Compute maneuvering speed below max gross using the formula $V_A\sqrt{\frac{W_2}{W_1}}$, where $V_A$ is the maneuvering speed at max gross, $W_2$ is actual weight, and $W_1$ is max gross.
We can derive this relationship — or for any other V-speed such as stall speed of landing speed that varies with weight — from the lift equation. In steady-state flight, weight equals lift so
$$W_1 = \frac{1}{2} C_L \rho v_1^2 S$$
and likewise for $W_2$ and $v_2$. Dividing the first by the second cancels the coefficients and leaves
$$\frac{W_1}{W_2} = \frac{v_1^2}{v_2^2}$$
Take the square root of both sides and solve for $v_2$ to arrive at the general formula
$$v_2 = v_1 \sqrt{\frac{W_2}{W_1}}$$
John Denker provides an intuition for why the relationship works the way it does.
Unlike $V_{NO}$, the maneuvering speed varies in proportion to the square root of the mass of the airplane. The reason for this is a bit tricky. The trick is that $V_A$ is not a force limit but rather an acceleration limit. When the manufacturers determine a value for $V_A$, they are not worried about breaking the wing, but are worried about breaking other important parts of the airplane, such as the engine mounts. These items don’t directly care how much force the wing is producing; they just care about the acceleration they are undergoing.
By increasing the mass of the airplane, you decrease the overall acceleration that results from any overall force. (Of course, if you increase the mass of cargo, it increases the stress on the cargo-compartment floor — but it decreases the stress on unrelated components such as engine mounts, because the acceleration is less.)
Later in the same section, Denker clarifies.
Finally, we should note that there are two different concepts that, loosely speaking, are called maneuvering speeds.
- The design maneuvering speed, which we can denote $V_{A(D)}$, is primarily of interest to aircraft designers, not pilots. The designer must choose a value for $V_{A(D)}$ and then build an aircraft strong enough to withstand certain stressful maneuvers at that speed. Higher values of $V_{A(D)}$ promote safety, by forcing the design to be stronger.
- The maneuvering speed limitation, which we can denote $V_{A(L)}$, is of interest to pilots. It is an operating limitation. It appears on a placard in the cockpit. Lower values of $V_{A(L)}$ promote safety, by restricting certain operations to lower, less-stressful airspeeds.
Denker, John S., See How It Flies, §2.14.2 “Maneuvering Speed,” accessed 16 Aug 2015.
The discussion above pertains to the maneuvering speed limitation, i.e., values a pilot would find in a POH or on placards. For example, the humble Cessna 152 POH shows $V_A$ decreasing with decreasing weight: 104 KIAS at the max gross of 1,670 pounds, 98 KIAS at 1,500 pounds, and 93 KIAS at 1,350 pounds. The reader will also note that these values fit the general relationship given at the beginning of this answer.