Two disks are mounted (like a merry-go-round) on low-friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with rotational inertia 3.30 kg·m2 about its central axis, is set spinning counterclockwise at 450 rev/min. The second disk, with rotational inertia 6.60 kg·m2 about its central axis, is set spinning counterclockwise at 900 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at 900 rev/min, what are their (b) angular speed and (c) direction of rotation after they couple together?
To solve this problem, we start by understanding the concept of conservation of angular momentum. According to this principle, the total angular momentum of a system remains constant unless acted upon by an external torque.
(a) When the two disks are coupled together, their total angular momentum before coupling must be equal to their total angular momentum after coupling.
Before coupling:
Disk 1:
Rotational Inertia, I₁ = 3.30 kg·m²
Angular Speed, ω₁ = 450 rev/min
Disk 2:
Rotational Inertia, I₂ = 6.60 kg·m²
Angular Speed, ω₂ = 900 rev/min
After coupling:
Angular Speed, ω (common for both disks) = ?
Using the conservation of angular momentum principle, we can calculate the total angular momentum before and after coupling.
The formula for angular momentum is L = Iω, where L is the angular momentum, I is the rotational inertia, and ω is the angular speed.
Before coupling, the total angular momentum is given by:
L₁ = I₁ω₁ + I₂ω₂
After coupling, the total angular momentum is given by:
L₂ = (I₁ + I₂)ω
Since angular momentum is conserved, we can equate the two equations:
L₁ = L₂
I₁ω₁ + I₂ω₂ = (I₁ + I₂)ω
Plugging in the given values:
(3.30 kg·m²)(450 rev/min) + (6.60 kg·m²)(900 rev/min) = (3.30 kg·m² + 6.60 kg·m²)ω
Simplifying the equation:
(1485 + 5940) kg·m²/min = (9.90) kg·m²ω
7425 kg·m²/min = 9.90 kg·m²ω
Dividing both sides by 9.90 kg·m²:
ω = 7425 kg·m²/min / 9.90 kg·m²
ω ≈ 750 rev/min
Therefore, the angular speed after coupling is approximately 750 rev/min.
(b) If the second disk is spinning clockwise at 900 rev/min, the negative sign should be used for that angular speed.
Before coupling:
Angular Speed, ω₁ = 450 rev/min
Angular Speed, ω₂ = -900 rev/min (due to clockwise rotation)
After coupling:
Angular Speed, ω = ?
Using the same conservation of angular momentum principle as before, we equate the total angular momentum before and after coupling.
L₁ = I₁ω₁ + I₂ω₂
L₂ = (I₁ + I₂)ω
Plugging in the given values:
(3.30 kg·m²)(450 rev/min) + (6.60 kg·m²)(-900 rev/min) = (3.30 kg·m² + 6.60 kg·m²)ω
Simplifying the equation:
(1485 - 5940) kg·m²/min = (9.90) kg·m²ω
-4455 kg·m²/min = 9.90 kg·m²ω
Dividing both sides by 9.90 kg·m²:
ω = -4455 kg·m²/min / 9.90 kg·m²
ω ≈ -450 rev/min
Therefore, the angular speed after coupling is approximately -450 rev/min. The negative sign indicates the new direction of rotation, which is clockwise.
(c) After coupling, the direction of rotation is determined by the net angular momentum of the system. If the sum of the individual angular momenta is positive, the system rotates counterclockwise (positive direction). If the sum is negative, the system rotates clockwise (negative direction).
In part (b), we found that the angular speed after coupling is approximately -450 rev/min. Since this value is negative, it indicates that the system rotates in the clockwise direction.