Objects can spin or orbit or rotate around balance point {rotation, force}. Rotations rotate around point {center of rotation} {rotation center} {balance point}. Forces or weights are at distances {radius, rotation} from balance point or rotation center. On levers, forces can act perpendicular, or at an angle, to radius. Weighing balances and seesaws are levers. Weights on strings can orbit, so string provides centripetal force, and spin provides centrifugal force.
Spinning objects, including Earth, bulge at equator {equatorial bulge} and flatten at poles, by centrifugal force.
Just as mass traveling in straight line has inertia that tends to keep velocity constant, mass rotating around axis has inertia {moment of inertia}| {inertia moment} that tends to keep angular velocity constant.
mass
Inertia depends directly on mass. Moment of inertia substitutes for mass when quantities use angular velocity instead of velocity. When mass rotates around rotation center at radius r, tangential momentum pt is mass m times tangential velocity vt: pt = m * vt. Angular momentum L is moment of inertia I times angular velocity w: L = I * w. Tangential velocity vt varies directly with angular velocity w: vt = w * r. Tangential momentum pt varies directly with angular momentum L: L = pt * r. Moment of inertia I depends on mass m and radius r: L = I * w = pt * r = m * w * r * r = (m * r^2) * w. Moment of inertia I is mass m times square of distance r from axis or point: I = m * r^2.
summation
Masses have volume, so object points have different radii from rotation center. Total moment of inertia is sum of moments of inertia at each radius.
Thin-ring moment of inertia equals total mass m times square of distance r from ring center to ring middle: m * r^2.
Disk or cylinder moment of inertia equals half total mass m times square of distance r from disk or cylinder center to outer edge: 0.5 * m * r^2.
Pipe or doughnut moment of inertia equals half total mass m times sum of squares of distances from pipe or doughnut center to inner edge ri and outer edge ro: 0.5 * m * (ri^2 + ro^2).
radius
Objects with moment of inertia around rotation center have moment of inertia around any axis parallel to rotation axis. New moment of inertia Inew is old moment of inertia Iold plus total object mass m times square of distance d between axes: Mnew = Mold + m * d^2.
Forces {torque}| can tend to cause motions around rotation centers.
acceleration
Torque causes angular acceleration. For example, force can act on a rigid rod that can turn around a balance point. Force can act perpendicular to rod or at another angle. Torque T is force F times radius r from balance point times sine of force-to-radius angle A: T = F * r * sin(A), which is cross product of force and radius vectors: T = F X r. Torque-vector direction is perpendicular to both force vector and radius vector and parallel to axis.
moment of inertia
Tangential force Ft equals mass m times tangential acceleration at, which equals angular acceleration aa times radius r: Ft = m * at = aa * r. If torque acts perpendicular to radius, torque T equals moment of inertia (I = m*r^2) times angular acceleration aa: T = Ft * r = m * at * r = m * aa * r * r = m * (r^2) * aa = I * aa.
examples
Frisbees and yo-yos have torques. Torque causes car front to fall when car stops. Torque causes car front to rise when car accelerates. To open door, push farthest from hinge to apply least force, because radius is greatest. Spins in ice-skating begin with torque. Gymnasts and divers apply torque. Torque causes spin on footballs, bullets, bicycle wheels, helicopter blades, and propellers. Scales use opposing torques to weigh objects.
equilibrium
When lever is not moving around balance point {equilibrium}, right Tr and left Tl side torques, F * r * sin(A), are equal: Tl = F1l * r1l * sin(A1l) + F2l * r2l * sin(A2l) + ... = F1r * r1r * sin(A1r) + F2r * r2r * sin(A2r) + ... = Tr.
In curved motion, force can go away from curvature center {centrifugal force}|, along radius direction. For example, Moon is in orbit around Earth.
In curved motion, force can go toward curvature center {centripetal force}|, along radius direction. For example, Moon is in orbit around Earth.
Moon falls toward Earth center by Earth gravity {free fall}|. Gravity is centripetal force. Orbital speed moves Moon tangentially in orbit. Tangential movement accelerates Moon away from Earth along radius. This acceleration is centrifugal force. Centrifugal force and centripetal force are equal, and motion rates away from and toward center are equal, so Moon maintains approximately same distance from Earth.
weightless
Moon in orbit has no weight, because centrifugal force equals centripetal force, just as astronauts in orbit are weightless. If jumping from height, in free fall, one feels weightless, because no force is opposing fall.
Kepler formulated three planetary-motion laws {Kepler's laws} {Kepler laws}.
first law
Radius from orbiting body to Sun sweeps out equal areas in equal times, because velocity is slow at large radius and fast at small radius.
second law
If object in orbit moves closer, speed increases as potential energy changes to kinetic energy and moves object back outward. If object in orbit moves farther away, speed decreases as kinetic energy changes to potential energy and moves object inward.
If spinning object becomes more compact, radius decreases and speed increases as potential energy changes to kinetic energy. If spinning object expands, radius increases and speed decreases as kinetic energy changes to potential energy.
third law
Acceleration cubed is directly proportional to time squared, because acceleration is highest at greatest curvature point, where velocity is highest.
Two objects in different orbits interact by gravity or electromagnetism to make torque that changes object spin axis {nutation}|.
Earth spins on an axis that is at an angle to axis of Earth orbit around Sun. Sun gravity causes torque on Earth axis and causes it to rotate {precession}| {precession of the equinoxes, Earth}, as angular velocity around axis interacts with angular velocity around orbit. Object spin and orbital motion interact to cause spin-axis precession.
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Date Modified: 2022.0225