Several problems with solutions and detailed explanations on systems with strings, pulleys and inclined planes are presented. Free body diagrams of forces, forces expressed by their components and Newton's laws are used to solve these problems. Problems involving forces of friction and tension of strings and ropes are also included.
Problem 1
A block of mass 5 Kg is suspended by a string to a ceiling and is at rest. Find the force F _{c} exerted by the ceiling on the string. Assume the mass of the string to be negligible.
Solution
a) The free body diagram below shows the weight W and the tension T_{1} acting on the block. Tension T_{2} acting on the ceiling and F_{c} the reaction to T_{2}. Hence action reaction (Newton's 3 rd law) : F_{c} = T_{2} We now consider the forces acting on the block (Free Body Diagram) Since the block is at rest W + T_{1} = 0 (Newton's second law, vector equation) W = (0 , W) T_{1} = (0, T_{1}) Hence : (0 , W) + (0, T_{1}) = 0 sum of y coordinates = 0 gives W = T_{1} T_{1} and T_{2} represent the tension of the string and their magnitudes are equal. Hence T_{2} = T_{1} T_{2} and F_{c} are action and reaction pairs and therefore their magnitudes are equal. Hence F_{c} = T_{2} = T_{1} = W = m g = 5×10 = 50 N 
Problem 2
In the figure below is shown the system below are shown two blocks linked by a string through a pulley, where the block of mass m _{1} slides on the frictionless table. We assume that the string is massless and the pulley is massless and frictionless.a) Find the magnitude of the acceleration of the two masses
b) Find the tension in the string
Solution
a)

Problem 3
In the two blocks of masses m _{1} and m _{2} and pulley system below, the pulley is frictionless and massless and the string around the pulley is massless. Find an expression of the acceleration when the block are released from rest.
Solution

Problem 4
Three cords are knotted at point P, with two of these cords fastened to the ceiling making angles α1, α2 and a block of mass m hangs from the third one as shown below.a) Find the magnitude of the tension in each cord in terms of α1, α2 and m so that the system is at rest.
b) Find numerical values to the three tensions found above for α1 = 45° , α2 = 30° and m = 1 Kg.
Solution
a)

Problem 5
The system below includes 3 blocks of masses m _{1} = 1 Kg, m2 = 2 Kg and m3 = 5 Kg linked by massless and frictionless strings and pulleys.a) Find the magnitude of the acceleration of the 3 blocks.
b) Find the magnitude of the tension of the string between m _{1} and m2.
c) Find the magnitude of the tension of the string between m2 and m3.
Solution
a)

Problem 6
In the system below, blocks of masses m _{1} = 10 Kg and m _{2} = 30 Kg are linked by a massless string through a frictionless pulley.a) Find the magnitude of the acceleration of the two masses if the coefficient of kinetic friction between the inclined plane and mass m _{1} is equal to 0.4.
b) Find the magnitude of the tension in the string.
Solution
a)
