Masses Connected by Vertical Strings

Masses connected by vertical strings is another problem that requires both Newton’s 2nd and 3rd laws. Consider two objects A and B, with masses of 2kg and 5kg respectively as shown below. Let’s examine two situations:

a) Objects A and B are stationary

b) Objects are accelerating up at a) If the objects are stationary, the net force on each object = 0. The forces involved will be the tension in the string. The tension in each string is equal to the weight that they support: = the weight of object A:    = the weight of object A + object B: . This is due to the string at supporting object A and B.  Therefore; and b) If the objects are accelerating up at , we can determine the net force acting on each object using . We can also determine the tension in each string using: . This equation involves vectors and down is best noted as positive. The acceleration will be negative if directed upwards or negative if directed downwards. Note that a positive or negative result for tension is not important because tension is a force that acts in both directions on a string. Essentially this means that the equation: gives us a magnitude for the tension.

If we consider the acceleration of each object we can determine the net force, acting on each object:   Observing the diagram below, we can see that: (we will note down as positive)          Example 1:

Two objects of masses 3kg and 10kg are connected by vertical strings as shown below. The objects are stationary. Calculate:

a) The net force acting on each mass

b) The tension in each string a) If the objects are stationary, the net force on each object = 0

b) The tension in each string is equal to the weight that they support: = the weight of the 3kg mass:    = the weight of both masses:   Therefore; and Example 2:

Two objects of masses 5kg and 12kg are connected by vertical cables as shown below. The objects are accelerating upwards at . Calculate:

a) The tension in each string: we will note down as positive:         Example 3:

Two objects of masses 2kg and 4.5kg are connected by vertical cables as shown below. The objects are accelerating downwards at . Calculate:

a) The tension in each string: we will note down as positive:         