A uniform disc of mass m and radius r is rotating with angular velocity w(omega) on a smooth horizontal surface. Another identical disc is moving translationally with velocity v as shown(see attachment). When they touch each other, they stick together. The angular velocity of center of mass of the system after contact will be

A)0 B)(2v-rw)/6 C)v/r D)w/6

To solve this problem, we need to consider the conservation of angular momentum and how it applies to the system of the two discs when they come into contact and stick together. Let's break down the situation step-by-step to find the angular velocity of the center of mass after they collide.

Understanding the System

We have two identical discs with mass m and radius r. The first disc is rotating with an angular velocity ω, while the second disc is moving with a linear velocity v. When these discs touch, they stick together, which means we can treat them as a single system afterward.

Conservation of Angular Momentum

Angular momentum is conserved in this scenario since there are no external torques acting on the system. The total angular momentum before they touch will equal the total angular momentum after they stick together.

Calculating Initial Angular Momentum

For the first disc (rotating), the angular momentum L₁ is given by:

• L₁ = I₁ * ω,
• where I₁ is the moment of inertia of the disc, which is (1/2) * m * r².

Plugging this into the equation, we find:

• L₁ = (1/2) * m * r² * ω.

The second disc (translating) has no angular momentum about its center of mass in this case, but when it touches the first disc, we can consider its motion relative to the center of mass of the system.

Finding the Angular Velocity of the Combined System

After the discs stick together, they will rotate about their combined center of mass. The total moment of inertia I of the system (two discs) is:

• I_total = I₁ + I₂ = (1/2) * m * r² + (1/2) * m * r² = m * r².

The total angular momentum before the collision (L_total) is:

• L_total = L₁ + L₂ = (1/2) * m * r² * ω + 0 = (1/2) * m * r² * ω.

After the collision, if we denote the new angular velocity of the combined system as Ω, the angular momentum after collision can be expressed as:

• L_final = I_total * Ω = (m * r²) * Ω.

Setting Up the Equation

Since angular momentum is conserved, we can set L_total equal to L_final:

• (1/2) * m * r² * ω = (m * r²) * Ω.

Now, we can simplify this equation by cancelling out the common terms:

• (1/2) * ω = Ω.

Final Results

Thus, the angular velocity of the center of mass of the system after the collision is:

• Ω = (1/2) * ω.

None of the options you provided directly match this result. However, if we consider the options given, we can see that the option closest to our result is B) (2v - rw) / 6, which seems to involve additional terms that may not relate directly to the angular velocity derived here.

In summary, the angular velocity of the combined discs after they stick together is half the original angular velocity of the rotating disc, assuming no other forces act on the system. If you need further clarification on any part of this, feel free to ask!