Question

A small 0.13 g bug stands at one end of a thin uniform bar that is initially at rest on a smooth horizontal table. The other end of the bar pivots about a nail driven into the table and can rotate freely, without friction. The bar has mass 0.3 g and is 2.49 cm in length. The bug jumps off in the horizontal direction, perpendicular to the bar, with a speed of 4.1 cm/s relative to the table. What is the angular speed of the bar just after the frisky insect leaps? The unit of angular speed is rad/s

Not a good question. You're trying to equate linear momentum to angular. They don't even have the same units.

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