Exercises: Pascal's principle

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 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 transmitting force and motion between the pistons.

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.

1. a) What pressure is exerted on the liquid?
2. b) According to Pascal's principle, what pressure reaches the large piston?

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 cover the nozzle of a water-filled syringe with a finger and try to push the plunger, as shown in the figure: it barely moves.

Explain what prevents it from moving and what your finger feels while you push.

State which of these machines operate by applying Pascal's principle and which do not, justifying each answer:

1. a) A garage hydraulic jack.
2. b) A mattress spring.
3. c) A dentist's elevating chair.
4. d) A bicycle cable brake.

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 $50\ \text{N}$.

1. a) Express both surface areas in $\text{m}^2$.
2. b) Calculate the pressure exerted on the liquid.
3. c) Calculate the force exerted by the large piston. By what factor has your force been multiplied?

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 $g = 10\ \text{m/s}^2$.

1. a) Calculate the weight of the car.
2. 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.
3. 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, the piston that squeezes the brake pad has a surface area of $10\ \text{cm}^2$.

1. a) Calculate the pressure reached by the brake fluid.
2. b) Calculate the force with which the brake pad on that wheel is squeezed.
3. c) Why is it advantageous for the surface area of the brake pad piston to be larger than that of the pedal piston?

A hydraulic press must lift cars weighing up to $2\,000\ \text{kg}$ by applying a force of $250\ \text{N}$ to the small piston, which has an area of $10\ \text{cm}^2$. Take $g = 10\ \text{m/s}^2$.

1. a) Calculate the pressure exerted on the liquid.
2. b) What force must the large piston provide?
3. c) What minimum surface area must the large piston have? Express it in $\text{m}^2$ and in $\text{cm}^2$.

The two pistons of a hydraulic press are circular, as shown in the figure: the small one has a radius of $1\ \text{cm}$ and the large one $10\ \text{cm}$.

1. a) Calculate the surface area of each piston using $S = \pi\cdot r^2$ and leave the result in terms of $\pi$, in $\text{cm}^2$.
2. b) How many times larger is the surface area of the large piston than that of the small one? Notice that $\pi$ cancels out.
3. c) If you push the small piston with $15\ \text{N}$, what force does the large one provide?

A dentist's hydraulic chair has a small piston of $20\ \text{cm}^2$ and a large piston of $500\ \text{cm}^2$.

1. a) If the pump exerts a force of $300\ \text{N}$ on the small piston, what force lifts the chair?
2. b) Take $g = 10\ \text{m/s}^2$: can the chair raise a patient of $70\ \text{kg}$? Justify your answer with a calculation.

In a hydraulic press, the surface area of the large piston is $50$ times that of the small one.

1. a) If you apply a force of $80\ \text{N}$ to the small piston, what force do you obtain at the large piston?
2. b) To achieve that force you have to pump and move through many centimetres with the small piston while the large one barely rises: explain what is multiplied in the press and what is "paid" in return.

The pressure gauge on a scrapyard press reads a pressure of $2,5\cdot10^{5}\ \text{Pa}$ in the circuit, and the piston that pushes the scrap metal has a surface area of $0,2\ \text{m}^2$. Take $g = 10\ \text{m/s}^2$.

1. a) With what force does the piston push?
2. b) If that force were used to lift a body, what mass would it correspond to?

A garage hydraulic lift has a small piston of $20\ \text{cm}^2$, to which a maximum force of $250\ \text{N}$ is applied, and a large piston of $0,1\ \text{m}^2$. Take $g = 10\ \text{m/s}^2$.

1. a) Calculate the maximum pressure the circuit can reach.
2. b) Calculate the maximum force that the large piston can exert.
3. c) What maximum mass can it lift?
4. d) A 4x4 vehicle of $1\,400\ \text{kg}$ cannot be raised: propose two technical changes to the lift to enable it to raise the vehicle and justify each using the hydraulic press formula.

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.

1. a) What force does the large piston exert?
2. b) The volume of displaced liquid is the same on both sides ($V = S\cdot h$): calculate how much the large piston rises.
3. c) Calculate $F_1\cdot h_1$ and $F_2\cdot h_2$, compare the results and explain why it is said that a hydraulic press multiplies force, but not energy.

A mechanic checks a car's brake circuit. The pedal pushes the master piston ($S = 3\ \text{cm}^2$) with a force of $150\ \text{N}$, and the piston for each brake pad has a surface area of $6\ \text{cm}^2$.

1. a) Calculate the pressure reached by the brake fluid.
2. b) Calculate the force with which each brake pad is squeezed.
3. c) If air enters the circuit, the pedal sinks almost to the floor and the car brakes much less effectively: explain what happens to the air when pressing the pedal, compared to the liquid, and why insufficient pressure reaches the pads.

You want to design a hydraulic jack capable of lifting vans weighing up to $1\,500\ \text{kg}$ by applying a maximum force of $200\ \text{N}$. The small piston will have a surface area of $10\ \text{cm}^2$. Take $g = 10\ \text{m/s}^2$.

1. a) What force must the large piston provide?
2. b) What maximum pressure will you exert on the liquid with your arm?
3. c) Calculate the minimum surface area that the large piston must have and express it in $\text{cm}^2$.

You build a home-made hydraulic press by connecting two cylinders with a tube: $S_1 = 10\ \text{cm}^2$ and $S_2 = 250\ \text{cm}^2$. On the small piston you place a weight of $5\ \text{kg}$. Take $g = 9,8\ \text{m/s}^2$.

1. a) Calculate the force exerted by the weight on the liquid.
2. b) What force appears at the large piston?
3. c) What mass could you balance on the large piston?
4. d) Explain why we say the system "multiplies" the weight, and calculate the multiplication factor.

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 $20$.

1. a) If you apply a force of $10\ \text{N}$ to the first press, what force does it deliver to the second?
2. b) What final force is obtained at the outlet of the second press?
3. c) By what factor has the original force been multiplied? Does it match the product $10 \cdot 20$?
4. 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).

The hydraulic arm of a crane truck pushes a piston of $0,05\ \text{m}^2$ with a pressure of $3\cdot10^{6}\ \text{Pa}$. Take $g = 10\ \text{m/s}^2$.

1. a) With what force does the piston push?
2. b) What maximum mass could it lift with that force?
3. c) The pump that generates this pressure is small and on its own exerts very little force: using Pascal's principle, explain how it is possible to lift such large loads.

You fill a water balloon, poke four symmetrical holes at the same height and squeeze with both hands, as shown in the figure: the four jets travel the same distance.

1. a) Using Pascal's principle, explain why the four jets emerge with the same force.
2. b) If you squeeze twice as hard, what do you predict will happen to the internal pressure and to the jets? Justify your answer.
3. c) Your classmate claims that water would come out stronger through a hole located lower down "because it is closer to the ground": evaluate whether they are correct using what you know about hydrostatic pressure inside the balloon itself.