Statement \(-1\) -If the cylinder rolling with angular speed- w. suddenly breaks up in to two equal halves of the same radius. The angular speed of each piece becomes \(2 \mathrm{w}\). Statement \(-2\) - If no external torque outs, the angular momentum of the system is conserved. \(\\{\mathrm{A}\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\) \(\\{\mathrm{B}\\}\) Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1 .\) \(\\{\mathrm{C}\\}\) Statement \(-1\) is true, statement- 2 is false \\{D \(\\}\) Statement- 2 is false, statement \(-2\) is true

Given a cylinder rolling with an angular speed w, suddenly breaking up into two equal halves with the same radius. The angular speed of each piece is said to become 2w. The moment of inertia of the whole cylinder can be expressed as \(I_{cylinder} = \frac{1}{2}MR^2\), where M is the total mass of the cylinder, and R is the radius. After breaking apart, the two equal halves of the cylinder have half the mass of the initial cylinder. Therefore, the moment of inertia of each half can be represented as \(I_{half} = \frac{1}{2}\left(\frac{M}{2}\right)R^2 = \frac{1}{4}MR^2\). According to the conservation of angular momentum, the initial angular momentum of the cylinder is equal to the sum of the angular momenta of the two halves: \(I_{cylinder}\omega = 2I_{half}\omega'\)

Statement -2 says that if no external torque acts on the system, the angular momentum of the system is conserved. By definition, if no external torque is involved, the angular momentum of such a system is indeed conserved. Therefore, statement -2 is true.

Now we need to check if statement -2 is the correct explanation for statement -1. For this, we can use the conservation of angular momentum equation derived in step 1: \(\frac{1}{2}MR^2\omega = 2\left(\frac{1}{4}MR^2\right)\omega'\) Simplifying the equation, we get: \(\omega = 2\omega'\) Hence, angular speed of each half becomes 2w. Since the conservation of angular momentum (statement -2) is the reason behind the resulting angular speed of the two halves (statement -1), it is the correct explanation for statement -1. So, our final answer is: \(\\{A\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\)