In my 5 Steps to a 5 AP Physics prep book, page 70 is a review of tension problems, also known as many-body problems. I give eight different situations in which blocks are connected by ropes. The goal of each problem is to find the tension(s) in the rope(s), and the acceleration of the system.
The approach that I advocate is to draw a separate free body diagram for each block, then write Newton's second law separately for each block. The acceleration and tension(s) are solved for by adding the Newton's second law equations together.
Ruth Mickle, of Atlanta, noted yesterday that she gets a different answer to problem 6 than is printed in the book. I agree -- for whatever reason, the answer in my book is wrong. Below I give a thorough solution.
The problem shows three blocks connected by strings over a pulley, as shown at the top of the post. Given that m is 1.0 kg, the question asks for the tensions and acceleration.
Start by drawing three free body diagrams. Note that the two ropes will have two tensions; I'll label these T1 and T2.
The acceleration will be toward the heavier blocks. Thus, the mass m will accelerate upward, and the other masses will accelerate downward. So when we write Newton's second law, for mass m we'll write "up forces - down forces = ma." For the other masses, we'll write "down forces - up forces = ma." Always start Newton's second law in the direction of the acceleration.