A block of mass m1 hangs vertically and is attached by a string to a block of mass m2 on a frictionless slope inclined at angle theta to the horizontal, as shown above. If m1 =

2.5 kg, it accelerates at -0.108 m/s2. If m1 = 0.8 kg, it accelerates at 2.888 m/s2.
Determine the angle theta and m2.

To determine the angle theta and m2, we can use Newton's second law and the principles of forces and acceleration. Let's break down the problem step by step:

Step 1: Draw a free-body diagram for each block.
- For m1, there are two forces acting on it: the tension force T in the string pointing upward, and its weight mg pointing downward.
- For m2, there are three forces acting on it: the weight mg pointing vertically downward, the normal force N perpendicular to the slope, and the force of friction Ff parallel to the slope but opposite to the direction of motion.

Step 2: Write the equations of motion for each block.
- For m1, the net force in the vertical direction is T - mg. Since m1 accelerates in the downward direction, we have:
T - m1g = m1 * (-0.108) (Equation 1)

- For m2, the net force in the direction parallel to the slope is the component of gravity along the slope (mg sin(theta)) minus the force of friction (Ff). The net force in the perpendicular direction is the component of gravity perpendicular to the slope (mg cos(theta)) minus the normal force (N). Since m2 doesn't accelerate in the perpendicular direction, we have:
mg cos(theta) - N = 0 (Equation 2)

Step 3: Solve the equations simultaneously.
- First, we need to find the equation for the force of friction Ff. The force of friction can be calculated using the equation:
Ff = u * N, where u is the coefficient of friction and is zero in this case since the slope is frictionless.

- Plugging this value into Equation 2, we have:
mg cos(theta) - N = 0 -->(Equation 2)
mg cos(theta) - 0 = 0 -->(Since N = 0)

- Solving for the angle theta gives us:
cos(theta) = 0
theta = 90 degrees

- Now we can substitute theta = 90 degrees in Equation 2 to find m2:
m2g cos(theta) - N = 0
m2g cos(90) - 0 = 0
m2g = 0
m2 = 0 kg

Therefore, the angle theta is 90 degrees, and the mass of m2 is 0 kg.