A train locomotive is pulling two cars of the same mass behind it (Fig. 4–51). Determine the ratio of the tension in the coupling (think of it as a cord) between the locomotive and the first car$\left(F_{T1}\right)$to that between the first car and the second car$\left(F_{T1}\right)$for any non-zero acceleration of the train.

FIGURE 4-51 Problem 27

The term ‘force’ can be defined as the action that tends to cause an object's acceleration. The relationship between the force and the object’s acceleration is linear.

Draw a free-body diagram.

As the locomotive is pulling both the cars of mass $m$, the tension in the coupling is given as

role="math" localid="1645427333239" $\begin{array}{c}{F}_{T2}=ma+ma\\ F_{T2}=2ma\dots \left(i\right)\end{array}$

Here, $a$is the acceleration of the system.

As the first car pulls the second car, the tension in the coupling will be

role="math" localid="1645427361010" $F_{T1}=ma\dots \left(ii\right)$

Dividing the equations (i) by (ii) to get the value of the ratio,

$\begin{array}{c}\frac{F_{T2}}{F_{T1}}=\frac{2ma}{ma}\\ \frac{F_{T2}}{F_{T1}}=\frac{2}{1}\\ F_{T2}=2F_{T1}\end{array}$

Thus, the tension in the coupling between the locomotive and the first car is twice that between the first and the second cars.