A2 Physics Notes OCR Unit 4: The Newtonian World


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1 A2 Physics Notes OCR Unit 4: The Newtonian World Momentum:  An object s linear momentum is defined as the product of its mass and its velocity. Linear momentum is a vector quantity, measured in kgms 1 : (where denotes linear momentum).  Kinetic energy is another quantity (albeit scalar) which relates an object s mass to its velocity. The difference between kinetic energy and momentum can be appreciated by considering two bodies in motion: one with a higher linear momentum and the other with a higher kinetic energy. When both are acted on by the same constant retarding force, the body with the higher kinetic energy will take a greater distance to stop (as ), whilst the body with the greater momentum will take a longer period of time (because ). Newton s Laws of Motion:  Newton s first law of motion states that a body will remain at rest or uniform motion along a straight line unless it is acted on by an external resultant force. Thus, a resultant force of zero will result in zero acceleration.  Newton s second law states that the rate of change of momentum of an object is directly proportional to the resultant force acting upon it, and that the change of momentum is in the direction of the resultant force.  Newton s second law can be expressed mathematically as, or. As, this gives. The Newton is the force required to give an object of mass 1kg an acceleration of 1ms 2 ; if forces are measured in Newtons, the constant of proportionality in the above equation is 1, so.  Newton s third law states that every action force has an equal and opposite reaction force of the same type. Conservation of Momentum:  The principle of conservation of momentum states that in a closed system, the total momentum in any direction remains constant. This means that momentum is conserved during all collisions.  The principle of conservation of momentum can be deduced from Newton s laws by considering a hypothetical collision between two bodies, A and B. During the collision, A exerts a force on B, and by Newton s third law, B exerts an equal and opposite force on A. The bodies are exerting forces upon each other, so each force acts for the same amount of time. Thus, as, the bodies experience and equal and opposite change in momentum. This means that the net momentum change is always zero. Not that this holds even when the bodies do not collide head on the horizontal and vertical components of the system s momentum will always remain constant. Collisions:  When an object experiences a force, the change in momentum experienced by the object is known as the impulse. Impulse is defined as the product of a force and the time it acts for. It can be given by the area under a forcetime curve.  In a collision, momentum is always conserved. Kinetic energy, however, is not: it is often dissipated as heat or sound. A collision in which kinetic energy is conserved is
2 known as a perfectly elastic collision. Collisions in which kinetic energy is not conserved are known as inelastic collisions. Circular Motion:  When a constant force acts perpendicular to an object s motion, it causes it to move in uniform circular motion: its trajectory traces a circular path around a fixed point. As there is never any component of the force in the direction of the body s motion, it moves with constant speed, but its direction (and thus its velocity) constantly changes. By convention, angles are measured in radians when considering uniform circular motion.  The time taken for a body travelling in uniform circular motion with radius to complete one revolution is known as the period (denoted ). The body s speed is thus given by. The rate of change of the body s angular displacement (angular displacement is denoted by ) is given by, and denoted.  The position vector of a body moving in uniform circular motion is ( ) (see fig. 1). Its velocity is thus given by ( ) as. The body s acceleration is given by ( ). This demonstrates that the direction of the acceleration is the opposite of that of the position vector, i.e. towards the centre. It also shows that the magnitude of the acceleration is equal to, or as.  By Newton s first law, the above indicates that there must be a constant resultant force directed towards the centre in order for an object to move in uniform circular motion. This force is known as the centripetal force.  Newton s second law gives the magnitude of the centripetal force as.  An example of uniform circular motion is a conical pendulum, where a mass is suspended by a string at an angle (of ) to the horizontal and traces a circular path around the vertical axis (such as in a swingball game). Vertically, the system is in equilibrium, so (where is the tension in the string and is the weight of the mass). For uniform circular motion,. Thus. A special case of a conical pendulum is a mass on a string moving horizontally in uniform circular motion. In this case, ;. Gravitational Fields:  A field is the region in which a force operates. Fields cause forces at a distance; there does not need to be any matter between two bodies exerting forces upon one another.  Gravitational field strength ( ) is defined as the gravitational force per unit mass acting at a fixed distance from a body ( Fig. 1: ). It is measured in Nkg 1 (note that this is the same as ms 2, but gravitational field strength is not linked to motion the concept of weight and gravitational field strength are the same).
3  Gravitational field lines can be used to represent a gravitational field. These are straight lines which pass through a body s centre of mass, and have directional markers pointing towards the body (as gravity is always attractive).  Newton s law of gravitation states that the gravitational force of attraction between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them: where and are the masses of the two bodies (by convention denotes the greater mass), is the distance between them, and is the gravitational constant (c m 3 kg 1 s 2 ).  As the gravitational field strength is equal to force divided by mass, it follows that.  Providing the distances involved are greater than the radii of the objects being considered, bodies may be modelled as point masses when performing calculations. Such calculations may include determining the mass of a body given the radius and orbital period of one of its satellites. Planetary Orbits:  In the seventeenth century, Johannes Kepler discovered that the orbits of planets in the solar system are elliptical, and that their velocities change during their orbit. However, the implications of these observations are slight enough that planetary orbits may be modelled by uniform circular motion.  From empirical data, Kepler formulated his third law: that the square of a planet s orbital period is proportional to the cube of its mean orbital radius:.  Kepler s third law may be derived from Newton s laws, using the equations and. They can be equated to give. As, ( ). Satellites:  A geosynchronous or geostationary orbit is one in which a satellite remains in the same position relative to the earth s surface. All satellites must centre their orbits on the centre of the earth in order to maintain an orbit. Geostationary satellites must be travelling from west to east be over the equator, and have an orbital period of 24 hours.  Using the quantitative version of Kepler s third law ( ( ) ) one can calculate the necessary height of a geostationary orbit from the mass of the earth: it is approximately 35900km.  Uses of geostationary satellites include GPS, telecommunications and weather monitoring. Issues with geosynchronous orbits stem from the height required: it is costly in terms of money and energy to place the satellite into orbit. The height also gives geostationary satellites a wide footprint, which means a high power output is required for their signals to be receivable.  Lowlevel satellites orbit at heights of around 500km, and are not geostationary. They are less expensive than geostationary satellites and take more detailed images. They are used by the military, for intelligence and monitoring climate change at the poles. Introduction to Simple Harmonic Motion:
4  Simple harmonic motion (SHM) describes oscillating systems such as a pendulum or a mass on a spring. An idealised oscillation with no external driving force or friction is known as a free oscillation. In reality, all oscillations require input of an external force to continue.  An oscillation can be defined by certain quantities: o Displacement ( ): the distance an object is from its rest position at a given point in time. o Amplitude ( ): the maximum displacement. o Frequency ( ): the number of oscillations per second. o Period ( ): the time taken for a single oscillation ( ). o Angular frequency: the angular displacement per second when the oscillation is represented by circular motion (one oscillation = one rotation). o Phase difference ( ): the angle between two oscillations at a given point in time, when the oscillations are represented by circular motion.  Simple harmonic motion is a model for an oscillation. The acceleration of an object undergoing simple harmonic motion is directly proportional to its displacement from equilibrium, and is in the opposite direction to the displacement (i.e. it is directed towards the equilibrium point). This can be summarised by the equation where is a constant.  The equation for simple harmonic motion can be expressed as a function of time in the form of a second order differential equation:. The general solution to this equation is where and are also constants. Because of the relationship between circular motion and SHM, the use of and allow SI units to be used:. Two special cases of the solution are and, corresponding to an oscillating system starting from equilibrium and from maximum displacement respectively.  The equations derived above can be used to plot a graph of displacement against time for SHM; such a graph resembles a sine curve. The gradient of this curve gives the velocity of the body. Differentiating gives the value of the velocity at a given time:. At the maximum velocity, so. For maximum velocity, the displacement of the body is zero. Energy in Simple Harmonic Motion:  In free oscillations the amplitude does not vary over time; energy is conserved within the oscillating system. However, energy is changed from one form to another.  At maximum displacement, and oscillating body has zero kinetic energy; all of its energy is potential energy. At zero displacement, the object has its maximum speed and thus its maximum kinetic energy. Between these two states it has a mixture of different types of energy.  In a system such as a free pendulum or a mass attached to a spring moving horizontally on a smooth surface, there is only one form of potential energy: gravitational or elastic respectively. However, in a vertically oscillating mass on a spring there are two types of potential energy involved. Interestingly, in this case, the dominant energy change is that from GPE to elastic potential energy and vice versa: the more visible change in kinetic energy is comparatively small.
5 Damping:  In practice, energy will always be lost from an oscillating system: its amplitude will decrease over time. In order to maintain a constant amplitude, energy must be supplied to the system.  The process of deliberately reducing the amplitude of an oscillation is known as damping: o If only light damping forces (such as air resistance) exist, the amplitude will gradually decrease with little visible change to the period. o If heavier damping forces (such as strong friction) are present, there will be a much more dramatic decrease in amplitude, and the period of the oscillation will increase slightly. The body will move back to its equilibrium position, the point after which the body is at rest is known as critical damping.  Situations where damping is used include in concert halls to minimise reverberation, and in the suspension of vehicles to avoid unnecessary bouncing. Resonance:  All objects which can oscillate have a natural frequency of vibration. This is isochronous, that is, it is independent of the amplitude of the oscillation. For a pendulum where is the length of the pendulum. For a horizontally oscillating spring, where is the spring constant.  One oscillation is capable of inducing an oscillation in another object. The frequency of the driving oscillation is known as the driver frequency. When the driver frequency matches the natural frequency of vibration of the second oscillator, the amplitude of the oscillation increases dramatically. This phenomenon is known as resonance. This is analogous to a child s swing: if a force is applied at the same point of the swing each time, the height of the swing increases quickly.  Resonance is exploited in radios, where the circuits have resonant frequencies matching that of a transmitted signal to amplify it. Microwaves give out radiation at a frequency equal to the natural frequency of vibration for water molecules. The resonance of the nuclei of certain atoms in a magnetic field is used in NMR spectroscopy and in MRI scanning.  Resonance can also have unwanted consequences: buildings and bridges can shake violently and even collapse in strong winds which cause their structures to resonate. Aircraft must be designed so oscillation of the wings is minimised during takeoff and landing. Feedback with speakers and microphones is a result of resonance. Solids, Liquids and Gases:  The phase of a body describes whether it is a solid, a liquid or a gas. This depends on the arrangement of its molecules: o In a solid, molecules or ions are tightly packed in a fixed position: they cannot move relative to one another. Their atomic spacing does not noticeably change with pressure or temperature. o In a liquid, molecules are closely packed together, but are able to move over one another: liquids are fluid. Like solids, their atomic spacing is largely independent of their temperature or pressure.
6 o In a gas, the molecules are very far apart compared to solids and liquids, and move randomly in a chaotic pattern. Temperature and pressure have a significant effect on their atomic spacing.  The density of a material is inversely proportional to the cube of the spacing between molecules. In general, solids are denser than liquids, which are (about 1000 times) denser than gases the atomic spacing in a gas is around 10 times greater than that in a liquid. There are some notable exceptions, such as H 2 O ice is less dense than liquid water.  The molecules in an object move at high speeds: in solids they oscillate whereas in liquids and gases they flow, moving randomly. The diffusion of a fragrance in gas or dye in a liquid can be used as evidence for this. Another piece of evidence is Brownian motion: this is the apparent vibration of small particles such as smoke or pollen when suspended in a liquid. This effect is a result of constant, random bombardment by liquid molecules. The Kinetic Theory and Gas Pressure:  The kinetic theory of gases uses a model known as an ideal gas. The ideal gas model has several assumptions which simplify calculations: o That a gas consists of a large number of particles with rapid, random motion. o That collisions between gas molecules, and with the walls of their container, are perfectly elastic. o That the gravitational force acting upon the molecules is negligible. o o That there are no intermolecular forces except during collisions. That the total volume of the molecules is negligible compared to that of the container.  The kinetic model can be used to determine the pressure exerted by an ideal gas, by modelling an ideal gas consisting of three sets of n molecules, each moving in one of the three directions perpendicular to the faces of a cube, with side length : o The molecules have mass and velocity ; when a molecule collides with a face of the cube its momentum changes from to ;. o The time taken for one molecule to collide twice with one face is given by. ; the fore the wall exerts on one molecule is. o Pressure is, so the pressure one molecule exerts upon a wall is where is the volume of the container. Thus, molecules exert a pressure of ; the gas exerts a pressure of on each face. o The total number of molecules in the container,, is equal to, thus. The pressure of the gas can be expressed as. (the total mass of the gas) and (the gas s density). Thus.  The above model is a simplified version, but is appropriate as, though a gas obviously does not consist of molecules only moving perpendicular to one another, their velocities can be resolved into components which are. It is more accurate to give the mean square speed of gas molecules, denoted. Thus,. Internal energy:  The Internal energy of a body is the sum of the randomly distributed kinetic and potential energies of all the molecules in the body. Note that the energies must be random: a body
7 inside a moving container, or a container in a gravitational field, has the same internal energy as an identical body which is stationary or not in a field.  For an ideal gas, the only internal energy is kinetic energy. However, actual materials have intermolecular attractive forces: the molecules have potential energy also.  There are three factors which affect the internal energy of a body: o Temperature: this is directly proportional to the molecules kinetic energy, so as it rises, internal energy rises. o Pressure: a change in pressure with no change in temperature results in no change of internal energy for an ideal gas. However, in a nonideal gas, changing the pressure causes a change in the molecular spacing, meaning that work is done by or against the intermolecular forces. This causes internal energy to fall or rise accordingly. o State: when an object is heated and reaches its melting or boiling point, the energy supplied is used to overcome the intermolecular forces and change the state of the body. This results in a gain in potential energy; internal energy rises. Note that this means that an object s temperature (i.e. internal kinetic energy) remains constant during a phase change: a temperaturetime graph for a body being heated will show plateaus at the melting and boiling point. Temperature:  Temperature is essentially a measure of internal kinetic energy. It determines the overall direction in which thermal energy flows: from an area of high temperature to one of low temperature. This causes the temperatures of two (or more) bodies to change as they transfer energy between one another: the hotter body will lose energy to the colder one at a greater rate than the colder one does to the hotter one. This continues until the bodies reach thermal equilibrium, where they are at the same temperature: there is no net energy transfer. Thermal equilibrium is dynamic: energy still flows in both directions, but at the same rate.  Early measures of temperature were defined by thermometers, and were unreliable because increments differed slightly between thermometers. Consequently, an absolute scale of temperature, the Kelvin scale, was established, which is independent of measuring equipment and measures upwards from absolute zero: the point at which a body has zero internal energy. 0K = o C; 0 o C = K. Specific Heat Capacity:  Thermal energy describes the energy supplied to an object which raises its internal energy. Like other forms of energy, it is measured in Joules.  Specific heat capacity is the thermal energy required to raise the temperature of an object of mass 1kg by 1K. It is measured in Jkg 1 K 1, and can be described using the equation, where is the specific heat capacity and is the temperature change.  The temperature change of an object of known mass when heated by a source of known power can be used to determine SHC: ;. is the gradient of a temperaturetime graph. Latent Heat:  When a body changes state, energy is given out or taken in. The value of this energy is determined by the body s specific latent heat.  The specific latent heat of fusion of a substance is the energy per kilogram required to change the substance from a solid to a liquid at constant temperature.  The specific latent heat of vaporisation of a substance is the energy per kilogram required to change the substance from a liquid to a gas at constant temperature. The Ideal Gas Equation:  If temperature is kept constant, the volume of a fixed mass of an ideal gas is inversely proportional to its pressure (this is known as Boyle s law). Furthermore, if pressure is kept
8 constant, volume is directly proportional to absolute temperature (Charles s Law), and if volume is constant, pressure and temperature are proportional (GayLussac s law). From this, one can conclude that the quantity is constant for a fixed mass of gas.  The above relationship holds true for an ideal gas. It is suitable for modelling most gases, except when the gas is very close to its boiling point or at a very high pressure. In these cases, the molecules occupy a significant volume of their container, so the assumptions of the kinetic theory are inappropriate.  The mole is a unit describing the amount of a substance. One mole contains, to three significant figures, particles (this is known as Avogadro s number, ); the same amount as in 12g of carbon The value of is proportional to the amount of gas (denoted ). Thus,, or, where is the molar gas constant (c. 8.31mol 1 kg 1 ). This is known as the ideal gas equation. Note that the molecular mass of the gas is independent of the quantities in the equation: only the number of moles is significant.  When working on a microscopic scale, it may be easier to consider the number of molecules rather than the number of moles. In this case,, where is the total number of molecules and is the Boltzmann constant (c , or ).  From the kinetic theory of gases, it can be derived that the pressure of a gas is given by the equation. Thus,. As, it follows that. The mean kinetic energy of a single molecule is thus given by (as gives the mass of one molecule), so. This gives the mean translational energy (i.e. internal kinetic energy) of an ideal as, showing the relationship between temperature and kinetic energy.
Unit G484: The Newtonian World
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