Pascal's principle
At a cookout, Emma squeezes a full water bottle that has several holes poked in its walls, and the water shoots out of all of them at once, with the same force, even out of the holes that are far from her hands. How does the water at the other end "know" she is squeezing, and why does it come out equally hard everywhere? We'll answer this at the end of the first section.
Pressure is transmitted throughout the liquid
If you fill a syringe completely with water, block the nozzle firmly with your finger and push the plunger, you notice that it barely gives at all, yet your finger instantly feels a strong push outwards. Why doesn't the water compress, and how is that force carried from one end of the container to the other without weakening?
Back to Emma's bottle: when she squeezes, she increases the pressure at the point where she presses, but that increase in pressure reaches every hole in full, wherever it is. That is why the water shoots out of all of them at once with jets of similar range, not just from the hole nearest her hand.
One detail: the liquid also has its own pressure from its weight, $P = P_0 + \rho g h$, which is a bit larger at deeper holes. Pascal's principle says that the increase you add by squeezing is the same everywhere. For a small bottle, the squeeze adds much more pressure than the few centimeters of water depth, so the jets look alike. Deeper holes do shoot out slightly faster because of the $\rho g h$ term, but that difference is small; the increase from squeezing is the same at every hole. Being "closer to the ground" is not what makes a jet stronger.
Write Pascal's principle in your own words: explain what happens to the pressure exerted at a point in an enclosed liquid and how it is transmitted through its interior.…
Write Pascal's principle in your own words: explain what happens to the pressure exerted at a point in an enclosed liquid and how it is transmitted through its interior. Then give an everyday example of its application.
You fill a plastic bottle with water, poke several holes in its sides and squeeze the bottle with your hands, as shown in the figure: water shoots out of all the holes…
You fill a plastic bottle with water, poke several holes in its sides and squeeze the bottle with your hands, as shown in the figure: water shoots out of all the holes at the same time, including those far from your hands.
Which principle explains this? What can you say about the pressure received by the water at each hole?
Hydraulic presses use a liquid (oil) and not air. Thinking about whether liquids and gases can be compressed, explain why a liquid is much more effective than a gas at…
Hydraulic presses use a liquid (oil) and not air. Thinking about whether liquids and gases can be compressed, explain why a liquid is much more effective than a gas at transmitting force and motion between the pistons.
The hydraulic press: multiplying a force
In a hydraulic press, a force of $100\ \text{N}$ is exerted on the small piston, which has a surface area of $0.01\ \text{m}^2$, as shown in the figure. a) What pressure…
In a hydraulic press, a force of $100\ \text{N}$ is exerted on the small piston, which has a surface area of $0.01\ \text{m}^2$, as shown in the figure.
- a) What pressure is exerted on the liquid?
- b) According to Pascal's principle, what pressure reaches the large piston?
In a hydraulic press, the small piston has a surface area of $5\ \text{cm}^2$ and the large one $100\ \text{cm}^2$. You push the small piston with a force of…
In a hydraulic press, the small piston has a surface area of $5\ \text{cm}^2$ and the large one $100\ \text{cm}^2$. You push the small piston with a force of $50\ \text{N}$.
- a) Express both surface areas in $\text{m}^2$.
- b) Calculate the pressure exerted on the liquid.
- c) Calculate the force exerted by the large piston. By what factor has your force been multiplied?
In a hydraulic press, the surface area of the large piston is $25$ times that of the small one. You push the small piston with $40\ \text{N}$ and cause it to descend by…
In a hydraulic press, the surface area of the large piston is $25$ times that of the small one. You push the small piston with $40\ \text{N}$ and cause it to descend by $50\ \text{cm}$, as shown in the figure.
- a) What force does the large piston exert?
- b) The volume of displaced liquid is the same on both sides ($V = S\times h$): calculate how much the large piston rises.
- c) Calculate $F_1\times h_1$ and $F_2\times h_2$, compare the results and explain why it is said that a hydraulic press multiplies force, but not energy.
Applications: jacks, brakes and lifts
When one press is followed by another, the multiplication factors are multiplied together: if the first multiplies by $k_1$ and the second by $k_2$, the combined system multiplies, in theory, by $k_1\cdot k_2$. In practice, chaining too many presses is not useful: each stage requires a longer piston stroke and adds possible leaks and losses, so the real benefit is smaller than the theoretical product.
You want to lift a $1\,200\ \text{kg}$ car using the garage hydraulic jack, as shown in the figure. The small piston has a surface area of $8\ \text{cm}^2$ and the large…
You want to lift a $1\,200\ \text{kg}$ car using the garage hydraulic jack, as shown in the figure. The small piston has a surface area of $8\ \text{cm}^2$ and the large one $400\ \text{cm}^2$. Take $g = 10\ \text{m/s}^2$.
- a) Calculate the weight of the car.
- b) Using $\frac{F_1}{S_1} = \frac{F_2}{S_2}$, calculate the force that must be applied to the small piston to lift it.
- c) Compare this force with the weight of the car and explain where the "extra force" comes from.
When braking, the pedal pushes the main piston of the brake circuit ($S = 2\ \text{cm}^2$) with a force of $200\ \text{N}$, as shown in the figure. On one of the wheels,…
When braking, the pedal pushes the main piston of the brake circuit ($S = 2\ \text{cm}^2$) with a force of $200\ \text{N}$, as shown in the figure. On one of the wheels, the piston that squeezes the brake pad has a surface area of $10\ \text{cm}^2$.
- a) Calculate the pressure reached by the brake fluid.
- b) Calculate the force with which the brake pad on that wheel is squeezed.
- c) Why is it advantageous for the surface area of the brake pad piston to be larger than that of the pedal piston?
An engineer connects two hydraulic presses in series: the first multiplies the force by $10$ and its large piston pushes the small piston of the second, which multiplies…
An engineer connects two hydraulic presses in series: the first multiplies the force by $10$ and its large piston pushes the small piston of the second, which multiplies the force by $20$.
- a) If you apply a force of $10\ \text{N}$ to the first press, what force does it deliver to the second?
- b) What final force is obtained at the outlet of the second press?
- c) By what factor has the original force been multiplied? Does it match the product $10 \times 20$?
- d) Explain why, in practice, many presses are not connected in series to achieve huge forces (think about the travel distance of the pistons and possible leaks).