Question

{"given":"A particle performs uniform circular motion with initial angular momentum $L$. The angular frequency is doubled ($\\omega' = 2\\omega$) and kinetic energy is halved ($K' = \\frac{K}{2}$). We need to find the new angular momentum.","key_observation":"For uniform circular motion, angular momentum $L = I\\omega$ and kinetic energy $K = \\frac{1}{2}I\\omega^2$. From these relations, we can express $L = \\sqrt{2IK}$. When both angular frequency and kinetic energy change, the moment of inertia must also change to satisfy the new conditions.","option_analysis":[{"label":"(A)","text":"$\\frac{L}{4}$","verdict":"correct","explanation":"Step 1: From the relation $K = \\frac{1}{2}I\\omega^2$, we can write $I = \\frac{2K}{\\omega^2}$\n$$I = \\frac{2K}{\\omega^2}$$\nStep 2: For the new conditions, $K' = \\frac{K}{2}$ and $\\omega' = 2\\omega$:\n$$I' = \\frac{2K'}{(\\omega')^2} = \\frac{2 \\cdot \\frac{K}{2}}{(2\\omega)^2} = \\frac{K}{4\\omega^2}$$\nStep 3: Comparing with original moment of inertia:\n$$I' = \\frac{I}{8}$$\nStep 4: New angular momentum:\n$$L' = I'\\omega' = \\frac{I}{8} \\cdot 2\\omega = \\frac{I\\omega}{4} = \\frac{L}{4}$$"},{"label":"(B)","text":"$2L$","verdict":"incorrect","explanation":"This would be the result if we incorrectly assumed that doubling the angular frequency directly doubles the angular momentum without considering the change in kinetic energy and moment of inertia."},{"label":"(C)","text":"$4L$","verdict":"incorrect","explanation":"This would be obtained by incorrectly applying $L \\propto \\omega$ without accounting for the constraint that kinetic energy is halved, which requires the moment of inertia to decrease significantly."},{"label":"(D)","text":"$\\frac{L}{2}$","verdict":"incorrect","explanation":"This is incorrect because it doesn't properly account for the simultaneous changes in both angular frequency and kinetic energy, leading to an incorrect calculation of the new moment of inertia."}],"answer":"(A)","formula_steps":[]}