A 33.5 kg block m1 is on a horizontal surface, connected to a 6.50 kg block m2 by a massless string. The pulley is massless and frictionless.

A force of 213.7 N acts on m1 at an angle of 33.7o. The coefficient of kinetic friction between m1 and the surface is 0.201. Determine the upward acceleration of m2.

To determine the upward acceleration of m2, we first need to calculate the net force acting on m1 and the force of friction between m1 and the surface. Then, we can use Newton's second law of motion to solve for the acceleration.

1. Calculate the net force acting on m1:
The force acting on m1 can be divided into horizontal and vertical components:
F_horizontal = Force * cos(angle) = 213.7 N * cos(33.7°)
F_vertical = Force * sin(angle) = 213.7 N * sin(33.7°)

2. Calculate the force of friction between m1 and the surface:
The force of friction can be calculated using the formula:
Force_friction = coefficient * Normal_force
The normal force is equal to the weight of m1:
Normal_force = m1 * g, where g is the acceleration due to gravity (9.8 m/s^2)
Force_friction = 0.201 * (m1 * g)
Note: Since the block is not sliding, we need to use the coefficient of kinetic friction.

3. Calculate the net force acting on m1 horizontally:
Net_force_horizontal = F_horizontal - Force_friction

4. Determine the net force acting on m2:
The tension in the string connecting m1 and m2 is equal to the force m1 exerts on m2.
Tension = m1 * acceleration
Force_net_m2 = Tension - m2 * g
Note: The net force acting on m2 is the tension pulling it upwards minus its weight.

5. Set up the equations of motion for m1 and m2:
Apply Newton's second law for both blocks:
m1 * acceleration = Net_force_horizontal
m2 * acceleration = Force_net_m2

6. Solve the system of equations:
Substitute the expressions for Net_force_horizontal and Force_net_m2:
m1 * acceleration = F_horizontal - Force_friction
m2 * acceleration = Tension - m2 * g

Substitute Tension = m1 * acceleration:
m1 * acceleration = F_horizontal - Force_friction
m2 * acceleration = m1 * acceleration - m2 * g

Combine the equations:
m2 * acceleration - m1 * acceleration = -m2 * g + (F_horizontal - Force_friction)
(m2 - m1) * acceleration = -m2 * g + F_horizontal - Force_friction

Solve for acceleration:
acceleration = (-m2 * g + F_horizontal - Force_friction) / (m2 - m1)

Now, substitute the given values for the masses, angle, force, and coefficient of kinetic friction into the equation and calculate the acceleration.