BASIC PHYSICAL PRINCIPLES

ABSTRACT
This section describes a number of basic physical principles that are requisite for understanding scientific articles. Issues covered are force, pressure, buoyancy, Archimedes law, energy and temperature

Force and Pressure

Definition

The action that is needed to move (accelerate) an object is called force. Force is expressed in Newton (N = kg m/s2). One Newton is the force that accelerates a mass at rest of 1 kilogram (kg) 1 meter/second2 (m/s2). This means that if a force of 1 N is exerted to a mass of 1 kg for 1 s, the mass ends up with a velocity of 1 m/s. The relation between force F (N), mass m (kg) and acceleration a (m/s2) is given by:

In this formula g is the gravity constant. It is 9.8 m/s². Every object we drop will accelerate with this acceleration towards the earth (if we neglect frictional forces due to presence of air)

Force is characterized by

• A direction in which the force acts.
• The strength expressed in N. A negative strength corresponds to an equal positive strength in the opposite direction.

Action and reaction

As long as the object is at rest (not accelerating: lying still or moving at constant speed), each force (action) is opposed by a force (reaction) equally large and pointing in opposite direction. When we hold the apple of 1 kg in our hand, gravity exerts an acting force of 9.8 N on the apple pointing downwards. Since we hold the apple, it exerts this force on our hand. We call this weight. We express weight in Newton. Our hand exerts a reacting force of 9.8 N on the apple. This force points upwards. Since the acting and reacting forces exerted on the apple are equal but in opposite direction, they cancel out. Net force is zero. Hence the apple does not accelerate. If we drop the apple, we actually remove the reacting force on it: the apple starts accelerating. Since the reacting force points in opposite direction, we state: action=-reaction. Note: in physics we clearly distinguish between mass and weight. Mass is a universal property of an object. It is a measure for inertia of the object: it defines how the object accelerates when a force is applied to the object (see equation (1)). Mass is expressed in kg. Weight of an object is the force the earth exerts on an object. It is expressed in Newton. Weight is not universal. If we take the object to the moon the weight of the object changes. Mass is the same everywhere.

Pressure

Pressurized air exerts a force on the walls of our cylinder. The force that is exerted on a unit area is called pressure. Pressure is expressed in Newton per square meter (N/m²). There are a number of derived pressure units. See the Glossary and Units section. Most important derived pressure unit that is used on this home page is the bar (10⁵ N/m²).

Hydrostatic pressure

Every diver knows that underwater pressure increases with depth. How much does the pressure increase? Consider a horizontal area A at depth d in a liquid with density ρ. The volume V of the column of liquid 'resting' on the area A is V=A d. The mass of the column is given by m=V ρ=A d ρ. The force this mass exerts on the area A (weight) is F_W = m g = A d ρ g. Since pressure is given as force per unit area, we have to divide F_W by A to get the pressure. Hence the pressure on the area at depth d is given by:

This pressure is called the hydrostatic pressure. This pressure is the pressure at depth d due to the weight of the liquid. It does not take into account the atmospheric pressure of the air. If we submerge a diver in water, we have to add the atmospheric pressure. To understand this, think of extending our hypothetic liquid column on area A above water into the atmosphere (to infinity, or at least a few 100 km). We have to add the weight of the air in the column to the weight of the liquid. In terms of pressure we have to add the atmospheric pressure at the liquid surface to the hydrostatic pressure to obtain the absolute pressure at depht d:

For a diver at depth d (diving from sea-level, P_atm=1.01325 bar) in seawater (ρ=1025 kg/m³) the pressure is roughly given by:

or

Buoyancy

What happens when we submerge an object in water? Why do some object sink, while others float? This question is answered by Archimedes' law:

Any object partially or wholly immersed in a fluid (or gas) is acted upon by an upward force which is equal to the weight of the fluid (or gas) displaced by the body.

This law is attributed to Archimedes, a Greek mathematician (287-211 b.c.). King Hiero II of Syracuse had a crown made by his goldsmith. Hiero suspected his goldsmith had replaced part of the gold that was given to him by an equal weight of silver. He asked Archimedes to show that the crown was made of pure gold. The solution occured to Archimedes when he stepped into a bath and saw the water level rise. He ran into the street, calling 'Eureka!'. The solution was to put an amount of pure gold with equal weight as the crown into a bowl filled with water to the rim. The gold would overflow the bowl. After that the gold was taken out and the crown was put in. If the crown contained silver, its volume would be larger than the gold he put in first. In that case the bowl would overflow more when he put in the crown.

In formula we can state the law of Archimedes:

In this formula F_A is the Archimedes force, V is the volume of the object (in m³), ρ_fluid/gas is the density (see below) of the displaced fluid or gas (in kg/m³) and g is the gravity constant.

Besides this upward pointing Archimedes force, gravity still acts on the submerged object. The gravity force points downwards. Depending on which of both forces is stronger, the object floats or sinks. Here comes in material density ρ. Density ρ is defined as the mass of a unit volume of a material:

ρ  =   m V

(7)

In this formula m is the mass of the object and V its volume.

We submerge an object with a volume of 2 dm³ (i.e. 2 liters). If the object is made of stone (granite, ρ_stone = 2.6 kg/dm³), its weight is, according to (2) and (7):

The upward pointing Archimedes force is (6):

Since the upward force is smaller than the downward force, the object sinks. The net downward force the object exerts on the bottom is F_W - F_A = 51.0-19.6 = 31.3 N. The upward force F_A - F_W or buoyancy (i.e. -31.3 N) is negative. Hence we speak of negative buoyancy: the stone object sinks.

If our object was made of wood with a density of 0.8 kg/dm³ (oak wood), the downward force due to gravity is:

The upward force stays the same, since the amount of displaced water stays the same. The net upward pointing force is F_A - F_W = 19.6-15.7 = 3.9 N. We speak of positive buoyancy: the object floats.

If our object was made of some plastic with equal density as water, the net upward force would be zero. The object has neutral buoyancy and would neither sink nor float. During the dive, the diver uses his BCD (Buoyancy Control Device) and the air volume in his lungs to control his net buoyancy to neutral.

In the picture on the right buoyancy is shown. Three cubes of equal size are put in a fish tank. The left cube is made of stone, the middle cube is made of a plastic with equal density as water, the right cube is made of wood. The gravity force is shown as blue arrows. They point down. Since the stone cube has largest density the gravity force acting on it is largest. The wooden cube has smallest density and, hence, smallest gravity force. Archimedes force is shown as green arrows. Since the stone and plastic cubes are completely under water, their Archimedes force are equal. The wooden cube floats and is partly above the water. It displaces less water then the other two cubes. Its Archimedes force is smaller than the Archimedes force of the other cubes.

Part of the weight of the stone cube is compensated by its Archemedes force. However, the Archemedes force is smaller than its weight. The stone cube sinks. The part of its weight that is not compensated by the Archimedes force is compensated by the reaction force of the bottom, shown in as a red arrow. The weight of the plastic cube is exactly compensated by its Archimedes force. The weight of the wooden cube is compensated by its Archimedes force, including the weight of the part above the water level. So for each of the three cubes the forces add up to zero: each object is at rest and does not move (acceleration a=0 means F=0, see equation (1)).

Actually, every object (not in water) experiences an Archimedes force due to the air it displaces. However, usually this force is very small compared to its weight. So, this force can be neglegted. However, in case of a balloon filled with Helium gas, the force is larger than the weight of the balloon since the density of Helium is smaller than that of air: the balloon 'floats' and goes up.

We now present two excercises to illustrate how Archimedes discovery can be applied.

Excercise 1

Problem If our oak wooden object (volume 2 dm3, density 0.8 kg/dm3) was a cube with side d, how much of its height would be above the water surface and how much would be below the surface? Solution Let a d be the height above the surface and (1-a)d the heigth below the surface, with 0≤a≤1. On the part below the surface the Archimedes force is acting, on the part above the surface not. The net upward force is zero

Solving this equation results in a=0.2. So in this case 20% of the height of the log is above the surface.

Excercise 2

Problem
We put a toy boat and a stone in a fish tank. The stone is lying on the bottom, whereas the boat floats at the surface. What will happen to the water level if we take the stone from the bottom and put it into the boat?

Solution
When the stone is on the bottom, part of its weight is compensated by Archimedes force. Most part of its weight however, is compensated by the reaction force of the bottom, since the stone lies on the bottom (it exerts force on the bottom). When the stone is in the boat, its entire weight is compensated by Archimedes force. For this, the boat has to lie deeper in the water when the stone is on board. More water is displaced in this situation. Displacement of water is upward (the only way the water can go). Hence the water level will rise.

Temperature and thermodynamics

Temperature

Matter is made of particles called molecules. Molecules move (gas, liquid), vibrate (solid) and collide. The measure for the amount of vibration and movement of molecules is temperature. The higher temperature, the more molecules vibrate. When we cool down material, molecules slow down. Temperature is roughly proportional to the average energy per molecule. We can cool down until all movement is gone. This is called the absolute zero temperature. The Kelvin temperature scale is based on this phenomenon: 0 Kelvin is the absolute zero. On the Celcius scale this corresponds to -273.15 degrees Celcius.

When two objects with different temperatures are brought into thermal contact with each other, heat (energy) flows from the object with higher temperature to the object with lower temperature. The higher temperature object cools down, the lower temperature object warms up. When temperature stops changing, the objects are in thermal equilibrium. An important statement is the zeroth law of thermodynamics:

If two systems are separately in thermal equilibrium with a third, then they must also be in thermal equilibrium with each other.

Themometers

The zeroth law of thermodynamics makes it possible to calibrate one system with respect to another system. Stated otherwise: this makes it possible to use thermometers. A thermometer is a small system M with a parameter θ (thermometric parameter) that changes when the system absorbs or gives off heat. This parameter may be any parameter like length, resistance, pressure, etc. An example is the well known mercury or alcohol thermometer. In this case the thermometric parameter θ is the length L of the liquid column in the capillary tube. If the thermometer M is brought into thermal contact with a system A, it absorbs or gives off heat until thermal equilibrium is established between the thermometer and the system. The thermometric parameter θ assumes a certain value θ = θ_A. The value θ_A is called the temperature of system A with respect to thermometer M.

If the thermometer is brought in contact (and equilibrium) with a second system B, the parameter θ may assume a value θ = θ_B. If θ_A ≠ θ_B, heat will be exchanged when system A and system B are brought into thermal contact with each other. If θ_A = θ_B no heat will be exchanged. System A and B are in thermal equilibrium.

In practice absolute temperature is used as thermometric parameter. Any thermometer using this thermometric parameter will read the same value when used to measure the temperature of a particular system.

Work, Heat and Energy

A force can do work. If we drop the apple from our example above, the force of gravity accalerates the apple. The amount of work W done by a force on to an object is defined as:

The parameter S is the displacement. If F is expressed in Newton and S in meter, the unit of work of work is Newton-meter (Nm). This unit is called Joule (J). So 1 Joule = 1 Nm.

According to equation (2) the force on the 1 kg apple in our example is F_W = m g = 1 x 9.8 = 9.8 N. If we let fall the apple 1.0 meter, the amount of work done by gravity is

Energy is defined as the ability to do work. Energy of an object is equal to the amount of work W it can do and is expressed in Joule as well. There are two types of energy:

• Potential energy - Energy of an object due to its position. Definition: Potential energy at a point of interest is the work we do in moving the object (without acceleration) from an intial location, arbitrarily assigned to be a zero of potential energy, to the point of interest.
• Kinetic energy - Energy an object possesses due to its motion.

If we define the apple's potential energy to be zero when it is lying on the floor (height h=0). The energy to lift to a height h against the force of gravity is

This is potential energy. Due to its position (height h), the apple has the ability to do work (i.e. the energy) equal to m g h with respect to its reference position (the floor).

If we drop the apple from height h, the apple is accelerated to velocity v. The potential energy is transferred to kinetic energy. We can show that the potential energy is given by

E k  =   1 2  m v 2  ( J )

(15)

When the apple reaches the floor, all the potential energy has been transferred into kinetic energy. No energy gets lost. This brings us at the Law of conservation of Energy: In a closed system the amount of energy is constant.

Heat is a special form of energy: heat is the energy of an object due to motion the particles (molecules) the object is made of. Heat is often denoted by Q and expressed in Joule.

The Law of conservation of Energy is stated formaly in the First Law of Thermodynamics. It states the internal energy increase ΔE of a system is equal to the heat Q added to the system minus the work W done by the system.

The Second Law of Thermodynamics states (Kelvin/Planck):

It is impossible to extract an amount of heat Q_H from a system and use it all to do work W. Some amount of heat Q_C must be exhausted to a colder system, where Q_H = W + Q_C.

or (Clausius):

Heat cannot flow from a colder system to a hotter system without work being done to accomplish this flow.

or:

The entropy S of a system remains constant or increases.

Entropy S is defined by the relation

dS  =   Q T

(17)

Entropy often is regarded as a measure for disorder. The Second Law of Thermodynamics states that Nature tends to increase disorder.

Heat flow, heat conductivity

Heat transfer at a surface

The heat transfer dQ/dt (in Watt = Joule/sec) through a thermal barrier with a surface A and thickness d is given by:

dQ dt   =   κ A ΔT d

(19)

In this equation ΔT is the temperature gradient accross the barrier. The coefficient κ is called the thermal conductivity and is expressed in W/m K.

Specific heat

As we have seen heat is the amount of energy an object possesses due to the movement of its molecules. Temperature is a measure for heat. The amount of heat (energy) that has to be supplied to an unit amount of a material to raise its temperature by one unit of temperature is called specific heat. Specific heat c is expressed in J/(kg K). If we have an object with mass m and we want to raise its temperature by ΔT (Kelvin), the amount of energy we have to supply is

Sometimes specific heat is defined per mole instead of kg. This heat is denoted as capital C. In next table for some materials the specific heat is given: