Question

For an airfoil section the pitching moment coefficient is determined about a reference point that is 0.3 times the chord behind the leading edge. It varies with the lift coefficient as shown in the table below. The distance of the aerodynamic center from the leading edge of the airfoil as a fraction of the chord is ________ (rounded off to one decimal place).

\(c_l\)	0.2	0.4	0.6	0.8
\(c_m\)	-0.02	0	0.02	0.04

Hint:

The slope of the cm versus cl line equals (href - hac); use two points from the table to find the slope, then solve for hac.

Answer 1

Correct Answer: 0.2

Step 1: Recall how the pitching moment coefficient changes with reference point.
For an airfoil, the moment about the aerodynamic center (ac) does not change with lift coefficient by definition, \(dc_{m,ac}/dc_l = 0\). About any other reference point located at a distance \(h\) chords behind the leading edge, the standard result is
\[ c_{m,ref} = c_{m,ac} + c_l\,(h - h_{ac}) \]
where \(h_{ac}\) is the location of the aerodynamic center as a fraction of chord. So the slope of the \(c_m\) versus \(c_l\) line directly gives \((h-h_{ac})\).

Step 2: Compute the slope from the table.
Using the first and last data points,
\[ \frac{dc_m}{dc_l} = \frac{0.04-(-0.02)}{0.8-0.2} = \frac{0.06}{0.6} = 0.1 \]
This is the same for any pair of points in the table, confirming the relationship is linear, as expected.

Step 3: Solve for the aerodynamic center location.
Here \(h = 0.3\) (the given reference point) and \(h-h_{ac}=0.1\), so
\[ h_{ac} = h - 0.1 = 0.3-0.1 = 0.2 \]

Final Answer:
The aerodynamic center lies 0.2 chords behind the leading edge.
\[ \boxed{h_{ac} = 0.2} \]