A uniform rod of length l is pivoted at point A. It is struck by a horizontal force which delivers an impulse J at a distance x from point A as shown in figure, impulse delivered by pivot is zero, if x is equal to :

Uniform Rod Impulse Problem

Answer: x = (2/3) l

Explanation:

• Let the mass of the uniform rod be m.
• The impulse J at distance x from pivot A causes both translational and rotational motion.
• For the pivot to experience zero impulse, the linear and angular momentum changes must be such that no reaction impulse is required at the pivot.
• The center of mass of the rod is at l/2 from A.
• The moment of inertia of the rod about pivot A is I = (1/3) m l².
• Using angular impulse-momentum and linear impulse-momentum relations, the condition for zero pivot impulse gives:

x = (2/3) l

At this distance, the impulse creates motion such that the pivot does not need to provide any reaction impulse.