9.1 What causes ocean tides?

• gravitational attraction of the sun and moon on earth —> ocean tides

• tides are generated by forces imposed on earth that are caused by a combination of gravity and motion among earth, moon, and sun

Tide-Generating Forces

• newton’s work on quantifying forces involved in earth-moon-sun system —> first understanding of underlying forces that keep bodies in orbit around each other

• the moon does NOT orbit earth

  • the two bodies rotate around a common center of mass: barycenter

• barycenter = balance pint of the system, located about 1700km beneath earth’s surface (NOT AT THE CENTER OF THE EARTH)

  • barycenter isn’t halfway between the two bodies bc earth’s mass >> moon mass

• if moon and earth are attracted to each other, why don’t they collide?

  • mutually stable orbit held together by a balance between centripetal (gravitational) and inertial (motion) forces which prevent the moon and earth from colliding or flying apart

Gravitational and Centripetal Forces in the Earth-Moon System

Gravitational Force

• newton’s law of universal gravitation: every object that has mass in the universe is attracted to every other object

  • F = G(m1)(m2) / r²

  • gravitational force is directly proportional to the product of the masses of the two bodies and is inversely proportional to the square of the distance between the two masses

    • G = gravitational constant, doesn’t change

• if mass increases, gravitational force increases

• if distance increases, gravitational force decreases

• closer and larger = greater gravitational attraction

• greatest gravitational attraction on Earth to the moon is at Z, the zenith

  • zenith = point closest to the moon

• weakest gravitational attraction on earth at N, the nadir

  • nadir = opposite the zenith, furthest point from the moon

• direction of gravitational attraction between most particles and the center of the moon is at an angle relative to a line connecting the center of earth and the moon

  • this angle causes the force of gravitational attraction between each particle and the moon to be slightly different

Centripetal Force

• centripetal force required to keep planets in their orbits is provided by the gravitational attraction between the center-of-mass of each of the planets and the sun

• centripetal force connects the center-of-mass of an orbiting body to its parent, pulling the object inward toward the parent, “seeking the center” of its orbit

  • i.e. you put a ball on a string, spin the ball around with the string in a fixed position, the ball is drawn toward the center of the string —> string exerts centripetal force on the ball, forcing the ball to seek the center of its orbit. if the string breaks, the ball is released and this doesn’t work anymore and the ball will fly off in a straight line tangent to the circle it was on

• earth and moon connected too, but by gravity, not a string

• gravity provides centripetal force that holds the moon in its orbit around earth

• iv all gravity in solar system could be shut off, centripetal force would vanish and momentum of celestial bodies would send them flying off into space along straight-line paths, tangent to their orbits

Resultant Forces

• particles of identical mass rotate in identical-sized paths due to the earth-moon rotattion system

• each particle requires an identical centripetal force to maintain it in its circular path

• gravitational attraction between the particle and the moon supplies the centripetal force but the supplied force is different than the required force (gravitational attraction varies w/ distance from the moon) except at the center of the earth

  • this difference —> tiny resultant forces = mathematical difference between two sets of arrows

• centripetal force is a force that would be required to keep the particles in a perfectly circular path, while gravitational force is the force actually provided for this purpose by the gravitational attraction between the particles and the moon

• resultant forces are established by constructing an arrow from the tip of the centripetal arrow to the tip of the gravity arrow, from where both begin

Tide-Generating Forces

• resultant forces are small, about 1 millionth the magnitude of earth’s gravity

• no tide-generating force where the resultant forces either point straight upward toward the sky or straight downward into the earth

  • in these cases, resultant force is oriented vertically relative to earth’s surface —> no tides occur here

    • this happens in 3 places:

      • zenith

      • nadir

      • along an “equator” connecting all points halfway between the zenith and nadir

• if resultant force has a significant horizontal component, it produces tidal bulges —> tide-generating forces

  • TGF are small but reach their max value at points on earth’s surface at a “latitude” of 45 deg relative to the “equator” between the zenith and the nadir

• TGF is inversely proportional to the cube of distance between each point on earth and the center of the tide-generating body (moon or sun)

  • although tide-generating force is derived from gravitational force, it’s not linearly proportional to it

  • ***distance has a larger effect on TGF than gravitational force!!

• TG create two simultaneous bulges:

  • one on the side of earth directed toward the moon (zenith)

    • bulge created bc the provided gravitational force is greater than the required centripetal force.

  • one on the side directed away from the moon (nadir)

    • bulge created bc the required centripetal force is greater than the provided gravitational force

  • **resultant forces are equal in magnitude —> bulges equal too

Recap: tides are caused by an imbalance between the required centripetal and provided gravitational forces acting on earth. this difference provides residual forces, the horizontal component of which creates two equal tidal bulges on opposite sides of earth

Tidal Bulges: The Moon’s

consider an ideal earth and ideal ocean:

• ideal earth has two tidal bulges, one toward moon and one away from moon (lunar bulges)

• ideal ocean has a uniform depth, with no friction between seawater and sea floor

• if the moon is stationary and aligned with the ideal earth’s equator, the maximum bugle will occur on the equator on opposite sides of earth.

• if you were standing on the equator, you would experience two high tides each day

• tidal period = time between high tides (12 hours on equator)

• if you move any latitude north or south of the equator, you’d experience the same tidal period but the high tides would be less high bc you’d be at a lower point on the bulge

actual earth

• most places on earth, high tides occur every 12hr 25 min bc tides depend on the lunar day, not the solar day

• lunar day = measured form the time the moon is on the meridian of an observer (directly overhead) to the next time the moon is on that meridian = 24 hr, 50 min 28 sec

• solar day = measured form time sun is on meridican of an observer to the next time the sun is on that meridian = 24 hrs long

• why is the lunar day 50 min longer than the solar day??

  • during 24 hr it takes earth to make a full rotation, moon has continued moving another 12.2 deg to the east in its orbit around earth, thus earth must rotate an additional 50 min to “catch up” to the moon

• alternating high tides are normally 50 min later each successive day, and the moon rises 50 min later each successive night

Recap: solar day (24 hrs) is shorter than a lunar day (24 hrs, 50 min, 28 sec). extra 50 min is result of moon’s movement in its orbit around earth

Tidal Bugles: The Sun’s Effect

• sun affects the tides too

• sun produces tidal bulges on opposite sides of earth:

  • one toward sun

  • one away from sun

• solar bulges = ½ size of lunar bulges

• even though sun is 27M times larger than the moon, it’s 390 times farther from earth than the moon

• tide-generating forces vary inversely as the cube of the distance between objects —> distance has a much larger effect

• sun’s TGF = 27/59 that of the moon (46%), and solar bugles are 46% the size of lunar bulges

• **simplicity: moon exerts over 2x as much gravitational pull of the sun on the tides

• even though the moon exerts >2x gravitatoinal pull of the sun on earth’s tides, the sun doesn’t exert a smaller gravitational force on earth as compared to the moon

  • sun’s total “pull” on all points on earth >> than that of the moon but the DIFFERENCE across earth is small bc diameter of earth is very small in relation to the distance from the sun

  • diameter of earth is quite large in relation to the distance to the center of the moon

**moon controls tides far more than the sun bc moon is much closer to earth, even though it’s smaller in size and mass compared to the sun. as a result, moon creates lunar bulges that are twice the size of the solar tidal bulges.

Earth’s Rotation and the Tides

• flood tide: water moves toward shore

• ebb tide: water moves away from shore

• earth’s rotation carries various locations into and out of the tidal bulges, which are in fixed positions relative to the moon and the sun

Recap: alternating high and low tides are created as earth constantly rotates inside fluid bulges that are supported by the moon and the sun.

CONCETP CHECK 9.1 - demonstrate an understanding of the forces that cause ocean tides.

(1) why are there tidal bulges on both sides of earth? why not on just the side of earth that faces the moon or the sun?

• side facing moon/sun has a stronger gravitational pull —> resultant force —> tidal bulge

• side facing away from moon/sun has a higher centripetal force —> resultant force —> tidal bulge

**resultant forces = in size, so bulges = in size too

(2) explain why the sun’s influence on earth’s tides is only 46% that of the moon, even though the sun is so much more massive than the moon.

• the sun is much farther away, 390M times farther, and TGF is inversely proportional to the cube of distance, so distance has a huge impact, even though the sun exerts a larger gravitational force on the earth

(3) why is a lunar day 24 hrs, 50 min, 28 sec long, while a solar day is 24 hrs long?

• the moon moves 12.2 deg while the earth is rotating, so the earth has to “catch up” to the moon to complete the lunar day.

(4) if earth did not have the moon orbiting it, would there still be tides? Why or why not?

• Yes, bc sun creates tides as well due to the resultant forces.