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1.

Three masses are placed on the \(x\)-axis: \(300~\text{g}\) at origin, \(500~\text{g}\) at \(x= 40~\text{cm}\) and \(400~\text{g}\) at \(x= 70~\text{cm}.\) The distance of the centre of mass from the origin is:
1. \(45~\text{cm}\)
2. \(50~\text{cm}\)
3. \(30~\text{cm}\)
4. \(40~\text{cm}\)

2. An electric fan rotating at \(1200\) rpm is switched off. If the fan stops after \(10\) seconds, the number of revolutions completed by the fan before it stops will be: (assume uniform retardation)

1.	\(100\)	2.	\(50\)
3.	\(40\)	4.	\(20\)

3. The moment of inertia of a horizontal ring about its vertical axis through the centre is \(mR^2\). The moment of inertia about its tangent parallel to the plane is:
1. \(\frac{3mR^2}{2}\)
2. \(\frac{mR^2}{4}\)
3. \(\frac{mR^2}{2}\)
4. \(\frac{3mR^2}{4}\)

4. A ball is spinning on a horizontal surface, about the vertical axis passing through its centre. Its angular velocity decreases from \(2\pi~\text {rad/s}\) to  \(\pi​​~\text {rad/s}\)  in \(10~\text {s}.\)  If the moment of inertia of the ball is \(0.5~\text{kg/m}^2,\) the torque acting on the ball is:

1.	\(-\frac{\pi}{100} ~\text{N-m}\)	2.	\(-\frac{\pi}{50} ~\text{N-m}\)
3.	\(-\frac{\pi}{20} ~\text{N-m}\)	4.	\(-\frac{\pi}{10}~\text{N-m}\)

5.

A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m are attached gently to the opposite ends of the diameter of the ring, the ring will then rotate with an angular velocity:

1.	\(\frac{\omega(M-2 m)}{M+2 m} \)	2.	\(\frac{\omega M}{M+2 m} \)
3.	\(\frac{\omega(M+2 m)}{M} \)	4.	\(\frac{\omega M}{M+m}\)

6.

A string is wrapped along the rim of a wheel of the moment of inertia \(0.10~\text{kg-m}^2\) and radius \(10~\text{cm}.\) If the string is now pulled by a force of \(10~\text N,\) then the wheel starts to rotate about its axis from rest. The angular velocity of the wheel after \(2~\text s\) will be:

1.	\(40~\text{rad/s}\)	2.	\(80~\text{rad/s}\)
3.	\(10~\text{rad/s}\)	4.	\(20~\text{rad/s}\)

7. A ring of mass of \(10~\text{kg}\) and diameter of \(0.4~\text m\) is rotated about its axis. If it makes \(2100\) revolutions per minute, then its angular momentum will be:
1. \(44~\text{kg m}^{2} \text{s}^{-1}\)
2. \(88 ~\text{kg m}^{2} \text{s}^{-1}\)
3. \(4.4~\text{kg m}^{2} \text{s}^{-1}\)
4. \(0.4~\text{kg m}^{2} \text{s}^{-1}\)

8.

Two persons of masses \(55~\text{kg}\) and \(65~\text{kg}\) respectively, are at the opposite ends of a boat. The length of the boat is \(3.0~\text{m}\) and weighs \(100~\text{kg}.\) The \(55~\text{kg}\) man walks up to the \(65~\text{kg}\) man and sits with him. If the boat is in still water, the centre of mass of the system shifts by:
1. \(3.0~\text{m}\) 
2. \(2.3~\text{m}\) 
3. zero
4. \(0.75~\text{m}\) 

9. Match the physical quantities given in Column-I with the physical dimensions in Column-II:

Column-I		Colum-II
(A)	Torque	(P)	\([M L^{-1} T^{-2} ]\)
(B)	Stress	(Q)	\([M L^{2} T^{-2} ]\)
(C)	Pressure Gradient	(R)	\([M L^{-2} T^{-2} ]\)
(D)	Angular momentum	(S)	\([M L^{-2} T^{-1} ]\)

Choose the correct option from the given ones:

1.	A−S, B−P, C−R, D−Q
2.	A−Q, B−P, C−R, D−S
3.	A−P, B−S, C−R, D−Q
4.	A−Q, B−P, C−S, D−R

10. Two uniform, thin, identical rods, each of mass \(M\) and length \(l\) are joined together to form a cross. What will be the moment of inertia of the cross about an axis passing through the point at which the two rods are joined and are perpendicular to the plane of the cross?
1. \(\frac{Ml^2}{12}\)
2. \(\frac{Ml^2}{6}\)
3. \(\frac{Ml^2}{4}\)
4. \(\frac{Ml^2}{3}\)