Solution

Statement

$2.1.55.$ Two small spheres, each of mass $m$, connected by a thread of length $l$, are moving with a velocity $v$ on a horizontal table in a direction perpendicular to the connecting thread (the thread is taut). The middle of the thread hits a nail. What is the tension force of the thread immediately after this?

Solution

Since the thread is taut:

$$T_1 = T_2 = T$$

As soon as the thread hits the nail, both bodies continue to move with velocity $v$ along a new trajectory with radius $r=\frac{l}{2}$:

$$T=ma_n$$

$$T=m\frac{v^2}{\frac{l}{2}}$$

We obtain the tension force of the thread immediately after this:

$$\boxed{T = \frac{2m v^2}{l}}$$