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?

1.	\(\vec{L}=-\frac{m vR}{\sqrt{2}} \hat{k}\) when the particle is moving from \(A\) to \(B\).
2.	\(\vec{L}=m v\left[\frac{R}{\sqrt{2}}+a\right] \hat{k} \) when the particle is moving from \(C\) to \(D\).
3.	\(\vec{L}=m v\left[\frac{R}{\sqrt{2}}+a\right] \hat{k}\) when the particle is moving from \(B\) to \(C\).
4.	\(\vec{L}=\frac{m vR}{\sqrt{2}} \hat{k}\) when the particle is moving from \(D\) to \(A\).

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:

Distance of the centre of mass of a solid uniform cone from its vertex is \(Z_0\). If the radius of its base is \(R\) and its height is \(h\) then \(Z_0\) is equal to:
1. \( \frac{{h}^2}{4{R}} \)
2. \(\frac{3 h}{4} \)
3. \(\frac{5 h}{8} \)
4. \(\frac{3{h}^2}{8{R}}\)

From a solid sphere of mass \(M\) and radius \(R\) a cube of maximum possible volume is cut. The moment of inertia of a cube about an axis passing through its center and perpendicular to one of its faces is:
1. \( \frac{{MR}^2}{32 \sqrt{2 \pi}} \)
2. \( \frac{{MR}^2}{16 \sqrt{2} \pi} \)
3. \( \frac{4 {MR}^2}{9 \sqrt{3} \pi} \)
4. \( \frac{4{MR}^2}{3 \sqrt{3} \pi}\)

A bob of mass \(m\) attached to an inextensible string of length \(l\) is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed \(\omega\) rad/s about the vertical. About the point of suspension:

A mass ‘\(m\)’ is supported by a massless string wound around a uniform hollow cylinder of mass \(m\) and radius \(R\). If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

                              
1. \(\frac{g}{2}\)
2. \(\frac{5g}{6}\)
3. \(g\)
4. \(\frac{2g}{3}\)

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:

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. \(2\sqrt{2}\)
 

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. \( \frac{M H^2}{3} \)