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 solve this problem, we can use the principles of conservation of momentum and conservation of angular momentum.

First, let's calculate the initial momentum of the first puck (m1) before the collision. The formula for momentum is given by:

Momentum = mass x velocity

m1 = mass of the first puck = 80.0 g = 0.080 kg (Convert grams to kilograms)
v1 = velocity of the first puck = 1.50 m/s

Initial momentum of the first puck (P1) = m1 * v1

Next, let's calculate the initial momentum of the second puck (m2) before the collision. Since the second puck is initially at rest, its momentum is zero.

m2 = mass of the second puck = 130.0 g = 0.130 kg (Convert grams to kilograms)

Initial momentum of the second puck (P2) = m2 * 0 (since it's at rest)

Now, let's calculate the total initial momentum before the collision (P_initial):

P_initial = P1 + P2

After the collision, the two pucks stick together and spin. Let's assume the final velocity of the combined pucks is V (in m/s) and the final angular velocity is ω (in rad/s).

To find V and ω, we can use the principle of conservation of momentum. The total momentum after the collision should equal the initial momentum before the collision:

P_initial = m_total * V

where m_total is the combined mass of both pucks.

m_total = m1 + m2

Now, let's find ω by using the principle of conservation of angular momentum. The initial angular momentum is zero since the second puck is at rest:

Initial angular momentum (L_initial) = 0

The final angular momentum (L_final) is given by:

L_final = I_total * ω

where I_total is the moment of inertia of the system, which can be calculated as the sum of the moments of inertia of each individual puck.

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

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

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

I_total = I1 + I2

Now, let's calculate I1 and I2 for each puck:

I1 = (1/2) * m1 * (r1^2)
I2 = (1/2) * m2 * (r2^2)

Finally, we can substitute these values into the equation for L_final:

L_final = (I1 + I2) * ω

To solve for the final velocity (V) and final angular velocity (ω), we have two equations:

P_initial = m_total * V
L_final = (I1 + I2) * ω

By solving these equations simultaneously, we can find the final values of V and ω after the collision.