# Tide-O-Matic

In most places on the earth, where the ocean meets the land, there are two high tides and two low tides each day. This Snack models the equilibrium theory of tides, showing why there are two tide cycles per day, why the heights of the tides change over the course of a month, and why the tides occur about an hour later each day.

- Earth–sun system model template, printed (in color, if possible) on white card stock or other stiff paper
- Earth–moon system model template, printed or copied onto a transparency
- Earth model template, printed on white card stock
- Scissors
- Two pushpins
- Two small, thin pieces of cork (a sliced-up wine cork works well)
- Optional: cardboard or card stock, at least 12 x 12 inches (30 x 30 centimeters)

- Cut out one of the rectangles containing the earth and sun from the earth-sun system model template that you have printed onto card stock.
- Cut out one of the earth models from the earth model template that you have printed onto card stock. Note that the perspective is centered over the North Pole.
- Insert a pushpin through the center of the earth model (where the + sign is) and attach it to the white circle with a + sign (which represents the earth) on the earth–sun system model.
- Place a piece of cork over the sharp end of the pin on the underside of the card stock.
- If you have cardboard, insert the second pushpin through the center of the sun and attach it to the center of the cardboard. Place another piece of cork over the sharp end of the pin on the underside of the cardboard.

Notice the tidal bulges on two sides of the earth, indicated by the yellow ellipse around the circle of the earth. Where are the tidal bulges located? When you revolve the earth around the sun, do the tidal bulges change?

Rotate the earth around the pushpin through one day. Through how many tidal bulges does any one place on the earth pass? (You can draw an arrow on the earth that approximates your meridian to help you keep track.)

- Cut out one of the rectangles that includes both the earth and the moon from the earth-moon system model template that you have printed onto a transparency.
- Remove the pushpin and cork from the earth–sun system model. Insert the transparency of the earth–moon system between the earth model and the earth-sun system model. Make sure all the + signs line up, then re-insert the pushpin so it passes through all three models.
- Replace the cork on the underside of the card stock to cover the sharp end of the pin.

Examine the graphics on the transparency. What do you notice? Where are the lunar tidal bulges located? When you revolve the moon around the earth, do these tidal bulges change their positions with respect to the moon?

Rotate the transparency so that the moon is in between the earth and the sun, modeling a new moon. Do the solar tides and lunar tides “line up"? Keeping the transparency still, rotate the earth through one day. Through how many high and low tides does any one location on the earth pass?

Are there other moon phases during which the solar and lunar tides align? Let’s find out. Starting at new moon, rotate the moon a quarter turn counterclockwise around the earth, modeling the first quarter. Do the solar and lunar tides still line up with each other?

Move the moon another quarter turn counterclockwise, modeling a full moon (when the earth is between the moon and sun). What do you notice?

Rotate the moon a final quarter turn counterclockwise, modeling the third quarter. What do you notice?

Now, pick a location on the earth (such as where you live), and orient it towards the sun. This position is called solar noon. Next, move the moon to the new moon position (between the earth and the sun).

Rotate the earth through one full solar day, so that your selected location goes around in a full circle, from solar noon to solar noon. However, as the earth rotates, the moon also revolves around the earth. So, is the moon still in the new moon position after 24 hours? Try to imagine how far the moon revolves in one 24-hour day. (Hint: It takes about 28 days for the moon to complete a full cycle from new moon to new moon.)

If the moon didn’t revolve around the earth and were fixed in a new moon phase, at what times would we experience our high and low tides?

Tides are caused by gravitational forces, which are based on the masses of and distances between objects. In Part 1 of this Snack, we explored solar tides. The force of gravity exerted by the sun pulls our oceans towards the sun. It’s easy to understand how this influence results in a tidal bulge, or high tide, on the sun side of the earth, but why is there also a high tide on the far side of the earth?

The force of gravity varies enough over the distance of the earth’s diameter that masses on either side of the earth experience different amounts of pull. While the gravitational force is proportional to the inverse of the distance squared between two objects, the tide-raising force is proportional to the inverse of the distance cubed. Therefore, the ocean on the side nearest the sun experiences the largest force, the ocean on the side farthest from the sun experiences the least force, and of course, the earth itself experiences a force somewhere in between, resulting in a “stretching out” of these three masses and tidal bulges on the near and far side of the planet.

As the earth rotates on its axis, it passes through two tidal bulges in one rotation (one day). As you revolve the earth around the sun (one year), the tidal bulges stay in line with the sun.

Just like the sun, the moon creates two tidal bulges on the earth—one on the side closest to the moon and one on the opposite side. These moon tides are created by the same gravity-based tide-raising forces that produce sun tides. Although the moon is much smaller than the sun, it is also much closer, so the moon’s tidal influence is twice that of the sun. As you rotate the paper earth model through a full day, each part of the earth rotates under the two tidal bulges, and therefore there are two high tides and two low tides per day on most parts of the earth. Watch our Dance of the Tides video for an illustration of this.

In Part 2 of this Snack, we looked at the influence of the sun and moon together.

During a new moon, the sun and moon are in line with the earth and their tidal influences add together, creating higher high tides and lower low tides. These additive tides are referred to as spring tides. Spring tides also occur during the full moon. Since the moon’s cycle takes about 28 days, spring tides occur every two weeks.

When the moon and the sun are at 90° to each other, during the first quarter or third quarter moon phases, their influences don’t add together, so the high tides and low tides are both less extreme. These tides are referred to as neap tides. Neap tides also occur every two weeks.

The sun and moon are the two main influences on the earth’s pattern of tides. Since the moon exerts the strongest influence because of its proximity, the main tidal pattern follows the phases of the moon.

As the earth rotates each day, the moon also moves in its orbit. It takes about a month (“moonth”) or 27.3 days for the moon to revolve around the earth. In one day, the moon moves 13°, viewed from the North Pole (13° per day = 360°/27.3 days) counterclockwise. Therefore, 24 hours after a new moon, the moon is 1/7 of the way to the first quarter. The larger tidal bulge follows the moon, so the earth has to rotate an additional 13° to reach high tide again. This takes about 54 minutes (approximately 24 hours x 13°/360°).

Thus, the high tide and all tides arrive about 54 minutes later each day.

The Tide-O-Matic provides a model of the *equilibrium theory* of tides. The equilibrium theory of tides makes the following assumptions: tides can be explained by celestial mechanics, ocean depth is uniform, there are no continents, the earth is not rotating, and friction is ignored.

The Tide-O-Matic is a good introductory model, but real tides in the earth's oceans are much more complicated. For example, the high tide arrives in Santa Cruz about one hour earlier than at San Francisco’s Golden Gate Bridge, 75 miles to the north, even though the moon is over both places at the same time. This delay is caused by friction as the ocean is dragged against the continental shelf and by the push of Coriolis forces from the spinning earth. Tidal bulges in the ocean do not always line up with the moon.

The *dynamic theory* of tides provides a more realistic model, but it also does not predict the tides perfectly. The dynamic theory of tides takes more factors into account; for example, the earth’s rotation, the shapes and locations of the continents, the ocean’s irregular bathymetry, and friction.

Our model also ignores the fact that the moon and the earth form a two-body system. The moon doesn’t simply orbit the earth around its center of mass; instead, the moon and earth orbit each other around the center of mass of this two-body system, called the *barycenter*. While the barycenter is located within the earth, it is not at the earth’s center of mass. The earth’s center of mass orbits the barycenter of the earth-moon system and is held in its orbit by the gravitational pull of the moon.

The gravitational force of the sun on a mass at the surface of the earth is 180 times greater than that of the moon. But tides are created by differences in gravitational pull. Because the sun is much farther away, the change in its gravitational force across the earth’s diameter is smaller (a mere 0.017%) compared to the change over the same distance created by the lunar force, which is 6.7%. As a result, the solar tide is smaller than the lunar tide. (In fact, 0.017% * 180 = 3.06%, which shows that the effect of the sun is about half that of the moon.)

Graph tidal data to generate questions that you can answer with the Tide-O-Matic:

- Pick a location and gather tide prediction data to graph (the Saltwater Tides website includes tide data for many areas).
- Print out blank sheets of our tide graph paper.
- Plot at least two weeks of high and low tide data, then connect graphed points with a smooth curve. If you have a large group, pair students off to plot a few days’ worth of data.
- Tape all the graphs together in sequential order, creating a continuous graph of several weeks of tide predictions.
- Observe this long graph and notice the following: there are two high tides in 24 hours, tides have different heights over the course of a month, and the tides arrive at a later hour each day.

All of these observations can be explained using the Tide-O-Matic. To make this Snack more inquiry-based, start with the graph and have students generate questions from what they notice in the data. If your students are struggling with understanding the high tide opposite the sun and moon, enact a live demonstration of Dance of the Tides to help them visualize it.

"NOAA Tides and Currents," revised September 15, 2013, https://tidesandcurrents.noaa.gov/.

Brent A. Ford and P. Sean Smith, “The Bulge on the Other Side of the Earth,” in *Physical Oceanography: Project Earth Science*, (Arlington, VA: National Science Teachers Association, 1995), 141-145.