The base of a record player is a disk that has a mass of 2 kg and a diameter of 34 cm. A child spins it so it rotates at a constant rate of 50 times a minute. Neglect friction and all that stuff that complicates problems! Then, the child drops on a record (also a disk) with mass 1.3 kg and a diameter of 32 cm.
How many times a minute will they spin together if the record player base and the record stick together (assume there is no slipping when the record lands on the base)? Remember, for a solid disk or cylinder, I = 1/2MR^2.
To find out how many times a minute the base and the record will spin together, we need to apply the principle of conservation of angular momentum.
Angular momentum, denoted as L, is given by the product of the moment of inertia (I) and the angular velocity (ω). In this case, both the base and the record have a moment of inertia, and the angular velocity will change when they stick together.
Let's find the initial angular momentum and the final angular momentum separately and equate them to solve for the final angular velocity.
1. Initial Angular Momentum:
The initial angular momentum is the sum of the individual angular momenta of the base and the record before they stick together.
The moment of inertia of the base (I_base) can be calculated using the formula I_base = (1/2) * mass_base * radius_base^2.
The moment of inertia of the record (I_record) can be calculated using the formula I_record = (1/2) * mass_record * radius_record^2.
Given:
Mass_base = 2 kg
Radius_base = 0.34 m (since diameter is given as 34 cm)
Mass_record = 1.3 kg
Radius_record = 0.32 m (since diameter is given as 32 cm)
Substituting these values into the equations, we can calculate I_base and I_record.
I_base = (1/2) * 2 kg * (0.34 m)^2
I_base = 0.5784 kg*m^2
I_record = (1/2) * 1.3 kg * (0.32 m)^2
I_record = 0.1664 kg*m^2
The initial angular momentum (L_initial) is the sum of the individual angular momenta:
L_initial = I_base * ω_base + I_record * ω_record
Since the base spins at a constant rate of 50 times per minute, ω_base can be calculated by converting 50 RPM (revolutions per minute) to radians per second (rad/s).
ω_base = (50 RPM) * (2π rad/1 min) * (1 min/60 s)
ω_base = 5π/3 rad/s
The record, before it lands on the base, has zero initial angular velocity, so ω_record = 0.
L_initial = 0.5784 kg*m^2 * (5π/3 rad/s) + 0.1664 kg*m^2 * 0 rad/s
L_initial = 3.0638π kg*m^2/s
2. Final Angular Momentum:
When the base and the record stick together, they rotate as one unit. The moment of inertia of the base and the record combined can be calculated using the formula I_combined = I_base + I_record.
I_combined = 0.5784 kg*m^2 + 0.1664 kg*m^2
I_combined = 0.7448 kg*m^2
The final angular momentum (L_final) is given by:
L_final = I_combined * ω_final
We want to find ω_final, the angular velocity of the combined system.
3. Equating Initial and Final Angular Momentum:
Since angular momentum is conserved, we can set L_initial equal to L_final and solve for ω_final:
L_initial = L_final
3.0638π kg*m^2/s = 0.7448 kg*m^2 * ω_final
Solving for ω_final:
ω_final = (3.0638π kg*m^2/s) / (0.7448 kg*m^2)
ω_final = 4.1134π rad/s
Now, we know the final angular velocity of the combined system is 4.1134π rad/s.
4. Convert to revolutions per minute (RPM):
To find out how many times per minute the base and the record spin together, we convert the angular velocity from rad/s to RPM.
ω_final_RPM = (4.1134π rad/s) * (1 min/60 s) * (1 rev/2π rad)
ω_final_RPM ≈ 3.286 rev/min
Therefore, the base and the record will spin together approximately 3.286 times per minute when they stick together without slipping.