a circular disc rotates on a thin air film with a period of 0.3s. its moment of inertia about its axis of rotation is 0.06kgm^2. a small mass is dropped onto th disc and rotates with it . the moment of inertia of the mass about the axis of rotation is 0.04kgm^2.determine the final period of rotating the disc and mass
To determine the final period of rotation for the disc and the added mass, we can apply the law of conservation of angular momentum.
The law states that the total angular momentum of a system remains constant if no external torques act on it. In this case, we assume that no external torques act on the system of the disc and the added mass.
The angular momentum of an object rotating around an axis is given by the equation:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Initially, before the mass is added, the disc rotates with its own moment of inertia I₁ and period T₁, given by:
I₁ = 0.06 kgm² (moment of inertia of the disc)
T₁ = 0.3 s (initial period of the disc)
The angular momentum of the disc before the mass is added is:
L₁ = I₁ω₁
Since the angular momentum is conserved, we can write:
L₁ = L₂
Where L₂ is the angular momentum of the system after the mass is added.
After the mass is dropped onto the disc, the moment of inertia of the system becomes the sum of the moment of inertia of the disc (I₁) and the moment of inertia of the added mass (I₂).
The angular momentum of the system after the mass is added is given by:
L₂ = (I₁ + I₂) ω₂
Since the angular momentum is conserved, we can equate L₁ and L₂:
I₁ω₁ = (I₁ + I₂) ω₂
This equation allows us to solve for ω₂, the angular velocity of the system after the mass is added.
Finally, the final period of the rotating system (disc + mass) can be calculated using the angular velocity:
T₂ = 2π/ω₂
Now we can substitute the given values and solve for T₂:
I₁ = 0.06 kgm²
T₁ = 0.3 s
I₂ = 0.04 kgm²
First, calculate the angular velocity ω₂:
I₁ω₁ = (I₁ + I₂) ω₂
0.06 kgm² * ω₁ = (0.06 kgm² + 0.04 kgm²) * ω₂
ω₂ = (0.06 kgm² * ω₁) / (0.1 kgm²)
Now, calculate the final period T₂:
T₂ = 2π/ω₂
T₂ = 2π / [(0.06 kgm² * ω₁) / (0.1 kgm²)]
Simplify the expression:
T₂ = (0.1 kgm² * 2π) / (0.06 kgm² * ω₁)
Evaluate the expression using the given value of ω₁ to find the final period T₂.