A block of mass m = 3 kg is attached to a spring (k = 28 N/m) by a rope that hangs over a pulley of mass M = 7 kg and radius R = 6 cm, as shown in the figure. 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, answer the following.

(a) Find the speed of the block after it falls 1 m.

(b) Find the maximum extension of the spring

Honestly, I don't know how to do this at all. Thanks for the help in advance :)

To solve this problem, you need to apply the principles of conservation of energy and the equations of motion.

(a) To find the speed of the block after it falls 1 m, we can consider the conservation of energy. The initial potential energy of the block will be converted into kinetic energy as it falls.

First, we need to find the potential energy of the block at the initial position. The gravitational potential energy can be calculated using the formula:
Potential energy = mass x gravity x height

Here, the height is given as 1 m and the mass of the block is 3 kg. The acceleration due to gravity is approximately 9.8 m/s^2.

Potential energy at the initial position = 3 kg x 9.8 m/s^2 x 1 m = 29.4 J

Next, we need to find the spring potential energy at the final position. The spring potential energy can be calculated using the formula:
Potential energy = (1/2) x spring constant x extension^2

Here, the extension is the final position of the block. Since the block falls 1 m, the extension becomes 1 m.

Potential energy at the final position = (1/2) x 28 N/m x (1 m)^2 = 14 J

By the principle of conservation of energy, the initial potential energy equals the final kinetic energy (due to the block's motion) and the spring potential energy combined:

Initial potential energy = Final kinetic energy + Final spring potential energy

29.4 J = (1/2) x (mass of the block) x (speed)^2 + 14 J

Now, we can rearrange the equation to solve for the speed:

(1/2) x 3 kg x (speed)^2 = 29.4 J - 14 J
(1/2) x 3 kg x (speed)^2 = 15.4 J
(speed)^2 = 15.4 J / ((1/2) x 3 kg)
(speed)^2 = 10.27 m^2/s^2

Taking the square root of both sides, we find:

speed = sqrt(10.27) m/s ≈ 3.20 m/s

Therefore, the speed of the block after falling 1 m is approximately 3.20 m/s.

(b) To find the maximum extension of the spring, we need to find the equilibrium position where the forces acting on the system are balanced.

At the equilibrium position, the gravitational force acting on the pulley should be equal to the tension in the string. The tension in the string can be calculated using the formula:
Tension = mass x gravity

Here, the mass of the block is 3 kg and the acceleration due to gravity is approximately 9.8 m/s^2.

Tension in the string = 3 kg x 9.8 m/s^2 = 29.4 N

The tension in the string is also equal to the force exerted by the spring. According to Hooke's Law, the force exerted by the spring is proportional to the extension:

Force = spring constant x extension

Substituting the given values, we have:

29.4 N = 28 N/m x extension
extension = 29.4 N / 28 N/m ≈ 1.05 m

Therefore, the maximum extension of the spring is approximately 1.05 m.

I hope this explanation helps you understand how to solve the problem!