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: