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define-pressure-in-fluids

๐Ÿš€ Pressure in fluids is defined as the force exerted per unit area. It is a scalar quantity and is measured in pascals (Pa). In a fluid at rest, the pressure at a point is the same in all directions. This concept is crucial in understanding how fluids behave under various conditions, including the effects of depth and temperature on pressure.

Theory Explanation

Definition of Pressure

Pressure (P) is defined as the force (F) applied perpendicular to the surface of an object per unit area (A) over which that force is distributed. Mathematically, it is expressed as P = F/A.

\[ P = \frac{F}{A} \]
Pressure in Fluids

In a fluid at rest, the pressure increases with depth due to the weight of the fluid above. The pressure at a depth (h) in a fluid of density (ฯ) is given by the equation P = ฯgh, where g is the acceleration due to gravity.

\[ P = \rho g h \]
Pascal's Law

Pascal's Law states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container. This principle is the basis for hydraulic systems.

\[ P_1 = P_2 \]

Key Points

  • ๐ŸŽฏ Pressure is defined as force per unit area (P = F/A).
  • ๐ŸŽฏ In fluids, pressure increases with depth due to the weight of the fluid above.
  • ๐ŸŽฏ Pascal's Law states that pressure changes in a fluid are transmitted equally in all directions.

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Examples:💡

Calculate the pressure at a depth of 10 meters in water. (Density of water = 1000 kg/mยณ)

Solution:

Step 1: Use the formula P = ฯgh, where ฯ = 1000 kg/mยณ, g = 9.81 m/sยฒ, and h = 10 m.

\[ P = 1000 \times 9.81 \times 10 \]

Step 2: Calculate P = 1000 ร— 9.81 ร— 10 = 98100 Pa.

Step 3: Thus, the pressure at a depth of 10 meters in water is 98100 Pa.

A hydraulic lift has a small piston with an area of 0.01 mยฒ and a large piston with an area of 1 mยฒ. If a force of 100 N is applied to the small piston, what is the force exerted by the large piston?

Solution:

Step 1: Calculate the pressure on the small piston: P = F/A = 100 N / 0.01 mยฒ = 10000 Pa.

\[ P = \frac{100}{0.01} = 10000 \text{ Pa} \]

Step 2: According to Pascal's Law, the pressure is the same in both pistons. Therefore, P = F_large / A_large.

Step 3: Rearranging gives F_large = P ร— A_large = 10000 Pa ร— 1 mยฒ = 10000 N.

Step 4: Thus, the force exerted by the large piston is 10000 N.

Common Mistakes

  • Mistake: Confusing pressure with force; students often think pressure is the same as force.

    Correction: Remember that pressure is force divided by area (P = F/A). Always check the units to ensure you are calculating pressure.

  • Mistake: Neglecting the effect of depth on pressure in fluids.

    Correction: Always use the formula P = ฯgh to calculate pressure at different depths, considering the density of the fluid and the depth.

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