The best size wire is fitted in a groove of a metric screw such that the wire touches the flanks of the thread on the pitch line as shown in the figure. The pitch (𝑝) and included angle of the thread are 4 mm and 60º, respectively. The diameter of the best size wire is ___________ mm (round off to 2 decimal places).

The correct answer is 2.31.

1. Understanding the Best Size Wire:
The "best size wire" is defined as a wire of a specific diameter that, when placed in the thread groove, makes contact with the flanks of the thread exactly on the pitch line. This is clearly shown in the provided figure, where the wire is tangent to both flanks at the points intersecting the horizontal dash-dot "Pitch line".

2. Given Data from the Image and Question:
Pitch of the screw thread, $p=4\text{mm}$
Included angle of the thread, $\alpha =60°$
From the geometry indicated in the figure:
- The horizontal distance from the centerline of the groove to the contact point on the pitch line is $0.25p$ (or $\frac{p}{4}$).
- The half-angle of the thread is $\frac{\alpha }{2}=30°$.

3. Formula for the Diameter of the Best Size Wire:
The relationship between the diameter of the best size wire ($d$), the pitch ($p$), and the thread angle ($\alpha$) is derived from the right-angled triangle formed by the wire center, the contact point, and the vertical groove axis:

$$d=\frac{p}{2}\sec \left(\frac{\alpha }{2}\right)$$

4. Step-by-Step Calculation:
Substitute the values of $p=4\text{mm}$ and $\alpha =60°$ into the formula:

$$d=\frac{4}{2}\sec \left(\frac{60°}{2}\right)$$

Simplify the expression:

$$d=2\times \sec (30°)$$

Since $\sec (30°)=\frac{1}{\cos (30°)}=\frac{2}{\sqrt{3}}\approx 1.1547$:

$$d=2\times 1.1547=2.3094\text{mm}$$

Rounding off to 2 decimal places, we get:

$$d\approx 2.31\text{mm}$$