Three identical spheres, each of mass \(M\), are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to \(2~\text{m}\) (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of the centre of mass. 
        
1. \(2(\hat{i}+\hat{j})\)
2. \(\hat{i}+\hat{j}\)
3. \(\frac{2}{3}(\hat{i}+\hat{j})\)
4. \(\frac{4}{3}(\hat{i}+\hat{j})\)

Subtopic:  Center of Mass |
 64%
From NCERT
NEET - 2020
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The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through:

         

1. \(C\)  2. \(D\)
3. \(A\) 4. \(B\)  
Subtopic:  Moment of Inertia |
 80%
From NCERT
AIPMT - 2012
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A circular platform is mounted on a frictionless vertical axle. Its radius \(R = 2~\text{m}\) and its moment of inertia about the axle is \(200~\text{kg m}^2\). It is initially at rest. A \(50~\text{kg}\) man stands on the edge of the platform and begins to walk along the edge at the speed of \(1~\text{ms}^{-1}\) relative to the ground. The time taken by man to complete one revolution is:
1. \(\frac{3\pi}{2}\text{s}\)
2. \(2\pi~\text{s}\)
3. \(\frac{\pi}{2}\text{s}\)
4. \(\pi~\text{s}\)

Subtopic:  Angular Momentum |
 64%
From NCERT
AIPMT - 2012
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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. \(40~\text{cm}\)

Subtopic:  Center of Mass |
 81%
From NCERT
AIPMT - 2012
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A thin circular ring of mass \(M\) and radius \(r\) is rotating about its axis with constant angular velocity ω. Two objects each of mass \(m\) are attached gently to the opposite ends of the diameter of the ring. The ring now rotates with angular velocity given by:

1. 2MωM+2m

2. (M+2m)ωM

3. MωM+2m

4. (M+2m)ω2m

Subtopic:  Angular Momentum |
 84%
From NCERT
AIPMT - 2010
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From a circular disc of radius \(R\) and mass \(9M,\) a small disc of mass \(M\) and radius \(R/3\) is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is:
1. \(MR^{2}\)
2. \(4MR^{2}\)
3. \(\frac{4}{9}MR^{2}\)
4. \(\frac{40}{9}MR^{2}\)

Subtopic:  Moment of Inertia |
 71%
From NCERT
AIPMT - 2010
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Given the following statements:

1. The centre of gravity (C.G.) of a body is the point at which the weight of the body acts.
2. If the earth is assumed to have an infinitely large radius, the centre of mass coincides with the centre of gravity.
3. To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G.
4. The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis.

Which one of the following pairs of statements is correct?
1. (a) and (b)
2. (b) and (c)
3. (c) and (d)
4. (d) and (a)

Subtopic:  Center of Mass |
From NCERT
AIPMT - 2010
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The angular speed of the wheel of a vehicle is increased from \(360~\text{rpm}\) to \(1200~\text{rpm}\) in \(14\) seconds. Its angular acceleration will be:

1. \(2\pi ~\text{rad/s}^2\) 2. \(28\pi ~\text{rad/s}^2\)
3. \(120\pi ~\text{rad/s}^2\) 4. \(1 ~\text{rad/s}^2\)

Subtopic:  Rotational Motion: Kinematics |
 69%
From NCERT
NEET - 2020
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A circular ring of mass M and radius R is rotating about its axis with constant angular velocity ω. Two particles, each of mass m are attached gently to the opposite ends of the diameter of the ring. The angular velocity of the ring will now become:

1. \(\frac{m \omega}{M   +   2 m}\)

2. \(\frac{m \omega}{M   -   2 m}\)

3.  \(\frac{M \omega}{M   +   2 m}\)

4. \(\frac{M   +   2 m}{M \omega}\)

Subtopic:  Angular Momentum |
 82%
From NCERT
AIPMT - 1998
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The moment of inertia of a uniform circular disc of radius 'R' and mass 'M' about an axis passing from the edge of the disc and normal to the disc is:

1. 12MR2

2. 72MR2

3. 32MR2

4. MR2

Subtopic:  Moment of Inertia |
 90%
From NCERT
AIPMT - 2005
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