Two discs of moments of inertia I1 and L2 about their...

Two discs of moments of inertia $I_{1}$ and $L_{2}$ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do vou account for this loss in energy? Take $\omega_{1} \neq \omega_{2}$.

Key Concepts

Frictional Effects

Friction is the resistive force that can cause energy to be transformed from mechanical to thermal energy during interactions between surfaces. In the context of rotating bodies brought into contact, friction facilitates the equalization of angular speeds while also leading to a reduction in the overall kinetic energy of the system.

Energy Dissipation in Inelastic Collisions

In inelastic collisions, not all kinetic energy is conserved; some energy is lost to non-mechanical forms such as heat or sound. This concept explains why, even though angular momentum is conserved, the combined system's kinetic energy can be lower than the sum of the initial energies due to energy dissipation during frictional contact.

Conservation of Angular Momentum

This principle states that in the absence of external torques, the total angular momentum of a system remains constant. It allows us to relate the initial and final conditions of interacting rotating bodies, as the sum of their angular momenta before contact must equal the angular momentum after they come into contact.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation and depends on the distribution of mass relative to the axis of rotation. It plays a key role in rotational dynamics by linking angular velocity and kinetic energy, and its values are crucial when analyzing how different bodies interact rotationally.

Rotational Kinetic Energy

Rotational kinetic energy is the energy associated with the rotation of an object, calculated by the formula (1/2)I?². Understanding this energy is important in problems involving rotating systems, particularly when comparing energy before and after interactions or collisions that might involve non-conservative processes.

Transcript

00:04 In this problem, two different disks are rotating at different speeds omega 1 and omega 2 and their moment of inertia about an axis which is perpendicular to their plane is i1 and i2 respectively.

00:19 So finally we have put them together in such a way that their axis of rotation coincides.

00:25 So we have to find what will be the new angular speed or combined angular speed of the disks.

00:32 So here we can observe that when we have put this disk over the another, then in that case, no external force was working or no external torque was present.

00:43 So whatever torque were present, that were the internal talks because that were the internal talks for these systems.

00:51 Therefore, due to internal talks, angular momentum of the system does not change.

00:56 So angular momentum of the system changes only because of the presence of external tors.

01:02 So, here what we can write, since no external torque is present in the system, therefore, angular momentum will be conserved.

01:42 It means that initial angular momentum will be equals to final angular momentum.

02:07 Now, what is the value of initial angular momentum? so initial angular momentum will be i1 omega 1 plus i2 omega 2 as these were the angular individual angular momentum now when they are kept over another then in that case total moment of inertia about the common axis will be some of the moment of inertia about the individual axis so i 1 plus i 2 and their combined angular speed is omega therefore or omega will be equals to i1 omega 1 plus i2 omega 2 divided by i 1 plus i 2 so this is the answer for first part now in second part we have to explain that we have to first determine that kinetic energy of the system is less than the sum of the kinetic energy of the two days we have to prove that fact so, what is the initial kinetic energy of the system? so, initial kinetic energy of the system will be half i1 omega -1 square plus half i2 omega -2 square.

03:29 So we can take half common here.

03:32 So half into i -1 -1 -square plus i -2 omega -2 square.

03:39 What is the final kinetic energy of the system? so, final kinetic energy will be half i...

Please give Ace some feedback