Expected learning outcomes
1. Confirm that the hydrostatic equations work in practice.
2. You appreciate the need to work only in m, kg, s and N.
3. You appreciate the degree of experimental accuracy possible.
Background:
Equations (1) and (2) below will be familiar from lectures and example classes and can be used to calculate hydrostatic forces on a vertical surface and to determine where they act.
F = ρghGA (1)
hP = hG+I_G/(Ah_G ) (2)
where;
F is the force in N
ρ is liquid density in kg/m3
g is gravity in m/s2
hG is the vertical depth from the water surface to the centroid of the wetted surface in m
A is the wetted area of the immersed surface in m2
hP is the vertical depth from the water surface to the centre of pressure where F acts in m
IG is the second moment of area in m4. For a rectangle IG = bd3/12
Theory:
With the beam horizontal, there are two equal and opposite moments acting about the pivot. Remember, a moment is a force multiplied by the perpendicular distance to the pivot.
The theoretical clockwise moment is the hydrostatic force Facting on the vertical, orange face multiplied by its perpendicular distance Y below the pivot.
The experimental anticlockwise moment is the weight W (= Mg) on the mass hanger multiplied by the perpendicular distance Z.
If equations (1) and (2) are valid then:
F * Y = W * Z (3)
If the experiment is conducted accurately you should find FY is within ±5% of WZ.
Experimental procedure
1. Note the dimensions:
Z = 275 mm
a = 100 mm
D = 100 mm
L = 75 mm
h = water depth
When h < 100 mm the rectangular vertical face is partially immersed; when h > 100mm the face is completely immersed.
2. If necessary, empty the tank then level it using the spirit level and adjustable feet.
3. With the mass hanger in place at the end of the balance arm, adjust the cylindrical, black counter balance until the beam is horizontal. This is indicated by the long white line on the black plastic marker near the mass hanger. This is the zero condition where M = 0 and h = 0.
4. Add a 50 g mass to the mass hanger. Pour water into the perspex tank until the hydrostatic force pushes the beam back up to the horizontal. (If you overfill the tank, drain water out using the black tap). Note the water depth (h) on the scale on the orange quadrant.
5. Add additional 50 g masses and record the corresponding water depths, h.
6. When a total of 400 g has been added to the hanger, take the masses off one at a time and again note the corresponding water depths, h. Look for evidence of hysteresis (ie a difference in the value of h when loading and unloading the beam).
Calculations:
The purpose of the calculations is see if the two moments WZ and FY agree. The calculations are split into partial immersion of the vertical face (h < 100 mm) and complete immersion (h > 100 mm).
Partial immersion (h < 100 mm)
1. Substitute a value of M into equation (4) and the corresponding water depth h into equation (5). Calculate the numerical values of equations (4) and (5) and compare them to see if they agree within ±5%.
2. The actual anticlockwise moment WZ due to the weight on the hanger is:
= M g Z Nm (4)
Substitute a value of M to calculate the numerical value of equation (4).
3. By combining equations (1) and (2) the theoretical clockwise moment FY is as shown in equation (5). Substitute the depth h (ie the depth corresponding to M in equation (4) above) to calculate the numerical value of equation (5):
= ρg L h2 (a + D – ) Nm (5)
where ρ = 1000 kg/m3 and g = 9.81 m/s2.
Complete immersion (h > 100 mm)
4. For each value of M (and the corresponding water depth h) compare the calculated moments obtained from equations (4) and (7).
5. As for partial immersion, the actual anticlockwise moment WZ due to the weight on the hanger is:
= M g Z Nm (4)
Again, substitute a value of M to calculate the numerical value of equation (4).
6. For each mass added with the vertical face completely immersed, calculate the value of hG from:
hG = h – (6)
7. By combining equations (1) and (2) the theoretical clockwise moment FY is:
= ρg hG L D (a + + ) Nm
(7)
8. Compare the results from equations (4) and (7). What is the percentage difference?
Both partial and complete immersion
9. For a particular mass, was there a difference between the water level when the mass was increasing compared to decreasing?
Dimensions
Z = 275 mm
a = 100 mm
D = 100 mm
L = 75 mm
h = water depth
When h < 100 mm the rectangular vertical face is partially immersed; when h > 100mm the face is completely immersed.