03. Moment of Inertia, Radius of Gyration
Rotational Motion

150088 Four spheres each of diameter ' \(2 a\) ' and mass ' \(m\) ' are placed in a way that their centers lie on the four comers of a square of side ' \(b\) '. Moment of inertia of the system about an axis along one of the sides of the square is-

1 \(\frac{8}{5} \mathrm{ma}^{2}\)
2 \(\frac{4}{5} \mathrm{ma}^{2}+5 \mathrm{mb}^{2}\)
3 \(\frac{4}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
4 \(\frac{8}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
Rotational Motion

150089 A uniform rod \(A B\) of length \(l\) and mass \(m\) is free to rotate about point \(A\). The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about \(A\) is \(\frac{\mathrm{ml}^{2}}{3}\), the initial angular acceleration of the rod will be
original image

1 \(\frac{2 g}{3 \ell}\)
2 \(\mathrm{mg} \frac{\ell}{2}\)
3 \(\frac{3}{2} \mathrm{~g} \ell\)
4 \(\frac{3 g}{2 \ell}\)
Rotational Motion

150090 The moment of inertia of a uniform ring of mass \(M\) and radius \(r\) about a tangent lying in its own plane is

1 \(2 \mathrm{Mr}^{2}\)
2 \(\frac{3}{2} \mathrm{Mr}^{2}\)
3 \(\mathrm{Mr}^{2}\)
4 \(\frac{1}{2} \mathrm{Mr}^{2}\)
Rotational Motion

150091 Four thin rods of same mass \(M\) and same length \(l\), form a square as shown in figure. Moment of inertia of this system about an axis through centre \(O\) and perpendicular to its plane is
original image

1 \(\frac{4}{3} \mathrm{M} l^{2}\)
2 \(\frac{\mathrm{M} l^{2}}{3}\)
3 \(\frac{\mathrm{M} l^{2}}{6}\)
4 \(\frac{2}{3} \mathrm{M} l^{2}\)
Rotational Motion

150092 The moment of inertia of a rod about an axis through its centre and perpendicular to it is \(\frac{1}{12} \mathrm{ML}^{2}\) (where, \(\mathrm{M}\) is the mass and \(\mathrm{L}\) the length of the rod). The rod is bent in the middle so that the two halves make an angle of \(60^{\circ}\). The moment of inertia of the bent rod about the same axis would be

1 \(\frac{\mathrm{ML}^{2}}{48}\)
2 \(\frac{\mathrm{ML}^{2}}{12}\)
3 \(\frac{\mathrm{ML}^{2}}{24}\)
4 \(\frac{\mathrm{ML}^{2}}{8 \sqrt{3}}\)
Rotational Motion

150088 Four spheres each of diameter ' \(2 a\) ' and mass ' \(m\) ' are placed in a way that their centers lie on the four comers of a square of side ' \(b\) '. Moment of inertia of the system about an axis along one of the sides of the square is-

1 \(\frac{8}{5} \mathrm{ma}^{2}\)
2 \(\frac{4}{5} \mathrm{ma}^{2}+5 \mathrm{mb}^{2}\)
3 \(\frac{4}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
4 \(\frac{8}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
Rotational Motion

150089 A uniform rod \(A B\) of length \(l\) and mass \(m\) is free to rotate about point \(A\). The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about \(A\) is \(\frac{\mathrm{ml}^{2}}{3}\), the initial angular acceleration of the rod will be
original image

1 \(\frac{2 g}{3 \ell}\)
2 \(\mathrm{mg} \frac{\ell}{2}\)
3 \(\frac{3}{2} \mathrm{~g} \ell\)
4 \(\frac{3 g}{2 \ell}\)
Rotational Motion

150090 The moment of inertia of a uniform ring of mass \(M\) and radius \(r\) about a tangent lying in its own plane is

1 \(2 \mathrm{Mr}^{2}\)
2 \(\frac{3}{2} \mathrm{Mr}^{2}\)
3 \(\mathrm{Mr}^{2}\)
4 \(\frac{1}{2} \mathrm{Mr}^{2}\)
Rotational Motion

150091 Four thin rods of same mass \(M\) and same length \(l\), form a square as shown in figure. Moment of inertia of this system about an axis through centre \(O\) and perpendicular to its plane is
original image

1 \(\frac{4}{3} \mathrm{M} l^{2}\)
2 \(\frac{\mathrm{M} l^{2}}{3}\)
3 \(\frac{\mathrm{M} l^{2}}{6}\)
4 \(\frac{2}{3} \mathrm{M} l^{2}\)
Rotational Motion

150092 The moment of inertia of a rod about an axis through its centre and perpendicular to it is \(\frac{1}{12} \mathrm{ML}^{2}\) (where, \(\mathrm{M}\) is the mass and \(\mathrm{L}\) the length of the rod). The rod is bent in the middle so that the two halves make an angle of \(60^{\circ}\). The moment of inertia of the bent rod about the same axis would be

1 \(\frac{\mathrm{ML}^{2}}{48}\)
2 \(\frac{\mathrm{ML}^{2}}{12}\)
3 \(\frac{\mathrm{ML}^{2}}{24}\)
4 \(\frac{\mathrm{ML}^{2}}{8 \sqrt{3}}\)
Rotational Motion

150088 Four spheres each of diameter ' \(2 a\) ' and mass ' \(m\) ' are placed in a way that their centers lie on the four comers of a square of side ' \(b\) '. Moment of inertia of the system about an axis along one of the sides of the square is-

1 \(\frac{8}{5} \mathrm{ma}^{2}\)
2 \(\frac{4}{5} \mathrm{ma}^{2}+5 \mathrm{mb}^{2}\)
3 \(\frac{4}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
4 \(\frac{8}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
Rotational Motion

150089 A uniform rod \(A B\) of length \(l\) and mass \(m\) is free to rotate about point \(A\). The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about \(A\) is \(\frac{\mathrm{ml}^{2}}{3}\), the initial angular acceleration of the rod will be
original image

1 \(\frac{2 g}{3 \ell}\)
2 \(\mathrm{mg} \frac{\ell}{2}\)
3 \(\frac{3}{2} \mathrm{~g} \ell\)
4 \(\frac{3 g}{2 \ell}\)
Rotational Motion

150090 The moment of inertia of a uniform ring of mass \(M\) and radius \(r\) about a tangent lying in its own plane is

1 \(2 \mathrm{Mr}^{2}\)
2 \(\frac{3}{2} \mathrm{Mr}^{2}\)
3 \(\mathrm{Mr}^{2}\)
4 \(\frac{1}{2} \mathrm{Mr}^{2}\)
Rotational Motion

150091 Four thin rods of same mass \(M\) and same length \(l\), form a square as shown in figure. Moment of inertia of this system about an axis through centre \(O\) and perpendicular to its plane is
original image

1 \(\frac{4}{3} \mathrm{M} l^{2}\)
2 \(\frac{\mathrm{M} l^{2}}{3}\)
3 \(\frac{\mathrm{M} l^{2}}{6}\)
4 \(\frac{2}{3} \mathrm{M} l^{2}\)
Rotational Motion

150092 The moment of inertia of a rod about an axis through its centre and perpendicular to it is \(\frac{1}{12} \mathrm{ML}^{2}\) (where, \(\mathrm{M}\) is the mass and \(\mathrm{L}\) the length of the rod). The rod is bent in the middle so that the two halves make an angle of \(60^{\circ}\). The moment of inertia of the bent rod about the same axis would be

1 \(\frac{\mathrm{ML}^{2}}{48}\)
2 \(\frac{\mathrm{ML}^{2}}{12}\)
3 \(\frac{\mathrm{ML}^{2}}{24}\)
4 \(\frac{\mathrm{ML}^{2}}{8 \sqrt{3}}\)
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Rotational Motion

150088 Four spheres each of diameter ' \(2 a\) ' and mass ' \(m\) ' are placed in a way that their centers lie on the four comers of a square of side ' \(b\) '. Moment of inertia of the system about an axis along one of the sides of the square is-

1 \(\frac{8}{5} \mathrm{ma}^{2}\)
2 \(\frac{4}{5} \mathrm{ma}^{2}+5 \mathrm{mb}^{2}\)
3 \(\frac{4}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
4 \(\frac{8}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
Rotational Motion

150089 A uniform rod \(A B\) of length \(l\) and mass \(m\) is free to rotate about point \(A\). The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about \(A\) is \(\frac{\mathrm{ml}^{2}}{3}\), the initial angular acceleration of the rod will be
original image

1 \(\frac{2 g}{3 \ell}\)
2 \(\mathrm{mg} \frac{\ell}{2}\)
3 \(\frac{3}{2} \mathrm{~g} \ell\)
4 \(\frac{3 g}{2 \ell}\)
Rotational Motion

150090 The moment of inertia of a uniform ring of mass \(M\) and radius \(r\) about a tangent lying in its own plane is

1 \(2 \mathrm{Mr}^{2}\)
2 \(\frac{3}{2} \mathrm{Mr}^{2}\)
3 \(\mathrm{Mr}^{2}\)
4 \(\frac{1}{2} \mathrm{Mr}^{2}\)
Rotational Motion

150091 Four thin rods of same mass \(M\) and same length \(l\), form a square as shown in figure. Moment of inertia of this system about an axis through centre \(O\) and perpendicular to its plane is
original image

1 \(\frac{4}{3} \mathrm{M} l^{2}\)
2 \(\frac{\mathrm{M} l^{2}}{3}\)
3 \(\frac{\mathrm{M} l^{2}}{6}\)
4 \(\frac{2}{3} \mathrm{M} l^{2}\)
Rotational Motion

150092 The moment of inertia of a rod about an axis through its centre and perpendicular to it is \(\frac{1}{12} \mathrm{ML}^{2}\) (where, \(\mathrm{M}\) is the mass and \(\mathrm{L}\) the length of the rod). The rod is bent in the middle so that the two halves make an angle of \(60^{\circ}\). The moment of inertia of the bent rod about the same axis would be

1 \(\frac{\mathrm{ML}^{2}}{48}\)
2 \(\frac{\mathrm{ML}^{2}}{12}\)
3 \(\frac{\mathrm{ML}^{2}}{24}\)
4 \(\frac{\mathrm{ML}^{2}}{8 \sqrt{3}}\)
Rotational Motion

150088 Four spheres each of diameter ' \(2 a\) ' and mass ' \(m\) ' are placed in a way that their centers lie on the four comers of a square of side ' \(b\) '. Moment of inertia of the system about an axis along one of the sides of the square is-

1 \(\frac{8}{5} \mathrm{ma}^{2}\)
2 \(\frac{4}{5} \mathrm{ma}^{2}+5 \mathrm{mb}^{2}\)
3 \(\frac{4}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
4 \(\frac{8}{5} \mathrm{ma}^{2}+2 \mathrm{mb}^{2}\)
Rotational Motion

150089 A uniform rod \(A B\) of length \(l\) and mass \(m\) is free to rotate about point \(A\). The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about \(A\) is \(\frac{\mathrm{ml}^{2}}{3}\), the initial angular acceleration of the rod will be
original image

1 \(\frac{2 g}{3 \ell}\)
2 \(\mathrm{mg} \frac{\ell}{2}\)
3 \(\frac{3}{2} \mathrm{~g} \ell\)
4 \(\frac{3 g}{2 \ell}\)
Rotational Motion

150090 The moment of inertia of a uniform ring of mass \(M\) and radius \(r\) about a tangent lying in its own plane is

1 \(2 \mathrm{Mr}^{2}\)
2 \(\frac{3}{2} \mathrm{Mr}^{2}\)
3 \(\mathrm{Mr}^{2}\)
4 \(\frac{1}{2} \mathrm{Mr}^{2}\)
Rotational Motion

150091 Four thin rods of same mass \(M\) and same length \(l\), form a square as shown in figure. Moment of inertia of this system about an axis through centre \(O\) and perpendicular to its plane is
original image

1 \(\frac{4}{3} \mathrm{M} l^{2}\)
2 \(\frac{\mathrm{M} l^{2}}{3}\)
3 \(\frac{\mathrm{M} l^{2}}{6}\)
4 \(\frac{2}{3} \mathrm{M} l^{2}\)
Rotational Motion

150092 The moment of inertia of a rod about an axis through its centre and perpendicular to it is \(\frac{1}{12} \mathrm{ML}^{2}\) (where, \(\mathrm{M}\) is the mass and \(\mathrm{L}\) the length of the rod). The rod is bent in the middle so that the two halves make an angle of \(60^{\circ}\). The moment of inertia of the bent rod about the same axis would be

1 \(\frac{\mathrm{ML}^{2}}{48}\)
2 \(\frac{\mathrm{ML}^{2}}{12}\)
3 \(\frac{\mathrm{ML}^{2}}{24}\)
4 \(\frac{\mathrm{ML}^{2}}{8 \sqrt{3}}\)