In the configuration shown (10m incline for m1 and 8m vertical length for m2 with a spring under it, both masses are linked by a rope and a pulley at the top), the 52.0 N/m spring is unstretched, and the system is released from rest. The mass of the block on the incline is m1 = 24kg and the mass of the other block is m2 = 5kg. (Neglect the mass of the pulley.). If the coefficient of kinetic friction between m1 and the incline is 0.24, then how fast (in m/s) are the blocks moving when m1 has slid 0.90 m parallel to the incline (and the 5.00 kg block has gained 0.90 m in elevation)?

Initial heights of the blocks: m1 -> H; m2 -> h

Initial energy of the “two blocks” system is
E1 = m1•g•H+m2•g•h
When the blocks covered s=0.9 m the energy of ”two blocks+spring” system is
E2= m1•g(H-s•sinα) +m1•v²/2 {m2•g(h+s) +m2•v²/2 +ks²/2.
From the law of conservation of energy
E1-E2 =W(fr)
W(fr) =μ•m1•g•cosα•s.
m1•g•H+m2•g•h- m1•g(H-s•sinα) - m1•v²/2 {m2•g(h+s) - m2•v²/2 -ks²/2 = μ•m1•g•cosα•s.
v=sqrt[{ 2gs(m1•sinα-m2)-kx²-2•μ•m1•g•cosα•s}/(m1+m2)]
I got v= 2.26 m/s. Check my calculations.

Sorry, how did you get the angle and what is x?

To determine the speed at which the blocks are moving, we can use the principle of conservation of mechanical energy. We'll break down the problem into individual steps and calculate the energy at each stage.

Step 1: Calculate the potential energy of block m1 at the starting position.
The potential energy is given by the equation:
Potential energy = mass * gravity * height

Given:
Mass of m1 (m1) = 24 kg
Gravity (g) = 9.8 m/s^2
Height (h) = 10 m

Potential energy of m1 = 24 kg * 9.8 m/s^2 * 10 m = 2352 J

Step 2: Calculate the potential energy of block m2 at the starting position.
The potential energy is given by the equation:
Potential energy = mass * gravity * height

Given:
Mass of m2 (m2) = 5 kg
Height (h) = 8 m

Potential energy of m2 = 5 kg * 9.8 m/s^2 * 8 m = 392 J

Step 3: Calculate the loss of potential energy due to the displacement of block m1.
The potential energy of m1 decreases by the amount of work done against friction.
The work done against friction is given by the equation:
Work done against friction = force of friction * displacement

Given:
Coefficient of kinetic friction (μ) = 0.24
Displacement (d) = 0.90 m

Force of friction = coefficient of friction * normal force

The normal force can be calculated using the equation:
Normal force = mass * gravity * cos(theta)
where theta is the angle of the incline.

Given:
Mass of m1 (m1) = 24 kg
Gravity (g) = 9.8 m/s^2
Angle of incline (theta) = angle of the incline

Using the equation:
Normal force = 24 kg * 9.8 m/s^2 * cos(theta)

The force of friction = 0.24 * normal force

Now, the work done against friction can be calculated as:
Work done against friction = 0.24 * normal force * displacement

Step 4: Calculate the gain in potential energy of block m2 due to its displacement.
The potential energy of m2 increases by the same amount of work done against friction as m1.
Therefore, the gain in potential energy of m2 is equal to the work done against friction.

Gain in potential energy of m2 = Work done against friction

Step 5: Calculate the final kinetic energy of the system.
The initial kinetic energy is zero as the system is released from rest.
The final kinetic energy is given by the equation:
Final kinetic energy = (Potential energy of m1 + Potential energy of m2) - (Loss of potential energy due to m1 + Gain in potential energy of m2)

Step 6: Calculate the speed at which the blocks are moving.
The final kinetic energy is equal to the kinetic energy of m2, as the blocks are connected and moving together.
The kinetic energy is given by the equation:
Kinetic energy = 0.5 * mass * velocity^2

Given:
Mass of m2 (m2) = 5 kg
Final kinetic energy (K.E) = Final kinetic energy calculated in Step 5

Using the equation:
Final kinetic energy = 0.5 * 5 kg * velocity^2

Solving for velocity, we find:
velocity^2 = (2 * Final kinetic energy) / (mass of m2)
velocity = sqrt((2 * Final kinetic energy) / (mass of m2))

Finally, substitute the calculated values and solve for velocity to get the answer in m/s.