M R^2 \)

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10

2015 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2015

Consider the motion of the Sun with respect to the rotation of the Earth about its axis. If \(\vec{F}_c\) and \(\vec{F}_{Co}\) denote the centrifugal and the Coriolis forces, respectively, acting on the Sun, then

A

\(\vec{F}_c\) is radially outward and \(\vec{F}_{Co} = \vec{F}_c\)

B

\(\vec{F}_c\) is radially inward and \(\vec{F}_{Co} = -2\vec{F}_c\)

C

\(\vec{F}_c\) is radially outward and \(\vec{F}_{Co} = -2\vec{F}_c\)

D

\(\vec{F}_c\) is radially outward and \(\vec{F}_{Co} = 2\vec{F}_c\)

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11

2017 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2017

A uniform solid cylinder is released on a horizontal surface with speed 5 m/s without any rotation (slipping without rolling). The cylinder eventually starts rolling without slipping. If the mass and radius of the cylinder are 10 gm and 1 cm respectively, the final linear velocity of the cylinder is ______ m/s. (up to two decimal places).

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12

2018 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2018

A uniform circular disc of mass \(m\) and radius \(R\) is rotating with angular speed \(\omega\) about an axis passing through its center and making an angle \(\theta = 30^\circ\) with the axis of the disc. If the kinetic energy of the disc is \(\alpha m\omega^2 R^2\), the value of \(\alpha\) is ______ (up to 2 decimal places).

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13

2019 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2019

During a rotation, vectors along the axis of rotation remain unchanged. For the rotation matrix \( \begin{pmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0 \end{pmatrix} \), the unit vector along the axis of rotation is

A

(1/2)(2î − ĵ + 2k̂)

B

(1/√3)(î + ĵ − k̂)

C

(1/√3)(î − ĵ − k̂)

D

(1/3)(2î + 2ĵ − k̂)

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14

2023 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2023

A symmetric top has principal moments of inertia $I_1 = I_2 = \frac{2\alpha}{3}, I_3 = 2\alpha$ about a set of principal axes 1, 2, 3 respectively, passing through its center of mass, where $\alpha$ is a positive constant. There is no force acting on the body and the angular speed of the body about the 3-axis is $\omega_3 = \frac{1}{8}$ rad/s. With what angular frequency in rad/s does the angular velocity vector $\vec{\omega}_1$ precess about the 3-axis?

A

2

B

3

C

5

D

7

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15

2026 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2026

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?

A

\(F_{cen} = 0\) and \(F_{cor} = 0\)

B

\(F_{cen} \neq 0\) and \(F_{cor} \neq 0\) and \(F_{cen} = \frac{F_{cor}}{2}\)

C

\(F_{cen} \neq 0\) and \(F_{cor} \neq 0\) and \(F_{cen} = 2F_{cor}\)

D

\(F_{cen} \neq 0\) and \(F_{cor} \neq 0\) and \(F_{cen} = F_{cor}\)

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16

2026 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2026

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

\(\omega_1 + \omega_3\)

B

\(\omega_1^2 + \omega_3^2\)

C

Angle between axis 2 and \(\vec{L}\)

D

\(\omega_2\)

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17

2026 · Physics · Classical Mechanics · Rigid-body Dynamics

Physics (PH) 2026

A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one’s index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed as it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle \(\theta\). The value of \(\theta\) in degrees (rounded off to one decimal place) is ____ (\(g = 9.8 m.s^{-2}\))

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