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.)

Question

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.)

Answer 1

To find the expression for the common speed of the two masses after they have moved a distance L, we can apply Newton's laws of motion and consider the forces acting on each mass.

Let's denote m1 as the mass on the left side and m2 as the mass on the right side. The tension in the string connecting the masses is the same throughout, so we can denote it as T.

For m1:
- There is a force of gravity acting downward (m1 * g), where g is the acceleration due to gravity.
- There is a frictional force opposing the motion given by the coefficient of kinetic friction (𝜇) multiplied by the normal force (m1 * g), which can be written as 𝜇 * m1 * g.

For m2:
- There is a force of gravity acting downward (m2 * g).

Since the masses are connected by a string over an ideal pulley, the tension force (T) is equal to the force of gravity for m2. Therefore, T = m2 * g.

Now, considering the motion of the system, we can write the net force equations for each mass:

For m1:
m1 * a = T - 𝜇 * m1 * g (1)

For m2:
m2 * a = m2 * g (2)

Since the masses are connected by a string, the acceleration (a) is the same for both masses. So, we can set equations (1) and (2) equal to each other:

m1 * a = T - 𝜇 * m1 * g
m2 * a = m2 * g

m1 * a = m2 * g

Now, we can solve for the acceleration (a):

a = (m2 * g) / m1

Finally, we can use this acceleration to find the common speed after the blocks have moved a distance L. The formula for speed (v) is given by:

v^2 = u^2 + 2 * a * d (3)

where u is the initial speed (which is zero in this case) and d is the distance traveled (L).

Plugging in the values of the acceleration and distance into equation (3), we get:

v^2 = 0 + 2 * ((m2 * g) / m1) * L
v^2 = (2 * m2 * g * L) / m1

Taking the square root of both sides, we find the expression for the common speed (v):

v = sqrt((2 * m2 * g * L) / m1)

So, the expression for the common speed of the two masses after they have moved a distance L, given m1, m2, L, and 𝜇, is sqrt((2 * m2 * g * L) / m1).

Answer 2

To find the expression for the common speed of the two masses after they have moved a distance L, we can apply Newton's laws of motion and consider the forces acting on each mass.

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