Consider the surface roughness profile as shown in the figure.

The center line average roughness (Ra, in µm) of the measured length (L) is

The correct option is "1".

To find the center line average roughness ($R_a$) from the given surface profile, we analyze the geometric representation of the profile shown in the image.

1. Understanding the Profile Parameters:
From the roughness profile figure, we observe the following features:

• The total evaluation length is $L$.
• This length is divided into four equal intervals, each having a width of:

  $w=\frac{L}{4}$

• The height deviations (roughness heights) relative to the center line (mean line at $y=0$) are:
  • For the first interval $\left(0\text{to}\frac{L}{4}\right)$: $y_1=1\text{µm}$
  • For the second interval $\left(\frac{L}{4}\text{to}\frac{L}{2}\right)$: $y_2=-1\text{µm}$
  • For the third interval $\left(\frac{L}{2}\text{to}\frac{3L}{4}\right)$: $y_3=1\text{µm}$
  • For the fourth interval $\left(\frac{3L}{4}\text{to}L\right)$: $y_4=-1\text{µm}$

2. Formula for Center Line Average Roughness ($R_a$):
The center line average roughness is the arithmetic average of the absolute values of the profile deviations from the mean line over the evaluation length $L$:

$$R_a=\frac{1}{L}{\int }_0^L\left|y\right(x\left)\right|dx$$

This integral is geometrically equivalent to calculating the total sum of the areas enclosed between the profile and the mean line, divided by the total evaluation length:

$$R_a=\frac{A_1+A_2+A_3+A_4}{L}$$

where $A_i=\left|y_i\right|\times w_i$ is the absolute area of each rectangular block.

3. Calculation:
Compute the absolute area of each of the four segments:

• $A_1=\left|1\right|\times \frac{L}{4}=\frac{L}{4}$
• $A_2=|-1|\times \frac{L}{4}=\frac{L}{4}$
• $A_3=\left|1\right|\times \frac{L}{4}=\frac{L}{4}$
• $A_4=|-1|\times \frac{L}{4}=\frac{L}{4}$

Summing these absolute areas gives:

$$\text{Total Area}=\frac{L}{4}+\frac{L}{4}+\frac{L}{4}+\frac{L}{4}=L$$

Substituting this back into the formula for $R_a$:

$$R_a=\frac{L}{L}=1\text{µm}$$

Thus, the center line average roughness ($R_a$) is 1 µm.