Section 6.14 Tension Force
Many common situations involve tying two objects by using a rope or a string. When the connecting string is taut, the string acts as a stretched spring and pulls in the the two bodies at the two ends. We say that the taut string provides a tension force on the two bodies. We will denote the magnitude of the tension force by \(F_T\) and sometimes just by \(T\text{.}\)
Consider two objects A and B connected by a taut string as shown in Figure 6.14.1. The dynamics of the two bodies are connected by force of tension in the string.
When tension force acts on A, we have the tension force vector \(\vec F_{T}^{\text{on A}}\text{,}\) and when it acts on B, we have the tension force vector \(\vec F_{T}^{\text{on B}}\text{.}\) The direction of \(\vec F_{T}^{\text{on A}}\) will be along the string away from A and that of \(\vec F_{T}^{\text{on B}}\) away from B. Even though these two tension forces are different forces, we will often use the same symbol for them and which one we mean will be clear from the context.
Unlike spring force, we do not have any formula for the magnitude of the tension force. If you have memorized formulas such as \(F_T = mg\text{,}\) then you need to forget that since there is no such law. The magnitude of tension is obtained by requiring the consistency of Newton's laws of motion applied to a given physical situation. When we study elasticity, we will be able to give a formula for tension in terms of Youn's modulus using Hooke's law. But, for now, we will just represent tension in a situation by a unknown symbol.
Dynamics of bodies connected by a tense string are connected. A great example of a conected system is Atwood machine shown in Figure 6.14.2, where two blocks are tied together by a string that goes over a pulley. The string connects the dynamics of three bodies in this system: the two blocks and the pulley.
If string does not slide on the pulley surface due to sufficient static friction, the strings on the two sides of the pulley may have two different magnitudes \(T_1 \) and \(T_2\) with \(T_1 \ne T_2\text{.}\) However, if pulley's mass is negligible compared to the masses of the blocks, we can assume \(T_1 = T_2\) and represent them with the same symbol \(T\text{.}\) We will study the dynamics of coupled systems in another section.
