The system of blocks of masses m1=3kg and m2 = 1.80 kg are connected by a string as shown. Mass m rests on a rough surface of unknown coefficient of kinetic friction. After the masses are released, they move with an acceleration of 2.80 m/s^2 (m1 moves to the right, m2 moves downward.)

Find the tension on the string.

Find the coefficient of kinetic friction.

To find the tension on the string, we can start by considering the forces acting on each block.

For m1 (the block with mass 3kg), the forces acting on it are:
1. The tension force T pulling it to the right.
2. The force of kinetic friction fk, acting to the left.
3. The weight force mg acting downward.

We can write the equation of motion for m1 as:
ΣF = m1 * a
where ΣF is the net force acting on m1, m1 is the mass of block 1 (3kg), and a is the acceleration (2.80 m/s^2).

ΣF = T - fk - mg

For m2 (the block with mass 1.80kg), the forces acting on it are:
1. The tension force T pulling it downward.
2. The weight force mg acting upward.

We can write the equation of motion for m2 as:
ΣF = m2 * a
where ΣF is the net force acting on m2, m2 is the mass of block 2 (1.80kg), and a is the acceleration (2.80 m/s^2).

ΣF = T - mg

Since both blocks are connected by the same string, the tension force T in both equations is the same.

Next, we need to determine the value of the friction force fk. To do this, we'll use the equation for kinetic friction:

fk = μk * N
where μk is the coefficient of kinetic friction and N is the normal force.

The normal force N can be calculated by considering the weight force acting on m1:
N = mg

Substituting the value of N into the equation for fk, we have:
fk = μk * mg

Substituting the expressions for ΣF, fk, and N into the equation of motion for m1, we get:
T - μk * mg - mg = m1 * a

Substituting the expressions for ΣF and N into the equation of motion for m2, we get:
T - mg = m2 * a

Now we have two equations with two unknowns (T and μk). We can solve these equations simultaneously to find the values of T and μk.

Let's set up the equations and solve them:

From the equation for m1:
T - μk * mg - mg = m1 * a

From the equation for m2:
T - mg = m2 * a

Substituting the given values:
m1 = 3kg
m2 = 1.80kg
a = 2.80 m/s^2
g = 9.8 m/s^2

We have the following equations:

T - μk * (3kg * 9.8 m/s^2) - (3kg * 9.8 m/s^2) = 3kg * 2.80 m/s^2

T - (μk * 29.4kg) - 29.4kg = 8.4kg m/s^2 (Equation 1)

T - 1.80kg * 9.8 m/s^2 = 1.80kg * 2.80 m/s^2 (Equation 2)

Now we can solve these equations simultaneously to find the values of T and μk.