# How does acceleration affect tension?

May 29, 2014

In a typical case of two objects, one pulling another with a rigid link in-between, the higher acceleration of the first results in a higher tension in a link.

Here is why.
Assume the mass of the first (pulling) object is ${M}_{1}$, the mass of the second (pulled) object is ${M}_{2}$, acceleration of a system of two objects is $a$ (the same for both objects since the link between them is rigid), the force moving the first object forward is $F$ and the tension force in a link between two objects is $T$.

The force of tension acts on the first object against its movement, it slows this object down. The force of tension acts on the second object towards the direction of movement since this is the only force that causes its moving forward.

These two forces, one acting on the pulling object and another acting on the pulled one, have opposite direction and the same absolute value. Let's choose the direction of the movement as the positive. Then the tension force acting on the first object is negative ($- T$) and the tension force acting on the second object is positive ($T$).

Force $F$, moving the first object forward is positive since it's directed towards the direction of movement.

The combination of forces acting on the first body is positive $F$ and negative $- T$. The resulting force equals, therefore, $F - T$.
The second Newton's law gives us:
$F - T = {M}_{1} \cdot a$

The only force acting on the second object is $T$, so the second Newton's law gives:
$T = {M}_{2} \cdot a$

Let's solve the system these two equations for $a$ and $T$, assuming that masses ${M}_{1}$ and ${M}_{2}$ as well as the main force $F$ moving the system forward are known.

The solution is (skipping the trivial manipulations):

$a = \setminus \frac{F}{\left({M}_{1} + {M}_{2}\right)}$

$T = \setminus \frac{F \cdot {M}_{2}}{\left({M}_{1} + {M}_{2}\right)}$

As we see, acceleration is proportional to the force $F$ as well as the tension. Increased acceleration may only be contributed to increased force $F$, which causes proportional increase of tension $T$. Therefore, the higher acceleration - the higher the tension.