Exam practice

The brake fluid must pass on the pressure from the pedal or lever to the brake pads fully and at once, so it must be a liquid that can’t be compressed (air or any gas would be squeezed instead, and the brakes would fail). It must also have a high boiling point, because braking makes the brakes very hot, and a bubble of vapor would act like air; a low freezing point, so it works in cold weather; and it must not damage the rubber seals or rust the metal parts. Ordinary water or oil can’t do all of this, so a special brake fluid is used.

Under the water, the cork displaces its whole volume of water. Cork is much less dense than water, so the weight of that water, the upthrust, is much greater than the cork’s weight. With the finger gone there is a net upward force, so the cork rises until only part of it is under water and the upthrust equals its weight.

The water gives our body an upthrust equal to the weight of water we displace. While floating, the upthrust equals our whole weight, so our muscles and the ground don’t have to hold us up. Our apparent weight is almost zero, so we feel light. (Our mass and real weight haven’t changed.)

Under water, the upthrust of the water holds up part of the stone’s weight, so we only lift its apparent weight. In the air, the upthrust is almost zero, so we must lift its full, real weight. Its weight hasn’t changed; we just lost the water’s help.

The Dead Sea is so salty that its water is 1.24 times as dense as pool water. A body must displace its own weight of water to float. In the denser Dead Sea water, a smaller volume is enough, so the body floats with much of it above the surface. A person whose average density is about 1,000 kg/m³ floats with only about $1000/1240 \approx 81\%$ of the body under the water there, compared with nearly all of it in a pool. People can lie on their back and read a newspaper.

Each passenger makes the boat sink a little deeper, so it displaces more water and gets enough extra upthrust to carry them. But once the water reaches the top edge of the boat, it can’t displace any more water, so the upthrust can’t grow any more. Extra weight beyond that is more than the greatest upthrust the boat can get; water comes in over the edge and the boat sinks. (In Nepal, overloaded boats and ferries on rivers and lakes have capsized in this way. Never get into a boat that is already full, and wear a life jacket.)

When pressure is applied at any point of a liquid enclosed in a container, it is transmitted equally in all directions throughout the liquid, and acts at right angles to the walls of the container.

Any two of: hydraulic brakes in cars and motorbikes; the hydraulic jack used to lift vehicles; hydraulic lifts (the dentist’s chair, garage car lifts); the hydraulic press; excavators (diggers) and cranes; a barber’s chair that pumps up.

A hydraulic machine is a force-multiplying device based on Pascal’s law. Two cylinders of different areas, with tightly fitting pistons, are joined and filled with a liquid. A small force on the small piston makes a pressure that is transmitted through the liquid to the large piston, where it produces a larger force: $F_2 = F_1 \times A_2/A_1$. Examples: the hydraulic jack, brake, lift and press.

When an object is partly or completely immersed in a fluid, it experiences an upthrust equal to the weight of the fluid it displaces.

Upthrust is the upward force exerted by a fluid on an object that is partly or completely immersed in it. It is equal to the weight of the fluid displaced ($U = V\rho g$), and its SI unit is the newton.

Two forces: the weight of the object (the pull of gravity), acting vertically downward, and the upthrust of the liquid, acting vertically upward. For a floating object they are equal in size, so the resultant force is zero.

A floating object displaces liquid whose weight is equal to its own weight. (Weight of the floating object = weight of the liquid displaced.)

Any two of: designing ships and boats so they displace enough water to float; submarines, which sink and rise by filling and emptying ballast tanks; the hydrometer, which measures the relative density of liquids; the lactometer, which tests whether milk has been watered down; hot-air and helium balloons; finding the density or purity of an object of any shape by weighing it in air and in water.

In a liquid: push an empty plastic bottle, its cap screwed on tight, down into a bucket of water. You feel it pushing back up, and when you let go it pops up to the surface. (Or hang a stone from a spring balance and weigh it in air, then in water: it weighs less in water, and the difference is the upthrust.)

In a gas: let go of a balloon filled with helium or hydrogen: it rises, because the air’s upthrust on it is more than its weight. (Or blow hot air into the mouth of a light paper bag from below, with a hair dryer: once the air inside is less dense, the bag rises.)

Draw a cube held under water, with arrows at right angles to each face, as in Figure 8.12. Then explain:

1. A liquid presses on every surface of an object in it, at right angles to the surface.
2. The forces on the opposite side faces (left–right, front–back) are equal and opposite, so they cancel.
3. Pressure increases with depth ($P = h\rho g$). The bottom face is deeper than the top face, so the upward force on the bottom is greater than the downward force on the top.
4. The difference between these two forces is a net upward force: the upthrust. For a cube of volume $V$, it works out as $V\rho g$, the weight of the liquid displaced.

No. The water does give the stone an upthrust, equal to the weight of the water it displaces (you can see it: the stone weighs less on a spring balance when it is in the water). The stone sinks because it is denser than water, so its weight is greater than the upthrust. The upthrust only slows the stone down; it can’t stop it.

Suggestion: put the stone on the larger piston and push on the smaller piston. The way it is now, the machine works backwards: to lift 500 N on the small piston, the push on the large piston must be 500 N × $A_2/A_1$, many times bigger than the stone’s weight.
How the force is increased (draw Figure 8.8): a force $F_1$ on the small piston of area $A_1$ makes a pressure $P = F_1/A_1$ in the liquid. By Pascal’s law, this pressure is transmitted equally through the liquid to the large piston of area $A_2$, where it makes a force $F_2 = P \times A_2 = F_1 \times A_2/A_1$. Since $A_2 > A_1$, $F_2 > F_1$. For example, if $A_2$ is 10 times $A_1$, a push of 50 N on the small piston lifts the 500 N stone on the large one.

No. It is true for a solid lump of the substance, but not for an object shaped to hold air. Iron (7,800 kg/m³) is much denser than water, yet a ship made of iron floats; aluminium foil (2,700 kg/m³) sinks as a crumpled ball but floats as a tray. A hollow, wide shape displaces a large volume of water, enough to give an upthrust equal to its weight. What decides floating is the average density of the whole object, including the air inside it, compared with the density of the liquid.

The hydrogen balloon rises; the air balloon falls to the ground. Both displace the same volume of air, so both get the same upthrust. But the air balloon contains air (slightly squeezed, so a little denser than the air outside) plus the rubber, so its weight is more than the upthrust. Hydrogen is about 14 times less dense than air, so the hydrogen balloon’s total weight is less than the upthrust, and it rises.

The object will rise, and then float with part of it above the surface, rising higher as more salt is dissolved. Dissolving salt increases the density of the water. The object, completely under, now displaces water of greater density, so the upthrust ($V\rho g$) becomes greater than its weight and it moves up. It comes to rest when only enough of it is under the surface to displace its own weight of the denser water, and the denser the water, the less of it needs to be under.

• a.25 N with the 5 N block, and 28 N with the 8 N block. The blocks float, so the water holds up their whole weight, and the water rests on the machine. No water leaves the beaker, so the machine reads the water plus the block: 20 + 5 = 25 N and 20 + 8 = 28 N.
• b.30 N both times. By the law of flotation, a floating block displaces its own weight of water. The beaker was full, so that water spills out over the edge. The 5 N block pushes out 5 N of water, and the 8 N block pushes out 8 N of water, so the weight on the machine stays 30 N.

The lactometer sinks deeper in a less dense liquid. Since even its whole stem went under, the milk is much less dense than pure milk (relative density about 1.03), so the milk has very likely been mixed with a lot of water. It is not pure; a milk collection center would reject it.

Given: $F_A$ = 20 N, $A_A$ = 0.5 cm², $A_B$ = 5 cm². Find $F_B$.
By Pascal’s law, the pressure is the same at both pistons:
$$\frac{F_A}{A_A} = \frac{F_B}{A_B} \quad\Rightarrow\quad F_B = F_A \times \frac{A_B}{A_A} = 20\ \text{N} \times \frac{5}{0.5} = 200\ \text{N}$$
(The areas can stay in cm² because only their ratio is used.) A force of 200 N is needed on piston B.

Given: $P$ = 30,000 Pa, $A$ = 0.1 m². Find $F$.
$$F = P \times A = 30{,}000\ \text{Pa} \times 0.1\ \text{m}^2 = 3000\ \text{N}$$
It can lift a load of 3,000 N, a mass of about $3000/9.8 \approx 306$ kg.

Real weight $W_1$ = 20 N.

• b.$U = 20 - 14 = 6\ \text{N}$
• c.$U = 20 - 10 = 10\ \text{N}$

Why different: in (b) only part of the object is under water (6/10, three-fifths of it), so it displaces less water; in (c) it is completely under and displaces more. Upthrust equals the weight of water displaced, so it is bigger in (c). The water collected in the small beaker in (c) weighs 10 N, and in (b) 6 N.