Pressure

Pressure is defined as the thrust per unit area of a surface. If a thrust F is applied on a surface having area A, then P = F/A. Its SI unit is N/m² or Pascal (Pa).

A hydraulic machine is a device based on Pascal's law which magnifies the effort applied on it. It uses an incompressible liquid to transmit pressure and convert a small applied force into a much larger output force.

A hydraulic brake is a braking system that uses liquid pressure to stop or slow down a vehicle. When the brake pedal is pressed, it pushes a piston in the master cylinder, creating pressure in the brake oil that is transmitted to the wheel calipers.

A hydraulic jack is a device that uses liquid pressure to lift heavy objects like cars with a small applied force, especially while changing their wheel. It works on Pascal's law using a pumping piston and a larger lifting piston.

The resultant upward force acting on an object when it is partially or wholly immersed in a fluid is called upthrust or buoyant force. Upthrust = Weight in air − Weight in fluid.

A hydrometer is an instrument used to measure the density or specific gravity of liquids. It works based on the principle of floatation — it floats higher in denser liquids and lower in less dense liquids.

Effort should be applied on Piston A (20 cm²) — the smaller piston. In a hydraulic machine, applying force on the smaller piston creates pressure that is transmitted equally to the larger piston. Since Piston B (200 cm²) has 10 times the area of Piston A, the output force at Piston B will be 10 times the input force applied at Piston A. This makes it easy to lift heavy loads with a small effort.

Pascal's Law states that: "When pressure is applied on an enclosed liquid, it is transmitted equally and perpendicularly in all directions."

This principle was first proposed by the French mathematician and philosopher Blaise Pascal in 1653 AD. It forms the basis of all hydraulic machines.

By Archimede's principle, the upthrust equals the weight of the liquid displaced. The volume of liquid displaced equals the volume V of the body immersed.

Where d = density of liquid, V = volume of body immersed (= volume of liquid displaced), and g = acceleration due to gravity.

A small iron nail sinks in water (density ≈ 1000 kg/m³) because the upthrust from water is not enough to support it. However, the same iron nail floats on the surface of mercury, which has a much higher density (≈ 13,600 kg/m³). Mercury exerts a much greater upthrust on the nail, which exceeds the nail's weight, allowing it to float.

This clearly shows that greater the density of the fluid, greater the upthrust it exerts. Similarly, an egg sinks in pure water but floats in concentrated salt solution because adding salt increases the density of water and hence the upthrust.

Archimede's Principle states: "When a body is wholly or partially immersed in a liquid, it experiences an upthrust which is equal to the weight of the liquid displaced by it."

The liquid that exerts the greatest upthrust has the greatest density. Upthrust = Weight in air − Weight in liquid. So the liquid in which the body weighs least gives the most upthrust.

Liquid C has the minimum density (the body weighs most in it, meaning least upthrust).

To sink at a desired depth, a submarine fills its ballast tanks with water. This increases the overall mass and density of the submarine above that of the surrounding water. As a result, the weight exceeds the upthrust and the submarine sinks. By controlling the amount of water in the ballast tanks, the submarine can remain at any desired depth. To rise, air is pumped into the ballast tanks to force out the water, reducing the submarine's density below that of water.

A hydraulic press is used to press, compress, or shape different materials with great force. Specific applications include compressing cotton bales, extracting oil from oil seeds, punching holes in metals, and giving specific shapes to metal sheets. It allows a small applied force to produce a very large output force by using Pascal's law with pistons of different cross-sectional areas.

Law of Floatation states: "The weight of a floating body is equal to the weight of the liquid displaced by it."

This means a body floats when it displaces a volume of liquid whose weight equals the body's own weight. It is a special case of Archimede's principle that applies specifically to floating objects.

A body sinks in a liquid when the upthrust exerted by the liquid is less than the weight of the body (Upthrust < Weight). This typically occurs when the density of the body is greater than the density of the liquid, so the body cannot displace a volume of liquid equal to its own weight. The resultant force acts downward, causing the body to sink.

A hydrometer works on the principle of floatation (Law of Floatation). Since the hydrometer's weight is constant, it must always displace liquid equal to its own weight. In denser liquids, a smaller volume is displaced, so the hydrometer floats higher. In less dense liquids, it sinks deeper. The density is read from the graduated scale on the stem at the liquid level.

When an object is immersed in a fluid, the fluid exerts pressure on all surfaces of the object — from the top, bottom, and sides. Since pressure increases with depth, the pressure on the bottom surface (at greater depth) is higher than on the top surface. This means the upward force on the bottom face is greater than the downward force on the top face. The forces on the lateral (side) surfaces are equal and opposite, so they cancel out. The net result is a resultant upward force called upthrust or buoyant force.

These two forces are opposite in direction. The body sinks if W > U, floats if W ≤ U, and is in equilibrium just below the surface if W = U.

Case B (fully immersed) produces maximum upthrust. Since upthrust is directly proportional to the volume of liquid displaced (U ∝ V), a fully immersed body displaces a greater volume of liquid than a partially immersed one of the same size. Therefore, case B, where the body is completely submerged, generates a larger upthrust than case A, where only part of the body is in the liquid.

By the law of floatation: weight of block = weight of liquid displaced. So density of block × V × g = density of liquid × (fraction × V) × g, giving: density of liquid = density of block / fraction submerged

Higher fraction submerged → lower liquid density. Lower fraction submerged → higher liquid density.

Increasing order: Y < X < Z

No, the fraction submerged will NOT change. When the block is cut into two equal halves, both the mass and volume are halved. The density of the wood (mass/volume) remains exactly the same. Since floatation depends on the ratio of the density of wood to the density of liquid — and neither changes — the fraction submerged in each liquid remains unchanged.

A hydrometer consists of a long narrow stem and a weighted bulb at the bottom filled with lead or mercury. When immersed in a liquid, it floats upright at a certain level.

Since the weight of the hydrometer is constant, by the law of floatation it always displaces liquid equal to its own weight. In a denser liquid, a smaller volume of liquid needs to be displaced to equal the hydrometer's weight, so it floats higher (the stem rises above the liquid surface). In a less dense liquid, more volume must be displaced, so it sinks lower. The density is read directly from the graduated scale on the stem at the liquid surface level.

Fish have a special internal organ called the swim bladder. When food is offered near the surface, the fish fills its swim bladder with more air or gas. This increases the fish's overall volume, allowing it to displace more water and experience a greater upthrust. Since the upthrust now exceeds the fish's weight, the net force acts upward and the fish rises toward the surface of the water. By controlling the amount of air in the swim bladder, the fish can precisely control its depth — exactly like a submarine's ballast tanks.

In condition A, the weight of air displaced by the balloon is greater than the weight of the balloon itself. The burner heats the air inside the balloon, causing it to expand and become less dense than the surrounding air. So the upthrust (= weight of displaced air) exceeds the total weight of the balloon, producing a net upward force that causes it to rise.

In condition B, the density of air inside the balloon increases. As the air inside the balloon cools (burner reduced or off), it contracts and becomes denser (heavier). The balloon's overall weight increases until it exceeds the upthrust from the surrounding air. The balloon can no longer displace air equal to its weight, so the net force acts downward and it descends.

The molecules in liquids are in close contact. Their intermolecular spaces are fixed and cannot be reduced by applying external force. Therefore, liquids cannot be compressed.

In a hydraulic machine, a small force applied on a small piston creates pressure that is transmitted equally (Pascal's law) to a large piston. Since the large piston has a much greater area (F = P × A), the force at the large piston is many times greater than the force applied at the small piston. Thus, it multiplies the applied force and is called a force multiplier.

When the bucket is fully submerged in water, the water exerts an upward upthrust on the bucket. This upthrust acts opposite to the weight, reducing the apparent weight of the bucket. Therefore, Apparent weight = Weight in air − Upthrust, which is less than the actual weight. So the bucket feels lighter when submerged.

In pure water, the density is lower and the upthrust produced is insufficient to support the weight of the egg, so it sinks. When salt is dissolved, the density of the solution increases, producing greater upthrust that equals or exceeds the egg's weight, allowing it to float.

When flat, the aluminum sheet has a large surface area that displaces a large volume of water, producing enough upthrust to support its weight. When crumpled into a compact ball, it displaces a much smaller volume of water, producing insufficient upthrust. Since its weight is unchanged but upthrust decreases, it sinks.

An iron nail is compact and solid; its density (≈7874 kg/m³) is much greater than water, so it cannot displace water equal to its own weight and sinks. An iron ship has a special hollow design with large empty spaces inside, which reduces its average density below that of water. As a result, it can displace water equal to its own weight and floats by the law of floatation.

Ocean water contains dissolved salts and is therefore denser than fresh river water. A denser liquid exerts greater upthrust on an immersed body. Since ocean water provides more upthrust on the swimmer's body, it is easier to stay afloat and swim in the ocean than in a river.

The Dead Sea has an extremely high concentration of dissolved salts, making its water much denser than ordinary seawater or fresh water. This very high density produces an enormous upthrust that easily exceeds the weight of a human body, causing people to float effortlessly without any swimming effort.

River water is less dense than ocean water (no dissolved salts). By the law of floatation, the ship must displace water equal to its own weight. Since river water is less dense, the ship needs to displace a greater volume of river water to equal its weight. Therefore, a larger portion of the hull sinks below the river surface.

A larger helium-filled balloon displaces a greater volume of air, experiencing a much larger upthrust. While its weight is also greater than a smaller balloon, the net upward force (upthrust minus weight) is larger for the bigger balloon, making it rise more rapidly in air.

Hydraulic machines work on Pascal's law: "When pressure is applied on an enclosed liquid, it is transmitted equally and perpendicularly in all directions."

A hydraulic machine consists of two cylinders with different cross-sectional areas (small and large), connected by a pipe. Each cylinder is fitted with an airtight, movable piston. Both cylinders and the connecting pipe are completely filled with an incompressible liquid such as water or oil.

Let A₁ = cross-sectional area of small piston, F₁ = force applied on it.
Let A₂ = cross-sectional area of large piston, F₂ = force exerted on it.

Pressure on small piston: P₁ = F₁/A₁

By Pascal's law, this pressure is transmitted equally: P₁ = P₂

Since A₂ > A₁, therefore F₂ > F₁. The force at the large piston is many times greater than the force applied at the small piston. Therefore, the hydraulic machine is a force multiplier.

It consists of two main cylinders of different diameters and a reservoir, all connected by tubes fitted with valves (Valve A and Valve B). The system is filled with hydraulic oil. The small cylinder is connected to a lever handle. The large cylinder has a platform on which the vehicle is placed. The large cylinder is also connected to the reservoir through a release valve.

Upthrust = Weight in air − Weight in medium. The medium in which the body weighs least gives the most upthrust. Body weighs least in Medium R (17 N), so R produced maximum upthrust.

Air is least dense → least upthrust → body weighs most in air → Q = Air (27 N).
Honey is densest → most upthrust → body weighs least → R = Honey (17 N).
Water is intermediate → P = Water (23 N).

By Archimede's principle, upthrust in water = weight of water displaced.
Upthrust in water = Weight in air − Weight in water = Q − P = 27 − 23 = 4 N.

Weight of body in air = 27 N. Given 1 kg weighs 10 N:

Upthrust = Weight in air − Weight in water = 50 − 35 = 15 N.
By Archimede's principle, upthrust = weight of water displaced.

Adding salt increases the density of water. A denser liquid exerts greater upthrust (U = dVg). Therefore, the apparent weight of the metal ball in water will decrease (it will feel even lighter in salt water than in plain water).

This experiment verifies Archimede's Principle: the upthrust experienced by the metal ball (15 N) equals the weight of the water displaced by it (15 N).

On the Moon, the acceleration due to gravity (g) is approximately 1/6 of Earth's. Since Upthrust = dVg, with smaller g, the upthrust will be less on the Moon. However, the weight of the ball (W = mg) also decreases by the same factor. Both forces decrease proportionally, so the ratio (upthrust/weight) stays the same. Archimede's principle still holds, but the actual numerical values of both upthrust and weight will be smaller on the Moon.

Object Y has more density than water. Y sinks to the bottom because its density exceeds the density of water, so the upthrust is less than its weight. Object X floats at the surface, meaning its density is less than that of water.

Archimede's Principle applies to object Y. Even though Y sinks, the water still exerts an upthrust on it equal to the weight of water displaced by Y. Since this upthrust is less than Y's weight, the net force is downward and Y rests at the bottom.

If the density of water increases, the upthrust on Y increases (U = dVg). If the upthrust becomes equal to or greater than Y's weight, Y will begin to rise from the bottom and may float. The position of Y moves upward.

If the volume of X is reduced while its mass stays the same, its density increases (density = mass/volume). X is currently floating (density < water). As density increases, X will be more submerged (float lower in the water). If the density eventually exceeds that of water, X will sink to the bottom.