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A uniform rod of length \( L \) and mass \( M \) is lying on a frictionless horizontal plane and is pivoted at one of its ends as shown in figure. There is no friction at the pivot. An inelastic ball of mass \( m \) is fixed with the rod at a distance \( L / 3 \) from \( O \). A horizontal impulse \( J \) is given to the rod at a distance \( 2 L / 3 \) from \( O \) in a direction perpendicular to the rod. Assume that the ball remains in contact with the rod after the collision and impulse \( J \) acts for a small time interval \( \Delta t \).
Find the magnitude of the impulse applied by the pivot during the time interval \( \Delta t \).
(1) \( \frac{m J}{(m+3 M)} \)
(2) \( \frac{m J}{(3 m+M)} \)
(3) \( \frac{M J}{(m+3 M)} \)
(4) \( \frac{M J}{(3 m+M)} \)
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