A record on a record player spins at 45.0 rpm. Two small 39.0 g balls of clay are dropped on the record and hit it simultaneously, at the edge of the record and at opposite ends from one another. Treat the record as a disk with radius 13.0 cm and mass 315 g, and the clay balls as particles.What is the record's angular velocity just after the clay balls hit it? Give your answer in rad/s.

To solve this problem, we need to use the principle of conservation of angular momentum. The initial angular momentum of the system is the angular momentum of the spinning record, and the final angular momentum is the sum of the angular momenta of the record and the clay balls after they hit the record.

Step 1: Convert the record's rotational speed from rpm to rad/s. The conversion factor is 2π rad/1 min = 2π/60 rad/s. So, the initial angular velocity (ω_initial) of the record is 45.0 rpm * 2π/60 rad/s = 4.71 rad/s.

Step 2: Calculate the initial angular momentum (L_initial) of the record. The moment of inertia (I) of a disk is 0.5mr^2, where m is the mass and r is the radius. Convert the mass and radius to kg and m, respectively. So, I = 0.5 * 0.315 kg * (0.13 m)^2 = 0.00265 kgm^2. Then, L_initial = I * ω_initial = 0.00265 kgm^2 * 4.71 rad/s = 0.0125 kg*m^2/s.

Step 3: Calculate the final angular momentum (L_final) of the system. The clay balls hit the record at the edge, so their moment of inertia is mr^2. So, the total moment of inertia of the system after the clay balls hit the record is I_total = I + 2mr^2 = 0.00265 kgm^2 + 20.039 kg(0.13 m)^2 = 0.00335 kgm^2. Since angular momentum is conserved, L_final = L_initial = 0.0125 kgm^2/s.

Step 4: Calculate the final angular velocity (ω_final) of the record. ω_final = L_final / I_total = 0.0125 kgm^2/s / 0.00335 kgm^2 = 3.73 rad/s.

So, the record's angular velocity just after the clay balls hit it is 3.73 rad/s.