Final answer:
To find the new angular speed after the putty sticks to the edge of the rotating vinyl record, the conservation of angular momentum is applied. The initial angular momentum of the record is being balanced with the final angular momentum of the system (record + putty), considering the increased rotational inertia due to the putty's mass at the edge of the record.
Step-by-step explanation:
To determine the angular speed of the vinyl record after the putty sticks to it, we can use the conservation of angular momentum, which states that the total angular momentum of a system remains constant if there is no external torque. Initially, only the record is spinning, and its angular momentum (L) can be calculated using the equation L = Iω, where I is the rotational inertia and ω is the angular speed.
When the putty sticks to the record, the new system (record + putty) will have a different rotational inertia because the mass distribution has changed. The putty adds to the moment of inertia by mr², where m is the mass of the putty and r is the radius of the record. So, the final rotational inertia is Ifinal = I + mr².
Using the conservation of angular momentum: Linitial = Lfinal or Iω = (I + mr²)ωfinal. Solving for ωfinal we get ωfinal = Iω / (I + mr²).
Plugging in the provided values:
ωfinal = (5.37×10⁻⁴ kg·m²)(4.27 rad/s) / ((5.37×10⁻⁴ kg·m²) + (0.0374 kg)(0.131 m)²).
After calculating the above expression, we obtain the new angular speed of the system.