The blocks of mass 20.0 kg and 10.0 kg are initially at rest on the floor and are connected by a massless string passing over a massless and frictionless pulley. An upward force F is applied to the pulley.
Find the accelerations a_A of block A and a_B for block B when F is
i) 124 N
ii) 294 N
iii) 424 N
∑F = ma
m_A = mass of block A = 20 kg
m_B = mass of block B = 10 kg
a_A = acceleration of block A
a_B = acceleration of block B
W_A = weight of block A = (20 * 9.8) = 198 N
W_B = weight of block B = (10 * 9.8) = 98 N
T = Tension
The Attempt at a Solution
Taking up to be the positive y direction and right to be the positive x direction, I've drawn free body diagrams for the pulley (F_pull in the +y direction; (W_A + W_B) in the -y direction), block A (T in the +y direction, W_A in the -y direction), and block B (T in the +y direction, W_B in the -y direction).
Also, I've stated that since there are no forces in the x direction, the net force is equal to the net force of the y-components of the system, and that since the string is massless, T is the same for both block A and block B.
I have set up the following equations:
∑F_A = T - W_A = T - 198 = m_A * a_A
∑F_B = T - W_B = T - 98 = m_B * a_A
T = (m_A * a_A) + 198 = (m_B * a_B) + 98
but I'm not sure where to go from here. I realize that, with respect to the floor, the system itself will be accelerating at different rates with the different given forces of i, ii, and iii. I also realize that blocks A and B will have the same acceleration with respect to the pulley once they are lifted from the floor, regardless of what force is pulling upwards on the pulley.
Any guidance as to what I need to do next would be very much appreciated.