The essential structure of a certain type of aircraft turn...

The essential structure of a certain type of aircraft turn indicator is shown. Springs $A C$ and $B D$ are initially stretched and exert equal vertical forces at $A$ and $B$ when the airplane is traveling in a straight path. Each spring has a constant of $600 \mathrm{~N} / \mathrm{m}$ and the uniform disk has a mass of $250 \mathrm{~g}$ and spins at the rate of $12000 \mathrm{rpm} .$ Determine the angle through which the yoke will rotate when the pilot executes a horizontal turn of 800 -m radius to the right at a speed of $720 \mathrm{~km} / \mathrm{h}$. Indicate whether point $A$ will move up or down.

Key Concepts

Centripetal Force and Acceleration

In circular motion, any object moving in a curved path experiences an inward-directed acceleration known as centripetal acceleration. This acceleration results from the centripetal force, which is necessary to keep an object following a curved trajectory. In contexts such as aircraft turning, the magnitude of this force is determined by the speed and the radius of the turn, influencing various components of the system that are sensitive to inertial forces.

Hooke’s Law and Spring Mechanics

Hooke’s Law describes the behavior of springs, stating that the force exerted by a spring is directly proportional to its displacement from its equilibrium position. This is essential when analyzing systems where springs are used to measure or counteract forces. Understanding the spring constant allows for calculating the force (or stretch) resulting from given loads, which is critical in designing and interpreting the response of instruments like turn indicators.

Gyroscopic Effects

Gyroscopic behavior comes into play with spinning disks or rotors in instruments, where the conservation of angular momentum gives rise to stability and precession phenomena. A spinning disk tends to maintain its orientation, but when external torques are applied, it exhibits precession — a tendency for its axis to rotate in a direction perpendicular to the applied torque. This property is important in understanding how such devices respond to undesired forces or twists, as seen in aircraft turn indicators.

Moment and Torque Balance in Rotational Systems

Analyzing the equilibrium of forces in rotating systems requires careful consideration of moments and torques. When forces act at various points on a disk or lever (such as the application of spring forces at distinct points), their contributions to rotational motion or deflections must be balanced. This concept is crucial in engineering applications where precise angular deflections are needed, such as in the calibration and functioning of the turn indicator mechanism.

Transcript

00:01 Hello everyone here it is given the essential structure of a certain type of aircraft turn indicator as shown in the figure having the spring ac and bd are initially stressed and exert equal vertical force at a and b when the airplane is traveling in a straight path each spring having the spring constant 600 newton per meter and a uniform disk having the mass 250 gram that is 0 .25 kg and spin at constant rate of 12 ,000 rpm it can be converted into radian and its value will be 1256 .6 radian per second and it will be along x -axis.

01:30 We have determined the angle through which the yop a b will rotate when the pilot execute so we have to calculate theta of a v when pilot execute when pilot execute horizontal turn of radius 800 meter to the right with 720 km per hour let us start solving we also indicate whether point a will move up or down let us start solving it.

02:58 We will draw the geometrical figure of the disk.

03:09 This is the centroid.

03:12 This is y -axis.

03:14 This is x -axis.

03:17 This is z -axis.

03:19 Here it is omega -x.

03:30 This is omega -y.

03:34 I -cap, j -cap.

03:37 Aircraft speed is given.

03:39 720 km per hour.

03:42 That is 200 meter per second.

03:45 And radius of the turn is given 800 meters.