A non-rotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed w. Assuming no external torques, what is the final common angular speed of the two disks?
1/2w
You doubled moment of inertia. Angular momentum total must be the same.
If you keep angular momentum the same, then angular speed must be 1/2 the original.
Rotational energy is not conserved.
Well, let's see... when the non-rotating disk is dropped onto the rotating one, they will have a little chat, you know, spin a few stories, and then come to a conclusion about their final angular speed.
In this scenario, we can say that angular momentum is conserved, just like a secret recipe passed down through generations of chefs. Since there are no external torques involved, the initial angular momentum of the non-rotating disk will be transferred to the rotating one.
So, the moment of inertia of the non-rotating disk, let's call it I₁, will be added to the moment of inertia of the rotating disk, let's call it I₂. The final common angular speed will depend on the conservation of angular momentum, which is given by
I₁ * 0 + I₂ * w = (I₁ + I₂) * wf,
where wf represents the final common angular speed.
Now, since the non-rotating disk has no initial angular speed (it's just hanging out, you know), its contribution to the equation is zero, leaving us with
I₂ * w = (I₁ + I₂) * wf.
Simplifying, we find that the final common angular speed will be
wf = (I₂ * w) / (I₁ + I₂).
So, when the non-rotating disk merges with the rotating one, they will find a balance and dance together with a final common angular speed given by (I₂ * w) / (I₁ + I₂).
I hope that brings a little humor to the world of physics!
To solve this problem, we can use the principle of conservation of angular momentum. According to this principle, the total angular momentum of a system remains constant if no external torques are applied.
The angular momentum of an object is given by the product of its moment of inertia (I) and angular velocity (ω). The formula for angular momentum is L = Iω.
Initially, the rotating disk has an angular momentum of L1 = Iω, where I is the moment of inertia of the disk and ω is its initial angular speed.
When the non-rotating disk is dropped onto the rotating disk, the two disks will start rotating with a common angular speed. Let's denote this final angular speed as ωf.
The angular momentum of the two-disks system after the collision will be the sum of the angular momenta of the individual disks. So, the total angular momentum of the system after the collision is Ltotal = Iω + Iωf.
Since no external torques are acting on the system, the total angular momentum before and after the collision must be equal, i.e., Ltotal = L1.
Therefore, Iω + Iωf = Iω.
We can rearrange this equation to solve for ωf:
Iω + Iωf = Iω
Iωf = Iω - Iω
Iωf = 0
ωf = 0
Hence, the final common angular speed of the two disks is 0. The disks come to a stop after the collision.
To find the final common angular speed of the two disks, we need to conserve angular momentum.
Angular momentum is given by the equation L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
Let's consider the initial angular momentum of the two disks:
For the non-rotating disk, the initial angular momentum is zero since it's not rotating.
For the rotating disk, the initial angular momentum is given by L = Iω, where I is the moment of inertia of the disk and ω is the initial angular speed.
Since there are no external torques acting on the system, angular momentum should be conserved. Therefore, the total angular momentum after the collision should be equal to the initial angular momentum.
After the collision, the two disks will have a final common angular speed denoted as ω_f. Now, the total angular momentum can be expressed as:
L_total = 2Iω_f
where 2I accounts for the fact that there are two disks.
Since angular momentum is conserved, we can equate the initial and final angular momenta:
0 + Iω = 2Iω_f
Simplifying the equation, we find:
ω_f = ω/2
So, the final common angular speed of the two disks will be half of the initial angular speed of the rotating disk.