A puck of mass 80.0 g and radius 4.10 cm slides along an air table at a speed of 1.50 m/s as shown in the figure below. It makes a glancing collision with a second puck of radius 6.00 cm and mass 130.0 g (initially at rest) such that their rims just touch. Because their rims are coated with instant-acting glue, the pucks stick together and spin after the collision.

To find the final velocity and angular velocity of the pucks after the collision, we can apply the principles of conservation of linear momentum and conservation of angular momentum.

1. Conservation of linear momentum:
The total linear momentum before the collision is equal to the total linear momentum after the collision. Since the second puck is initially at rest, the total initial momentum is only due to the first puck:

Initial momentum = mass_1 * velocity_1

Given:
mass_1 = 80.0 g = 0.08 kg
velocity_1 = 1.50 m/s

2. Conservation of angular momentum:
After the collision, the two pucks stick together and rotate about their center of mass with a common angular velocity. The final angular momentum is given by:

Final angular momentum = I * angular velocity

where I is the combined moment of inertia of the two pucks and angular velocity is the common angular velocity.

Now, we need to find the moment of inertia:

The moment of inertia of a solid disk rotating about an axis through its center is given by:

I = (1/2) * m * r^2

where m is the mass of the disk and r is its radius.

1st puck:
mass_1 = 80.0 g = 0.08 kg
radius_1 = 4.10 cm = 0.041 m

2nd puck:
mass_2 = 130.0 g = 0.13 kg
radius_2 = 6.00 cm = 0.06 m

The combined moment of inertia is the sum of the individual moments of inertia:

Total moment of inertia = (1/2) * m_1 * r_1^2 + (1/2) * m_2 * r_2^2

Now, let's calculate the values:

I = (1/2) * (0.08 kg) * (0.041 m)^2 + (1/2) * (0.13 kg) * (0.06 m)^2

After finding the moment of inertia, we can use the principle of conservation of angular momentum:

Initial angular momentum = Final angular momentum

The initial angular momentum is zero since the pucks are initially at rest and not rotating. Therefore, the final angular momentum is also zero.

Now, we can set up our equation:

0 = I * angular velocity

Solving for angular velocity will yield the final angular velocity of the spinning pucks.

Once we have the final angular velocity, we can use it to find the final linear velocity using the formula:

Final linear velocity = final angular velocity * final radius

where the final radius is the combined radius of the two pucks.