Two blocks are attached together with a piece of string. Block #1 (3 kg) slides along a

rough horizontal surface and block #2 (2 kg) hangs off the end of the surface. If the blocks
accelerate at 2.5 m/s2
in the directions shown, determine the tension in the string and the
coefficient of kinetic friction (µk) between block #1 and the surface.

To determine the tension in the string and the coefficient of kinetic friction between block #1 and the surface, we need to analyze the forces acting on the blocks.

Let's start by listing the forces acting on each block:

For Block #1:
1. The tension force in the string is pulling Block #1 to the right.
2. The force of kinetic friction is opposing the motion and acting to the left.

For Block #2:
1. The weight of Block #2 (mg) is acting downward.

Since the blocks are connected and moving together, the tension in the string is the same for both blocks. Let's call this tension force T.

Now, let's apply Newton's second law to Block #1:

ΣF = ma

Where,
ΣF is the net force acting on the block,
m is the mass of the block (3 kg in this case),
and a is the acceleration of the block (2.5 m/s^2 in this case).

The net force on Block #1 is given by:
ΣF = T - friction force

The friction force can be calculated using the equation:
friction force = µk * normal force

Here, the normal force is equal to the weight of Block #1:
normal force = mg

Substituting the value of the normal force, we have:
friction force = µk * mg

So the equation for the net force becomes:
ΣF = T - µk * mg

Now, let's apply Newton's second law to Block #2:

ΣF = ma

Since Block #2 is hanging vertically, the net force acting on it is the tension force T minus the weight (mg) acting downward:

ΣF = T - mg

Since the blocks are connected, their acceleration is the same. Therefore, we can equate the two expressions for ΣF and set them equal to each other:

T - µk * mg = T - mg

Notice that the masses cancel out, so we can rearrange the equation to solve for the tension T:

T - T = µk * mg - mg

0 = µk * mg - mg

0 = mg(µk - 1)

Since the mass (m) is nonzero, µk must be equal to 1 for the equation to hold true. This means that the coefficient of kinetic friction (µk) between Block #1 and the surface is 1.

Next, let's solve for the tension T:

0 = mg(µk - 1)

0 = 3kg * 9.8 m/s^2 * (1 - 1)

0 = 0

Since the tension T does not affect the equation, its value cannot be determined from the given information.

In conclusion, the coefficient of kinetic friction (µk) between Block #1 and the surface is 1, but the tension in the string (T) cannot be determined without more information.