Pascal's Principle
At the tire shop, the mechanic lifts an entire pickup truck with a hydraulic jack by moving a lever with just one hand. And if you squeeze a full water bottle that has several holes in it, the water shoots out of all of them at once, even the ones far from your hands. How does the water at the other end find out that you are squeezing it? And how can a small force move a pickup truck?
Pressure is transmitted throughout the liquid
Fill a syringe with water, cover the tip with your finger, and push the plunger: it barely moves, but your finger immediately feels a strong push outward. Why doesn't the water compress, and how does that force reach the other end without getting weaker?
When you squeeze the bottle, you increase the pressure at the point where you squeeze, but that increase reaches every hole in full, wherever they are. That is why the water comes out of all of them at once with streams of similar reach, not just from the one closest to your hand.
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 liquid transmits a pressure of $20\,000\ \text{Pa}$ to the large piston, which has a surface area of $0.5\ \text{m}^2$. With what force does…
In a hydraulic press, the liquid transmits a pressure of $20\,000\ \text{Pa}$ to the large piston, which has a surface area of $0.5\ \text{m}^2$. With what force does the liquid push this 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
To lift an object, the force that the large piston must provide is, at minimum, its weight. The weight $P$ is the force with which the Earth attracts the object: $P = m\,g$, with $m$ in kg, $g \approx 9.8\ \text{m/s}^2$ (or the value given in the exercise, such as $10\ \text{m/s}^2$), and $P$ in newtons. For example, a $1\,000\ \text{kg}$ car weighs $P = 1\,000 \times 10 = 10\,000\ \text{N}$, and that is the value you use as $F_2$ to solve for the force $F_1$ or the area $S_2$ that you need. Conversely, if you know $F_2$, the maximum mass it can lift is $m = F_2/g$.
When one press feeds another, the multiplication factors multiply each other: if the first multiplies by $k_1$ and the second by $k_2$, the whole system multiplies, in theory, by $k_1 k_2$. In practice it is not a good idea to chain many together: each stage requires more piston travel and adds leaks and losses.
When pressing the brake pedal, a car brakes all four wheels at the same time. Explain the role of the brake fluid and which physical principle makes it possible for the…
When pressing the brake pedal, a car brakes all four wheels at the same time. Explain the role of the brake fluid and which physical principle makes it possible for the force from your foot to reach all wheels almost instantly.
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) Reason 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).
Back to the beginning: the water in the bottle "finds out" because the pressure you apply is transmitted undiminished throughout the liquid, and the jack lifts the pickup truck because that same pressure acts on a piston with much more area, in exchange for pumping several times.