The system shown in the figure below consists of a m1 = 5.52-kg block resting on a frictionless horizontal ledge. This block is attached to a string that passes over a pulley, and the other end of the string is attached to a hanging m2 = 2.76-kg block.The pulley is a uniform disk of radius 7.86 cm and mass 0.592 kg. Calculate the speed of the m2 = 2.76-kg block after it is released from rest and falls a distance of 2.08 m. Calculate the angular speed of the pulley at this instant.

To calculate the speed of the m2 block and the angular speed of the pulley, we can use the conservation of mechanical energy principle.

1. Calculate the potential energy of the m2 block at the initial position:
Potential Energy (m2) = m2 * g * h
where m2 is the mass of the block, g is the acceleration due to gravity (9.8 m/s^2), and h is the initial height (2.08 m).

2. Calculate the potential energy of the m1 block at the initial position:
Potential Energy (m1) = m1 * g * h
Since the ledge is frictionless, the potential energy of m1 will be converted into kinetic energy of m2 and rotational kinetic energy of the pulley.

3. Calculate the total initial potential energy:
Total Initial Potential Energy = Potential Energy (m1) + Potential Energy (m2)

4. Calculate the final kinetic energy of the m2 block:
Final Kinetic Energy (m2) = Total Initial Potential Energy
The final kinetic energy of m2 will be the same as the initial potential energy because energy is conserved.

5. Calculate the speed of the m2 block:
Final Kinetic Energy (m2) = (1/2) * m2 * v^2
where v is the speed of the m2 block.

6. Solve the equation for v:
v = sqrt((2 * Final Kinetic Energy (m2)) / m2)

7. Calculate the final angular kinetic energy of the pulley:
Final Angular Kinetic Energy (pulley) = Total Initial Potential Energy - Final Kinetic Energy (m2)

8. Calculate the final angular speed of the pulley:
Final Angular Kinetic Energy (pulley) = (1/2) * I * ω^2
where I is the moment of inertia of the pulley and ω is the angular speed of the pulley.

9. Solve the equation for ω:
ω = sqrt((2 * Final Angular Kinetic Energy (pulley)) / I)

Note: The moment of inertia of a disk can be calculated as (1/2) * mass * radius^2.

Plug in the values given in the problem statement and follow the steps above to find the speed of the m2 block and the angular speed of the pulley.