A particle is moving with a uniform speed in a circular orbit of radius \(R\) in a central force inversely proportional to the \(n^{\text{th}}\) power of \(R\). If the period of rotation of the particle is \(T\), then:
1. \(T \propto R^{3 / 2} ~\text{for any } n\)
2. \(T \propto R^{\frac{{n}}{2}+1} \)
3. \({T} \propto {R}^{({n}+1) / 2} \)
4. \( T \propto R^{n / 2} \)

A uniform rod of length \(l\) is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is \(T(x)\) at a distance \(x\) from the axis, then which of the following graphs depicts it most closely?

1.		2.
3.		4.

A smooth wire of length \(2\pi r\) is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed \(\omega\) about the vertical diameter \(AB\), as shown in figure, the bead is at rest with respect to the circular ring at position \(P\) as shown. Then the value of \(\omega^2\) is equal to:

                 
1. \(\frac{g\sqrt{3}}{r}\)
2. \(\frac{2g}{r}\)
3. \(\frac{g\sqrt{3}}{2r}\)
4. \(\frac{2g}{\sqrt{3}r}\)

A bead of mass \(m\) stays at point \(\text{P (a,b)}\) on a wire bent in the shape of a parabola \(y=4Cx^2 \) and rotating with angular speed \(\omega\) (see figure). The value of \(\omega\) is (neglect friction) :

    
1. \( \sqrt{\frac{2 g C}{a b}} \)
2. \( 2 \sqrt{2 g C}\)
3. \( \sqrt{\frac{2 g}{C}} \)
4. \( 2 \sqrt{g C} \)