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- Assuming the pulleys are frictionless, the tension in the string is the same all the way round.
- Each pulley is in equilibrium under the action of 3 forces: the tensions in the two parts of the string, and a reaction force exerted by the table. This reaction force, then, is equal and opposite to the resultant force exerted by the string.
So, let the reaction force at each pulley have two components at right angles to each other in convenient directions (e.g. for pulley A along BA and along DA) and then resolve in each of these two directions.
PS Indeed that works OK.
I presume that you have found the tension in the string, , to be
Then let the components of the reaction force at A be and along DA and AB, respectively. Then resolve in these two directions, and get:
Solve for and . Then find the magnitude of the reaction (and hence the resultant force exerted by the string on the pulley at A) by calculating N
Similarly at B, but it's even easier because the components of the reaction force are equal to each of the tensions along BA and BC. The resultant force therefore has magnitude N.