# How does mass affect rate of turn?

After seeing effect of speed and bank angle on rate of turn, how would one mathematicly calculate the effect of changing mass. For example, in a 30 degree turn at 100 knots, how would the rate of turn be affected if identical planes, one at 10,000 lbs, one at 9000 lbs were compared. Granted, the pitch and power settings may be slightly different, generating different amounts of lift. Would the heavier plane turn faster due to its greater horizontal lift component, or would it's heavier mass keep it more in the same direction?

The second law of motion states that: $$F = m * a$$

Rate of turn will be determined by the acceleration: $$\frac{F}{m} = a$$

If your vertical lift is 10,000 lb, in a 30 degree bank, your lateral force is $$10,000 * tan(30)$$, your acceleration is $$\frac{10000 * tan(30)}{10000}$$, or $$tan(30)$$.

If your vertical lift is 9,000 lb, in a 30 degree bank, your lateral force is $$9,000 * tan(30)$$, your acceleration is $$\frac{9000 * tan(30)}{9000}$$, or $$tan(30)$$.

Thus you can see the rate of turn depends on the bank angle, not the mass.

You have stumbled on an intriguing and fundamental aspect of Physics, the equivalence of Gravitational Mass and Inertial Mass.

Gravitational mass is the "m" in the equation that is used to determine the "force" of gravity due to one mass M, acting on another body with mass m.

$$F=G\frac{m M}{r²}$$

Inertial mass is simply the mass in the famous equation based on Newton's second Law, F=ma, that is used to calculate the acceleration that results from applying a force F to a body with mass m.

The fact that these two completely unrelated (or what seem to be unrelated) masses, are identical, and, (to the degree that physicists have been able to measure them), are always identical, was and had been a deep mystery for a long time. This mystery, (which even Newton was confounded by and unable to explain), was only explained by Einstein, when he published his General Theory of Relativity in 1915. The explanation requires a more elaborate explanation than can be provided here, but if you're interested, check out this link.

But this equivalence is the answer to your question. As the mass of an aircraft increases, the "force" of gravity, and the lift required to counteract that force increase by exactly the same amount as the force required to generate the same sideways acceleration (turn rate).

It's going to be identical (in the ideal case).

I'm presuming that you want to maintain a constant-altitude, constant-speed, constant-bank turn. To do that, the vertical component of lift must be equal to weight, or 1g. Therefore a fixed bank will give you a horizontal component of lift that is a fixed ratio of the vertical lift. For 30 degrees, this is 0.57 times the vertical component or 0.57g horizontally.

Increased mass(weight) requires additional lift for flight. And that additional lift balances the inertia of the increased mass.

Obviously for a plane with no other changes, this increase in required lift will cause it to reach limits at a lower bank angle as the mass increases.