Answer :
Answer:
Therefore, the final angular velocity of the stuck-together disks is approximately 13.0 rad/s.
Explanation:
To solve this problem, we can use the principle of conservation of angular momentum. According to this principle, the total angular momentum before the collision is equal to the total angular momentum after the collision.
The angular momentum of an object can be calculated as the product of its moment of inertia and its angular velocity. The moment of inertia of a disk is given by the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass of the disk, and r is the radius of the disk.
Let's calculate the initial angular momentum of the large disk before the collision:
I1 = (1/2) * m1 * r1^2
= (1/2) * 6.2 kg * (0.25 m)^2
= 0.775 kg·m^2
The initial angular velocity of the large disk is given as 155 RPM. To convert it to radians per second (rad/s), we multiply it by (2π/60):
ω1 = 155 RPM * (2π/60)
= 16.251 rad/s
The initial angular momentum of the large disk is then:
L1 = I1 * ω1
= 0.775 kg·m^2 * 16.251 rad/s
= 12.6 kg·m^2/s
Now, let's calculate the initial angular momentum of the small disk:
I2 = (1/2) * m2 * r2^2
= (1/2) * 2.4 kg * (0.18 m)^2
= 0.1944 kg·m^2
Since the small disk is dropped onto the large disk, it starts with an initial angular velocity of 0 rad/s.
L2 = I2 * ω2
= 0.1944 kg·m^2 * 0 rad/s
= 0 kg·m^2/s
The total initial angular momentum before the collision is the sum of the individual angular momenta:
L_initial = L1 + L2
= 12.6 kg·m^2/s + 0 kg·m^2/s
= 12.6 kg·m^2/s
After the collision, the two disks stick together and rotate with a common final angular velocity. Let's denote this final angular velocity as ωf.
The final moment of inertia of the combined system can be calculated by adding the moments of inertia of the individual disks:
I_final = I1 + I2
= 0.775 kg·m^2 + 0.1944 kg·m^2
= 0.9694 kg·m^2
Using the principle of conservation of angular momentum, we equate the initial and final angular momenta:
L_initial = L_final
L1 + L2 = I_final * ωf
Substituting the known values:
12.6 kg·m^2/s = 0.9694 kg·m^2 * ωf
Solving for ωf:
ωf = (12.6 kg·m^2/s) / (0.9694 kg·m^2)
≈ 13.0 rad/s
Therefore, the final angular velocity of the stuck-together disks is approximately 13.0 rad/s.
