The centre of mass of a solid hemisphere of radius \(8\) cm is \(x\) cm from the centre of the flat surface. Then value of \(x\) is:
1. \(3\)
2. \(7\)
3. \(9\)
4. \(1\)

A circular hole of radius \(\dfrac{a}{2}\) is cut out of a circular disc of radius, \(a\) as shown in the figure. The centroid (centre-of-mass) of the remaining circular portion with respect to the point, \(O\) will be:

A uniform thin bar of mass \(6~\mathrm{kg}\) and length \(2.4\) meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is:
1. \(0.8\mathrm{~kg} \mathrm{~m}^2\)
2. \(0.4\mathrm{~kg} \mathrm{~m}^2\)
3. \(0.1\mathrm{~kg} \mathrm{~m}^2\)
4. \(8\mathrm{~kg} \mathrm{~m}^2\)

Four point masses, each of mass m are fixed at the corners of a square of side \(l\). The square is rotating with angular frequency \(\omega\), about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is:

 
1. \( 2 m l^2 \omega \)
2. \( m l^2 \omega \)
3. \( 3 m l^2 \omega \)
4. \( 4 m l^2 \omega \)

Shown in the figure is a hollow ice cream cone (it is open at the top). If its mass is \(M\), radius of its top, \(R\) and height, \(H\), then its moment of Inertia about its axis is:

   
1. \( \frac{M\left(R^2+H^2\right)}{4} \)
2. \( \frac{M R^2}{3} \)
3. \( \frac{M R^2}{2} \)
4. \( \frac{M H^2}{3} \)

Consider two uniform discs of the same thickness and different radii \(R_1=R\) and \(R_2=\alpha R\) made of the same material. If the ratio of their moments of inertia \(I_1\) and \(I_2\), respectively, about their axes is \(I_1:I_2=1:16\) then the value of \(\alpha\) is:
1. \(\sqrt{2}\)
2. \(4\)
3. \(2\)
4. \(2\sqrt{2}\)
 

For a uniform rectangular sheet shown in the figure, if \(I_O\) and \(I_{O'}\) be moments of inertia about the axes perpendicular to the sheet and passing through \(O\) (the centre of mass) and \(O'\) (corner point), then:

      
1. \(I_{O'}=I_O\)
2. \(I_{O'}<I_O\)
3. \(I_{O'}>I_O\)
4. can't say

A particle of mass \(m\) is moving along side of a square of side '\(a\)', with a uniform speed \(v\) in the x-y plane as shown in the figure:

              
Which of the following statements is false for the angular momentum \(\vec L\) about the origin?

A cord is wound around the circumference of the wheel of radius \(r.\) The axis of the wheel is horizontal and the moment of inertia about it is \(I.\) A weight \(mg\) is attached to the cord at the end. The weight falls from rest. After falling through a distance \('h',\) the square of the angular velocity of the wheel will be:
1. \( \dfrac{2 m g h}{I+2 m r^2} \)

2. \( \dfrac{2 m g h}{I+m r^2} \)

3. \( 2 g h\)

4. \( \dfrac{2 g h}{I+m r^2} \)

Two masses \(A\) and \(B,\) each of mass \(M\) are fixed together by a massless spring. A force acts on the mass \(B\) as shown in the figure. If mass \(A\) starts moving away from mass \(B\) with acceleration \(a,\) then the acceleration of mass \(B\) will be: