A uniform stick of mass m = 12 kg and length l = 1 m spins around on a frictionless horizontal plane, with its CM stationary. A mass M is placed on the plane and the stick collides elastically with it, as shown (with the contact point being the end of the stick). What should M (in kg) be so that after the collision, the stick has translational motion, but no rotational motion?
Physics · Mixed Chapter
JEE Mains

Question

A uniform stick of mass m = 12 kg and length l = 1 m spins around on a frictionless horizontal plane, with its CM stationary. A mass M is placed on the plane and the stick collides elastically with it, as shown (with the contact point being the end of the stick). What should M (in kg) be so that after the collision, the stick has translational motion, but no rotational motion?

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let the initial angular speed be ω and impulse transferred to the ball be J. Angular impulse about the rod’s centre cancels its angular momentum: J(l/2)=Iω=(ml²/12)ω. Thus J=mlω/6. Step 2: The rod recoils translationally at speed J/m opposite the initial tip velocity; the struck ball moves at J/M. Step 3: Perfect elasticity implies relative speed of separation equals the initial tip speed ωl/2, so J/M+J/m=ωl/2. Step 4: Substitute J=mlω/6: m/M+1=3, yielding M=m/2=6 kg.