Choosing a Weir Notch Shape for Wilmot Brook

• Teacher preparation. Cut the plates and notches before class using the cutting guide below, and run Plates B and C once at the fast pour. Standard formulas for sharp-edged notches predict that Plate B will settle near 15 mm and that Plate C will overflow. Small notches often read 1 to 2 mm higher than the formulas predict, because surface tension holds water against the plate. If Plate B spills in your trial, the pour is faster than 400 mL every 10 seconds, so time the cup and slow the pace. Plan the setup and the six runs for the first session and the analysis items for the second.
• Do not drink any water used in this lab. Have an adult cut the plates and the notches, since the plastic edges can be sharp. Wipe spills right away.
• Each plate fits snugly across the tub. Each notch is cut down from the top edge of its plate and is 20 mm deep. The bottom of each notch should sit at least 5 cm above the tub floor, and the top edge of the plate should sit at least 2 cm below the tub rim.

• Have an adult cut the far end wall of the tub down below the bottom of the notches, or cut a wide spout there, so water that passes the plate spills straight into the dish tub.
• Stand Plate A upright across the middle of the tub. Press modeling clay along both sides and the bottom so water cannot sneak past the plate.
• Tape the ruler upright on the upstream side, about 8 cm back from the plate, with its zero mark level with the lowest point of the notch. Water right beside the plate dips as it speeds up, so a reading taken there would be too low.
• Place the sponge in the upstream end of the tub, where it may float once water is added. Always pour onto the sponge, or onto the water just above it, so the stream does not make waves near the ruler.
• Set the tub in the dish tub so water leaving the far end is caught.
• Before starting the stopwatch, pour water into the upstream side until it just reaches the lowest point of the notch.
• Run the slow flow first. Pour 100 mL onto the sponge in a thin, steady stream that takes the full 10 seconds, while a partner calls out each 10-second mark and fills the second cup. Switch cups at each call so the stream never stops.
• Read the water height above the lowest point of the notch at 70 seconds and again at 90 seconds, to the nearest millimeter. If the two readings differ by more than 1 mm, keep pouring and read every 10 seconds until two readings in a row agree. Record the last reading.
• Run the fast flow next with the same plate. Pour 400 mL every 10 seconds in the same way and read the height by the same method. Also record whether any water ran over the top edge of the plate beside the notch.
• Between runs, pour the water caught in the dish tub back into the bucket so the supply lasts.
• Empty the tub, swap in Plate B, and repeat both flows. Then do the same with Plate C.
• Pour the water down the sink, peel off the clay, dry the tub, and return the materials.

Record your own readings in the table below.

The Wilmot Brook gauge in this lab is fictional, and so are its three rules. Rule 1: the height must change by at least 4 mm between the slow flow and the fast flow, so a reader can see the change. Rule 2: at the fast flow the height must stay below 20 mm, the top of the notch, and no water may run over the top edge of the plate, because water spilling across the whole plate no longer gives a true reading. Rule 3: at the slow flow the height must be at least 5 mm, because a thinner sheet clings to the plate and a ruler reading of it is mostly error.

Sample data. These values were worked out from the standard formulas engineers use for sharp-edged notches, for these exact plates and pours, and rounded to the nearest millimeter. Your own readings may run a millimeter or two higher. Plate A read 2 mm at the slow flow and 4 mm at the fast flow. Plate B read 9 mm and 15 mm. Plate C read 9 mm at the slow flow. At the fast flow, water ran over the top of Plate C beside the slot, and the height stopped rising at 21 mm. Use the sample data for items 1 through 11.

1. Plate A read 2 mm at the slow flow and 4 mm at the fast flow. What is the difference?
2. Plate B read 9 mm at the slow flow and 15 mm at the fast flow. What is the difference?
3. Plate C read 9 mm at the slow flow and 21 mm at the fast flow. What is the difference?
4. Rule 1 asks for a change of at least 4 mm. The three differences were 2 mm, 6 mm, and 12 mm. How many plates pass Rule 1?
5. Rule 2 asks the fast flow height to stay below 20 mm with no water over the plate top. The three fast flow heights were 4 mm, 15 mm, and 21 mm, and only Plate C spilled over its top. How many plates pass Rule 2?
6. Rule 3 asks the slow flow height to be at least 5 mm. The three slow flow heights were 2 mm, 9 mm, and 9 mm. How many plates pass Rule 3?
7. Use a decision table with one row for each plate and one column for each rule. Mark each box pass or fail. Which single plate passes all three rules?
8. The slow pour delivered 100 mL every 10 seconds, and the fast pour delivered 400 mL every 10 seconds. How many milliliters per minute is the fast pour, and how many times the slow pour is that?
9. A ruler reading can be off by about 1 mm. What percent of Plate A's slow reading of 2 mm is that 1 mm? What percent of Plate B's slow reading of 9 mm is it, to the nearest whole percent? Use your answers to explain why the gauge needs Rule 3.
10. Plates B and C both read 9 mm at the slow flow, and both then received a flow 4 times larger. Plate B rose to 15 mm, while Plate C rose to at least 21 mm and overflowed. Use the shape of each notch to explain why Plate C's water rose so much higher.
11. The notch on Wilmot Brook cannot be deeper than 20 mm. Name one change to Plate C that would keep its slow reading at 5 mm or more and keep its fast reading below 20 mm. Say which reading in the sample data made you choose that change, and explain why the change cannot go too far.