Two masses are connected by a string over an ideal pulley as shown. The coefficient of kinetic friction between m1 and the table surface is 𝜇. Find an expression for their common speed after the blocks have moved a distance L. You can assume that you know: m1, m2, L, and 𝜇. (i.e. It’s ok for those variables and any known constants to appear in your answer.)

To find the expression for the common speed of the two masses after they have moved a distance L, we can use the principles of Newton's laws and the concepts of work and kinetic friction.

Let's break down the problem:

1. Identify the forces acting on each mass:
- Mass m1: Tension force T and kinetic friction force f1.
- Mass m2: Tension force T and gravitational force mg.

2. Write the equations of motion for each mass:
- For mass m1:
The net force acting on m1 is given by:
ΣF1 = T - f1 = m1 * a1, where a1 is the acceleration of mass m1.

- For mass m2:
The net force acting on m2 is given by:
ΣF2 = T - mg = m2 * a2, where a2 is the acceleration of mass m2.

3. Determine the acceleration and the common speed:
- We need to find the acceleration of the system. Since both masses are connected and move together, their accelerations will be the same. Therefore, a1 = a2 = a.

- Rearranging the equations above, we can solve for T:
T = m1 * a + f1 (Equation 1)
T = m2 * a + mg (Equation 2)

Setting the right sides of Equations 1 and 2 equal to each other, we get:
m1 * a + f1 = m2 * a + mg

- Using the expression for kinetic friction force: f1 = μ * N, where N is the normal force acting on m1. We know that N = m1 * g (where g is the acceleration due to gravity), so:
m1 * a + μ * m1 * g = m2 * a + m2 * g

- Now, solving for a, we get:
(m1 + μ * m1) * a = (m2 + m1) * g
a = ((m2 + m1) * g) / (m1 + μ * m1)

- Once we have the acceleration, we can use it to find the common speed of the masses after they have moved a distance L. The relationship between acceleration, distance, and speed is given by the equation:
v^2 = u^2 + 2 * a * s

Where v is the final speed, u is the initial speed (which we assume to be zero), a is the acceleration, and s is the distance (L in this case). Plugging in the values, we get:
v^2 = 0 + 2 * a * L
v = sqrt(2 * a * L)

Therefore, the expression for the common speed of the masses after moving a distance L is given by:
v = sqrt(2 * ((m2 + m1) * g) / (m1 + μ * m1) * L)

Make sure to substitute the given values of m1, m2, L, and μ into the equation to get the numerical value.