Tension force in a string – class notes

A lot of problems we discuss involve ropes and strings. Most of the time, we assume that they are massless, or more precisely, that the mass of a string is much smaller than the other masses in the problem.

As you know from everyday life, if you grab a weight by a string, you feel the full force of weight transmitted through the string. This is illustrated in Figure 1

magnitude of the tension exerted at one end of a massless string is the same as on the other side

Here, with the help of FBD (Free Body Diagram), we will see that the magnitude of the force (i.e. tension) exerted at one end of a massless rope (or string) is the same as on the other side of the rope (or string)

Here we have a person holding a string that’s attached to a weight.

The different colored arrows correspond to the forces acting on different objects.

The red arrows correspond to the forces acting on the weight. The blue corresponds to the forces acting on the string, and the green arrow, the force acting on the hand.

We want to figure out what is force acting on the hand.

So we consider each object separately.

Say, the force exerted by gravity on the weight W.
Call the force applied by the weight on the string F_WS.
Similarly, the force applied by the string(tension force) on the weight = F_SW.

And, say the force applied by the weight on the string = F_WS
Similarly, the force applied by the string on the weight = F_HS

These names are listed in the figure.

Then, Considering the forces acting on the weight and considering zero acceleration of it:

W – F_sw = 0 ……… (1)

W = F_sw ……… (1)

Now, let’s consider the forces acting on the string:

F_WS = F_HS ……… (2)

From Newton’s 3rd law, we can say:
F_sw = – F_ws …… (3)
F_HS = – F_SH …… (4)

From the above equations, we get
W = F_sw = – F_ws = – F_HS = – (- F_SH ) = F_SH
W = F_SH ……….(5)

Weight at the bottom end of the string = Tension applied by the string on the hand

As you can see from the above equation, the tension transmits the force of gravity that acts on the weight, to a force acting on the hand. This is a general feature of massless strings and is true even if there is acceleration.

The magnitude of the force (i.e. tension) exerted at one end of a massless rope is the same as on the other side, which can be seen from Equation 5 above.