Pascals principle – problems and solutions

1. __Known :__

The area of A_{1 }= 10 cm^{2}

The area of A_{2} = 100 cm^{2}

Force 2 (F_{2}) = 100 Newton

__Wanted :__ Force 1 (F_{1})

__Solution :__

P = F / A

*P = **pressure**, F = **force**, A = **area*

P_{1 }= F_{1} / A_{1 }

P_{2} = F_{2} / A_{2}

P_{1} = P_{2}

F_{1} / A_{1 }= F_{2 }/ A_{2}

F_{1} / 10 cm^{2 }_{ }= 100 N / 100 cm^{2}

F_{1} / 10_{ }= 1 N

F_{1} = (10)(1 N)

F_{1} = 10 Newton

2. If the area of A_{1} = 0.001 m^{2} and the area of A_{2} = 0.1 m^{2} , external input force F_{1} = 100 N, then the external output force F_{2} ?

__Known :__

The area of A_{1} = 0.001 m^{2}

The area of A_{2} = 0.1 m^{2}

External input force F_{1} = 100 Newton

__Wanted ____:__ External output force (F_{2})

__Solution :__

P_{1} = P_{2}

F_{1} / A_{1 }= F_{2 }/ A_{2}

100 N / 0.001 m^{2} = F_{2} / 0.1 m^{2}

100 N / 0.001 = F_{2} / 0.1

100,000 N = F_{2} / 0.1

F_{2} = (0.1)(100,000 N)

F_{2} = 10,000 N

3. Car’s weight = 16,000 N. What is the external input force F…

__Known :__

Car’s weight (w) = 16,000 N

Area of B (A_{B}) = 4000 cm^{2} = 4000 / 10,000 m^{2} = 4 / 10 m^{2} = 0.4 m^{2 }

Area of A (A_{A}) = 50 cm^{2} = 50 / 10,000 m^{2} = 0.005 m^{2 }

__Wanted :__ Force F

__Solution :__

F / A_{A} = w / A_{B}

F / 0.005 m^{2 }= 16,000 N / 0.4 m^{2 }

F / 0.005 = 16,000 N / 0.4

F / 0.005 = 40,000 N

F = (40,000 N)(0.005)

F = 200 Newton

4.

Area of A is 60 cm^{2} and area of B is 4,200 cm^{2}, determine the external input force of F.

__Known :__

Area of A (A_{A}) = 60 cm^{2}

Area of B (A_{B}) = 4200 cm^{2}

Weight w (w) = 3500 Newton

__Wanted :__ F_{1}

__Solution :__

Force of F calculated using the equation of Pascal’s principle :

F_{1} / A_{1} = F_{2} / A_{2}

F_{1} / 60 cm^{2 }= 3500 N / 4200 cm^{2}

F_{1} / 60 = 35 N / 42

F_{1} = (60)(35) / 42

F_{1} = 2100 / 42

F_{1} = 50 Newton

5. The hydraulic lift has a large cross section and a small cross section. Large cross-sectional area is 20 times the small cross-sectional area. If on the small cross section is given an input force of 25 N, then determine the output force.

__Known :__

Small cross section area (A_{1}) = A

Large cross-sectional area (A_{2}) = 20A

Input force (F_{1}) = 25 N

__Wanted :__ Output force (F_{2})

__Solution :__

**What is Pascal’s Principle?****Answer**: Pascal’s Principle states that any change in pressure applied at any point in a fluid is transmitted undiminished throughout the fluid in all directions.

**How does a hydraulic lift work based on Pascal’s Principle?****Answer**: In a hydraulic lift, a small force applied to a small-area piston creates a pressure that is transmitted undiminished to a larger-area piston. This results in a larger force on the larger piston, enabling it to lift heavy objects.

**Why is the pressure change undiminished in Pascal’s Principle?****Answer**: Because fluids are incompressible, when pressure is applied to one part of the fluid, it cannot compress, so it transfers the pressure change uniformly to every other part of the fluid.

**How does an increase in depth affect fluid pressure, considering Pascal’s principle?****Answer**: As depth increases in a fluid, the pressure also increases due to the weight of the fluid above. This is consistent with Pascal’s principle as the pressure increase is experienced uniformly at a particular depth.

**Why do our ears pop when ascending or descending quickly in an airplane or diving deep underwater?****Answer**: Rapid altitude or depth changes lead to rapid pressure changes in the surrounding medium (air or water). Our ears pop as a way to equalize the internal pressure with the external pressure, in line with Pascal’s Principle.

**If a sealed syringe with no air bubbles is depressed, why is it hard to compress the fluid inside?****Answer**: Fluids are virtually incompressible. When you try to decrease the volume of the fluid inside the syringe, the pressure increases, resisting the compression, according to Pascal’s Principle.

**How does Pascal’s Principle explain the phenomenon of a dam bursting if a small hole is created at its base?****Answer**: Due to the weight of the water, the pressure at the base of the dam is high. Even a small opening allows this high pressure to be exerted on the surrounding structures, which can lead to catastrophic failure of the dam.

**Why is it easier to squirt liquid from a nearly full squeeze bottle than a nearly empty one?****Answer**: A full bottle has a larger amount of liquid which, when squeezed, transmits the applied pressure more effectively throughout the fluid due to Pascal’s Principle. An almost empty bottle has more air, which is compressible, so the same pressure doesn’t produce as forceful a squirt of liquid.

**In hydraulic systems, is the output force always greater than the input force?****Answer**: Not necessarily. The ratio of the output force to the input force is determined by the ratio of the areas of the output and input pistons. The system is designed to amplify force by having a larger output piston area than input, but it’s the area ratio that determines force amplification.

**Considering Pascal’s Principle, why do deep-sea creatures have sturdy and compact bodies?**

**Answer**: Deep-sea creatures experience enormous pressures due to the weight of the overlying water column. To survive, these creatures have evolved to have sturdy and compact bodies that can withstand these high pressures.