Motions {translation, motion}| can be in straight lines.
Motions {uniform motion}| can cover equal distances in equal times or over equal values.
Motions {rectilinear motion}| can be in straight lines.
Two motion components, in different directions, can change motion direction {curved motion}|. Typically, one component is tangential to curve, and one component is normal to curve. Distance {arc length, motion} traveled along curve depends on curvature, which depends on curvature radius r and angle A subtended by arc: arc length = r * A.
Motion can involve speed and/or direction change {acceleration}|. Acceleration has amount and direction, so acceleration is vector. Acceleration a is velocity change dv divided by time change dt: a = dv / dt. Gravity acceleration g is 9.8 meters per second per second: g = 9.8 m/s^2.
Acceleration can change over time {jerk}|: j = da/dt.
Jerk can change over time {jounce} {snap}: dj/dt.
Motion can be through angle around point or axis {rotation, motion}.
comparison
All linear distance, velocity, acceleration, and time relations are true for angular counterparts.
vectors
Angular quantities are vectors, perpendicular to curve plane. If right-hand fingers point in motion direction, vector points in thumb direction.
examples
Top, gyroscope, wheel, gears, banked track, and airplane dive and turn illustrate angular motion.
universe
Rotation is not relative but is absolute against distant-galaxy and universe reference frame.
Motions {spin, object}| can be around object axis or point.
Objects can move around points or axes {revolution, physics}| {orbit, revolution}. Object comes back to starting point after angle 360 degrees (2*pi radians), after traveling circumference distance.
Speed can be constant around circumference {uniform circular motion}|.
Angular speed w and/or direction can change over time t {angular acceleration}|: a = dw / dt. Angular acceleration a depends on angle A passed per second per second: a = (d^2)A / dt^2, where (d^2) is second derivative, and d is derivative.
Angle distance {angular distance}| {total angle} A equals current angular distance A0 plus current angular velocity w times time t plus one-half times angular acceleration aa times time t squared: A = A0 + w * t + 0.5 * aa * t^2, whish is analogous to linear distance equation.
Rotation velocity {angular velocity}| w, in radians per second, is angle change A per time unit t: w = dA / dt. Average angular velocity w equals 360 degrees (2*pi radians) divided by period T: w = 2 * pi / T. Average angular velocity w equals 360 degrees (2*pi radians) times frequency f: w = 2 * pi * f.
A number of orbits or revolutions happens over time {frequency, physics}|. Frequency f is period-T reciprocal: f = 1/T. For example, electric current alternates at 60 cycles per second in USA.
Acceleration {radial acceleration}| can be along perpendicular to curve. For circular motion, object pulls back toward center to make circle, and radial acceleration ar equals tangential velocity vt squared divided by radius r: ar = vt^2 / r. If radial acceleration is more, orbit is ellipse. If radial acceleration is less, orbit is spiral.
Motions {radial velocity}| can be along perpendiculars to curves. Radial velocity equals zero for circular motion, because distance from circle center is constant.
If object rotates around point or axis, object makes one complete revolution during time {period, rotation} {rotation period}|. Complete revolution sweeps through angle of 360 degrees (2*pi radians) and travels circumference distance. Period T is frequency f reciprocal: T = 1/f.
Acceleration {tangential acceleration}| can be along tangent to curve. Tangential acceleration at equals angular acceleration a times curvature radius r: at = a * r.
Motions {tangential velocity}| can be along tangents to curves. Tangential velocity vt equals angular velocity w times curvature radius r: vt = w * r.
Objects can rotate around horizontal axis perpendicular to motion axis {pitch, motion}|. Airplanes can pitch around wings, horizontal to body.
Objects can rotate around motion axis {roll, rotation}|. Airplanes can roll around airplane body.
Objects can rotate around vertical axis perpendicular to motion axis {yaw}|. Airplanes can yaw around tail, vertical to body.
Motion can be back and forth {vibration, motion}| {oscillation}.
period
Vibrations take time to complete one vibration.
frequency
Vibrations have number of vibrations per time unit. Period T relates to frequency f: f = 1/T.
wavelength
Moving vibration travels distance during one period.
velocity
Movement velocity v equals wavelength l times frequency f: v = l*f. Vibration velocity maximizes at center. Vibration velocity is zero at maximum displacement.
acceleration
Acceleration is zero at center. Acceleration maximizes at maximum displacement.
displacement
During vibration, object is at distance from equilibrium or center point. Amplitude is maximum displacement. Period does not depend on amplitude. Large amplitudes have large acceleration, and small amplitudes have small acceleration, so period stays the same.
phase
Two vibrations can have same angle for same displacement {in phase} or not {out of phase}.
rotations
Vibrations are similar to rotations but are back and forth, instead of around axis. Rotation looks like vibration if viewed from orbital plane.
trigonometric function
Sine or cosine functions can model vibration. Sines and cosines have varying displacement, which has maximum amplitude. Sines and cosines have varying phase angle. Angle A equals frequency f times time t times 360 degrees expressed in radians 2*pi: A = 2 * pi * f * t. If time is zero, angle is zero. If time is period, angle is zero. If time is half period, angle is 180 degrees, and sine is zero. If time is one-quarter period, angle is 90 degrees, and sine is one. Sine equals zero if angle is zero. Sine maximizes if angle is 90 degrees.
Angle A equals displacement x divided by wavelength l times 360 degrees expressed in radians 2*pi: A = 2 * pi * x / l. If displacement is zero, angle is zero. If displacement is wavelength, angle is zero. If displacement is half wavelength, angle is 180 degrees, and sine is zero. If displacement is one-quarter wavelength, angle is 90 degrees, and sine is one.
Displacement x equals amplitude A times sine: x = A * sin(2 * pi * f * t) or A * sin(2 * pi * x / l). Vibrations can shift angle: x = A * sin(2 * pi * f * t + Ao), where Ao is starting angle.
string vibration
Vibrating strings are stationary waves and have partial differential equations. Second partial derivative of function y with respect to time t equals constant (a^2) times second partial derivative of function y with respect to distance x. (D^2)y / Dt = (a^2) * (D^2)y / Dx, where (D^2) is second partial derivative, D is partial derivative, a is constant, t is time, x is distance, and y is function of time and distance. To be dimensionless, constant a equals period T in seconds divided by seconds: a = T / second. Because endpoints are stationary, function y at x = zero equals zero: y(t,0) = 0. Function y at x = one wavelength equals zero: y(t,1) = 0. For stationary waves, partial derivative of y with respect to time t, at t equals zero, equals zero: Dy(0,x) / Dt = 0. Function y at t = zero equals function of x: y(0,x) = f(x), which is odd and periodic.
Sine or cosine function has varying magnitude, which can have maximum {amplitude, vibration}|.
Like rotation, vibration has time {period, vibration}| to complete one vibration.
Sine or cosine function has varying angle {phase, vibration}|.
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Date Modified: 2022.0225