A block of mass m = 2.67 kg is attached to a spring (k =

32.3 N/m) by a rope that hangs over a pulley of mass M =
5.34 kg and radius R = 6.41 cm, as shown in the figure.
a) Treating the pulley as a solid homogeneous disk, neglecting
friction at the axle of the pulley, and assuming the system
starts from rest with the spring at its natural length, find the
speed of the block after it falls 1.00 m.
b) And find the maximum extension of the spring.

To find the speed of the block after it falls 1.00 m and the maximum extension of the spring, we can break down the problem into smaller steps.

Step 1: Calculate the acceleration of the system.
First, find the net force acting on the system. The only forces acting on the system are the gravitational force and the force from the spring.

The gravitational force on the block is given by:
F_gravity = m*g, where g is the acceleration due to gravity (9.8 m/s^2).

The force from the spring can be determined using Hooke's law:
F_spring = -k*x, where k is the spring constant and x is the displacement of the spring.

Since the rope is attached to the pulley, the displacement of the spring is equal to the distance the block has fallen. Therefore, x = 1.00 m.

The net force on the system is the sum of the gravitational force and the force from the spring:
F_net = F_gravity + F_spring
F_net = m*g - k*x

Next, calculate the moment of inertia of the pulley, which is a solid homogeneous disk:
I_pulley = (1/2) * M * R^2

Now, find the torque caused by the net force to accelerate the pulley:
τ = F_net * R

The torque is also equal to the moment of inertia multiplied by the angular acceleration, α:
τ = I_pulley * α

Since the pulley starts from rest, its initial angular velocity is 0. Therefore, α can be determined as:
α = (final angular velocity - initial angular velocity) / time

Step 2: Calculate the final angular velocity of the pulley.
The final angular velocity can be determined using the kinematic equation:
ω = ω0 + α*t

The initial angular velocity ω0 is 0, and the time t can be determined by dividing the distance fallen by the speed of the block:
t = x / v

Step 3: Calculate the speed of the block.
Using the equation of motion for the block in free fall:
v = g*t

Step 4: Calculate the maximum extension of the spring.
The maximum extension occurs when the block comes to rest momentarily. At this point, the net force on the system is 0. Therefore, set the net force equation to 0 and solve for x:
0 = m*g - k*x

Solving for x, we get:
x = m*g / k

With these steps, you can find the speed of the block after it falls 1.00 m and the maximum extension of the spring.