3.1. Calculating Pressure Difference between Points M and N

Before we go to the next concept, and that is, we have shown a manometer here in this figure. We have to calculate the pressure difference between points, M and N, this is point M and this is point N. The best way as I told you before, if you want to calculate the pressure difference or pressure at that point you have to start at one point where the pressure is known and traverse to the other point where you have to calculate the pressure.

In fluid mechanics, calculating the pressure difference between two points is crucial for understanding how fluids move. When you want to find the pressure at a certain point, you can 'traverse' from a point where the pressure is known. This means you will add or subtract pressure values based on whether you are moving up or down in terms of depth in the fluid. If you go upward in the fluid column, you subtract the pressure; if you go downward, you add the pressure.

Imagine you are swimming in a pool. When you dive deeper, you feel the water pressure increasing on your ears and body. If you swim upward and come to the surface, you feel less pressure. This illustrates how the pressure changes based on depth - similar to how we calculate pressure differences in fluids.

Here, what we are going to do, we are going to equate the pressure at this level. okay. So, equating the pressures at both the limps as I told you, while discussing the figure along the horizontal plane, and what is that plan, xx. So, pm - gamma w 0.06 + 0.035 will give us pN – gamma w 0.12 + 0.06 - gamma not into 0.035.

To find the pressure difference, we set up an equation relating the pressures at points M and N. The equation takes into account the weights of the fluid columns above each point. Here, 'pm' is the pressure at point M, and 'pN' at point N. The term 'gamma w' is the unit weight of water, multiplied by the heights of the fluid columns (0.06, 0.035 for different segments) we are considering. By relating these terms, we derive a formula that allows us to find the difference between the two pressures.

Think of a seesaw: if one side has a heavier weight (representing a higher pressure), it will tip the seesaw in that direction. By equating the weights (pressures) on both sides, you can find out how much heavier one side is compared to the other, just like we are finding the pressure difference between points M and N.

Here, gamma w is unit weight of water, which is equal to 9790 Newton per meter cube. And this value has been assumed, okay. This is a standard value. What is gamma not, is unit weight of oil.

In fluids, the unit weight is a measure of how much weight a specific volume of the fluid has. For water, this is typically 9790 Newtons per cubic meter. Different liquids have different unit weights based on their density. Here, we are comparing water's weight to the weight of oil, which is lighter. The specific weight of oil can be determined using its specific gravity, and which relates it to the weight of water by multiplying by its unit weight.

Imagine comparing a gallon of water to a gallon of olive oil. Even though they take up the same space, water is heavier because it has a higher density. This is similar to our calculations, where we need to take into account the weight of different fluids when measuring pressure differences.

If we use these 2 values in equation here what we are going to get pM - pN is equal to 9790 into 0.095 - 9790 into 0.18 – 8125.7 into 0.035 and this is going to give us - 1116.5 Newton per meter square or - 1.1165 kilo Pascal.

By substituting the known values of unit weights and heights into the pressure difference equation we set up earlier, we can calculate the precise difference in pressure. The resulting value gives us a negative number (-1.1165 kilo Pascal), indicating that the pressure at point N is greater than that at point M by this exact measure. This is crucial for applications in fluid mechanics, as it helps us understand how fluids behave in various conditions and configurations.

Think of it like measuring how much harder you need to push down on one side of a balloon to keep it from popping. If one side of the balloon can take more pressure (like pressure at point N), then you need to apply less force on the other side to match it (pressure at point M). Thus, the negative value reflects that one side comfortably withstands more pressure.

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