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: Write the moment-transfer relation as two equations.
Using $c_{m,ref} = c_{m,ac} + c_l(h - h_{ac})$ for two of the given data points, with $h = 0.3$:
At $c_l = 0.2$: $-0.02 = c_{m,ac} + 0.2(0.3-h_{ac})$
At $c_l = 0.8$: $0.04 = c_{m,ac} + 0.8(0.3-h_{ac})$

Step 2: Eliminate $c_{m,ac}$ by subtracting the equations.
$0.04-(-0.02) = (0.8-0.2)(0.3-h_{ac})$
$0.06 = 0.6(0.3-h_{ac})$
$0.3-h_{ac} = 0.1$

Step 3: Solve for $h_{ac}$.
$h_{ac} = 0.3-0.1 = 0.2$

Step 4: Check with the other data points.
Substituting back, $c_{m,ac} = -0.02 - 0.2(0.1) = -0.04$. Checking with $c_l=0.4$: $c_{m,ref} = -0.04+0.4(0.1) = 0$, which matches the table, confirming the answer.

Final Answer:
\[ \boxed{h_{ac} = 0.2} \]