If the Short Rook starts its life on b1, c1, f1, or g1, it gets in the way of the other pieces and impedes its whole army.
If the short Rook starts the game on a1 or h1, it is slow getting into the game; if it is an R5 or better, it is good that it is slow to develop because this helps to preserve it for the endgame.
The R4 (or shorter), however, is a minor piece, like a Bishop or a Knight, and it should get into the game as soon as possible in order to fight with the enemy's minor pieces. This tension between the strategic role of a minor piece and the slow development suitable for a major piece helps to make the R4 (or shorter) an interesting piece to have.
Similar calculations show the R2 has the mobility of 0.566 Rooks, whereas a Knight has a value of 0.666666 Rooks, or if you use the beginners' method of counting values it is worth 0.6 Rooks. Comparing average mobility with piece values is dangerous; sometimes the mobility is a very good guide, and sometimes it is misleading. In this case it seems to be pretty good, so here is a list of all the values:
RATIO OF PIECE TO ================= PIECE R7 R1 R(N-1) WIN ===== ===== ===== ===== ==== R1 0.335 1.000 ***** .... R2 0.566 1.690 1.690 .... R3 0.726 2.166 1.282 0.22 R4 0.836 2.495 1.152 0.33 R5 0.911 2.721 1.091 0.38 R6 0.964 2.878 1.057 0.45 R7 1.000 2.986 1.038 0.5The R(N-1) column shows how much more mobility each piece has than the one before it; so for example R3 has 1.282 times the mobility of R2, and so on.
The WIN column shows the winning percentage when a weak computer program plays a few thousand games with White having short Rooks, Black having full Rooks, and of course the program believes thay are of equal value. These numbers are very precise and repeatable, but I don't know exactly what they mean, or even if they do mean anything.
The calculated values correspond somewhat to the observed values of the short-Rooks R3, R4, and R5, but the calculations tend to greatly overestimate the values of the shorter pieces (R1, R2, and R3; and, to a lesser extent, R4).
It is better to think of the R4 as a strong Bishop than as a weak Rook. In other words, you should treat the R4 as a "minor piece", in the same class as the N or B.
It is interesting that the practical difference between R4 and R5 is so great, even though the mobility calculations show the R5 to be only "worth" 1.1 times as much. The main reason seems to be that if you lift the R5 a mere two squares, it can reach all the way to the other side of the board.
For example, moving an R5 from e1 to e3 allows it to attack an enemy Rook on e8, but an R4 on e1 would have to advance at least to e4; it is much easier to have e3 defended by friends and not attacked by enemy Pawns or by enemy minor pieces than it is to have e4 similarly available.
Another way of looking at this is that when a R4 is on one of the four center squares it can make all the moves that a Rook could make; a R5 can make all the R moves from 16 squares, and a R6 can do it from 36.
In a position with a White K on d2, a White R on e4, a Black K on d7, a Black Rook e8, and Black Pawns a7 and h7, White to play can draw simply by means of 1. Re4-a4 Re8-a8 2. Ra4-h4 and so on.
If White had a R6 on e4, the game would quite likely be lost.