According to Wikipedia, "In the turn coordinator, the gyro is canted 30 degrees from the horizontal, so it responds to roll as well as yaw." I don't understand this point. How does rotating the gimbal 30 degrees allow the turn coordinator to respond to roll? The angular momentum and torque vectors are the same in both situations. In this case, angular moment is pointed out of the screen and torque from rolling the aircraft pointed horizontal to the right. What am I missing?
Short answer:
What am I missing?
In two words: gyroscopic precession.
The angular momentum and torque vectors are the same in both situations.
Yes they are, but the orientation of the instrument's gimbal axis is not the same in each case, so the indication of the instrument will not be the same in each case.
In more detail:
In analyses like these, it is always useful to "take it to the extreme".
Consider two hypothetical cases:
One, the gimbal axis is purely "fore-and-aft" in the aircraft's reference frame.
And two, the gimbal axis is purely "up-and-down" in the aircraft's reference frame.
In both cases the gimbal axis is perpendicular to the spin axis of the gyro wheel.
So if one case makes the instrument respond only to yaw, then the other case must make the instrument respond only to roll. And intermediate cases where the gimbal axis is somewhere between "fore-and-aft" and "up-and-down" in the aircraft's reference frame, must make the instrument respond to both yaw and roll.
(The instrument will not respond to pitch in any of these cases, because pitching motions are in the same plane of rotation as the gyro wheel's rotation about its axle.)
Seen in this light, we can boil your question down to the following: "If the gimbal axis were purely "up-and-down" in the aircraft's reference frame, why would the instrument respond to rolling motions of the aircraft, but not to yawing motions of the aircraft?"
When the aircraft rolls to the right, this exerts a downward force on the right-hand side of the gyro wheel and an upward force on the left-hand side of the gyro wheel. Rotate these forces 90 degrees in the direction of the gyro's spin, as we always must when considering how a gyro wheel reacts to torque1, and we have an apparent forward force on one side of the gyro wheel and an apparent aftwards force on the other side of the gyro wheel.2 In other words the gyro wheel will experience an apparent yaw torque. This will make the gyro wheel pivot around the "up-and-down" gimbal axis. Yawing the aircraft, on the other hand, will create a forward force on one side of the gyro wheel and an aftwards force on the other side of the gyro wheel, so the gyro will experience an apparent upward force on one side of the gyro wheel and an apparent downward force on the other side of the gyro wheel.3 In other words the gyro wheel will experience an apparent roll torque. In this hypothetical case where the gimbal axis is purely vertical, this apparent yaw torque experienced by the gyro wheel cannot result in any rotation around the gimbal axis, so the instrument does not respond to yaw rotation of the aircraft.
Regarding the diagrams included in the question--
The diagram appears to be the question author's own work, and makes use of the right-hand rule convention for illustrating a torque or an angular momentum as a single vector. We'll assume that the gyro's gimbal axis is parallel to the red bar in each diagram, which indicates that the aircraft's nose is toward the right side of each diagram, and we're viewing each diagram from somewhere out beyond the aircraft's right wingtip. The blue dot labelled "angular momentum" represents a vector coming straight out of the page or screen toward the viewer, which illustrates that the gyro wheel is spinning counterclockwise as seen from our vantage point, i.e. with the top side moving toward the pilot and the bottom side moving away from the pilot. By the right-hand rule convention, the direction of the roll torque illustrated by the single vector pointing toward the right side of the diagram, is toward the right (i.e. clockwise) in the aircraft's reference frame. If this is meant to illustrate the apparent roll torque experienced by the gyro wheel after taking into account the gyroscopic precession effect, then it is clear that tilting the gimbal axis (parallel to the red bar) toward the vertical does not make the gyro sensitive toward this apparent roll torque-- rather, it makes the gyro less sensitive to this apparent roll torque. But remember that, as explained above, an apparent roll torque on the gyro wheel is actually the result of a yawing motion of the aircraft. If we want to illustrate the result of a rolling motion of the aircraft, we need to remember that this will make the gyro wheel experience an apparent yaw torque. By the right-hand rule convention, we should illustrated this apparent yaw torque as an arrow pointing straight up or down, not forward or aft. The diagram as drawn illustrates the apparent right roll torque exerted on the gyro wheel by a right yawing motion of the aircraft. That doesn't really help us understand how tilting the gimbal axis toward the vertical makes the instrument become sensitive to rolling motions of the aircraft.
Footnotes:
Due to the principal of torque-induced precession, also known as gyroscopic precession.
The direction of the apparent roll torque experienced by the gyro wheel when the aircraft yaws depends on the direction of spin of the gyro wheel. In principle the instrument could be built with the gyro wheel spinning in either direction. In practice there are pros and cons to each direction. For reasons beyond the scope of the answer, the gyro wheels of turn rate indicators (where the gimbal axis is purely "fore-and-aft" in the aircraft's reference frame) and turn "coordinators" (where the gimbal axis is canted 30 to 40 degrees towards the "vertical" in the aircraft's reference frame) usually spin in opposite directions. Read more about that in this answer to a related question on ASE.
The direction of the apparent yaw torque experienced by the gyro wheel when the aircraft rolls depends on the direction of spin of the gyro wheel. See footnote 2 for more.
