Gradium Prep.
Try the tutor
JEEOlympiad

A uniform thin rod of mass 3 kg and length 2 m lies at rest on a…

Question

A uniform thin rod of mass 3 kg and length 2 m lies at rest on a frictionless horizontal table. A point particle of mass 1 kg, moving on the table with speed 8 m/s in a direction perpendicular to the rod, strikes one end of the rod and sticks to it. After the collision the combined body rotates with angular speed equal to p/q rad/s in lowest terms, where p and q are positive integers. Find p + q.

✓ Verified answer: 31checked by our engine — not a guess

Step-by-step solution

After sticking, momentum and angular momentum are conserved (no external horizontal force/torque).

Locate the combined centre of mass: from the rod's centre it is d = m(L/2)/(M+m) = 1*(1)/4 = 0.25 m toward the impact end.

Distances from the system CM: particle r_p = L/2 - d = 1 - 0.25 = 0.75 m; rod centre r_rod = 0.25 m.
Moment of inertia about the system CM: I = [M L^2/12 + M r_rod^2] + m r_p^2 = [3*4/12 + 3*0.0625] + 1*0.5625 = [1 + 0.1875] + 0.5625 = 1.75 kg m^2 = 7/4.
Angular momentum about the system CM before impact = m v r_p = 1*8*0.75 = 6 kg m^2/s.
So omega = 6 / (7/4) = 24/7 rad/s.
Thus p/q = 24/7 and p + q = 31.

Final answer31

Stuck on a problem like this?

Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.

Solve my doubt →

More Physics Mechanics solutions

A wedge of mass M = 4 kg with a smooth inclined face making 45…JEE · PhysicsA block of mass 2 kg moving at 10 m/s slides onto the left end of a…JEE · PhysicsA particle is projected from the foot of a fixed inclined plane whose…JEE · PhysicsA thin uniform rod of mass M and length L is hinged at one end about…JEE · PhysicsA small particle is placed at rest on the very top of a fixed smooth…JEE · PhysicsOn a frictionless horizontal line, a ball A of mass m moves with…JEE · PhysicsA solid uniform sphere (moment of inertia 2/5 M R^2 about its centre)…JEE · PhysicsA uniform flexible chain of total length 2 m lies on a frictionless…JEE · Physics