A thin uniform rod of length L and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse P is imparted to the rod at a distance x=L/n from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of n is . [figure]
Answer
18
Solution
Key idea: rod moves as if hinged at O; the string (perpendicular to P) gives no impulse just after the blow, since the rod starts from rest.
Centre of rod is at 3L2 from O. Rigid rotation with : vcm= 3L2 .
Linear impulse: P=Mvcm= 3ML 2.
Angular impulse about the centre: Px=Icm = ML212 .
x= ML2 /123ML /2= L18 ⇒ n=18.
Answer: 18