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# MechanicsNotes.pdf

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Date Created: 07/03/14

Chapter 2 Mechanics 21 Kinematic Concepts 0 The technical terms used below should not be confused with their everyday use 0 For example speed and velocity are not the same thing same thing for displacement and distance 0 Vector quantities always have a direction regardless of where the direction is going 0 In other words velocity and speed are NOT the same thing this is important if the object is not travelling in a straight line 0 The units of acceleration come from its definition the change in velocity 0 Whenever the motion of an object changes there is acceleration 0 When an object is travelling at constant speed at one moment but changes its direction later there is a case of acceleration because the direction changed therefore there is a change in velocity which means the object had to accelerate to the other direction 0 A deceleration means slowing down it could mean negative acceleration if the velocity was positive Kinematic Commonly Definition Example SI Unit Vector or Terms used symbol Scalar Displacement S The distance The displacement M Vector moved in a from London to particular Rome is 143 1O6m direction to the southeast Velocity V or u The rate of The velocity during a Ms391 Vector change of flight from London to displacement Rome is 160ms391 to the southeast Speed V or u The rage of The speed during a Ms391 Scalar change of flight from London to distance Rome is 16Oms391 Acceleration a The rate of The average Ms392 Vector change of acceleration of a velocity plane on the runway during takeoff is 35ms392 in a forward direction This means that on average the velocity of the plane increases by 35ms391 per every second Average value the mean value over a period of time is different compared to the instantaneous value the value at one particular point of time If two objects are moving in the same line but are travelling at different speeds then their relative velocities can be determined by using simple additions and subtractions Three types of graphs are useful in mechanics 1 Displacement vs time graph distance vs time graph 2 Velocity vs time graph speed vs time graph 3 Acceleration vs time graph 2 common methods to find physical quantities and properties from the graph are 1 The gradient of the line o Can give either the average value by using the straight line part of the gradient o Or can give the instantaneous value by finding the tangent to the graph at one point 2 The area under the line o For a velocity vs time graph the area under the line can give the displacement travelled over time since the product of velocity and time is displacement o For an acceleration vs time graph the area under the line can give the total velocity of the object over time since the product of acceleration and time is velocity Displacement vs time graph a The gradient of a displacement vs time graph gives the velocity b The area under a displacement vs time graph is not useful Velocity vs time graph a The gradient of a velocity vs time graph gives the acceleration b The area under a velocity vs time graph gives the displacement Acceleration vs time graph a The gradient of an acceleration vs time graph gives the jerk or the rate change of acceleration of an object but it is not useful at the moment b The area under an acceleration vs time graph gives the velocity There are 5 useful equations of uniform motion however these equations are only useful when the acceleration is constant There are also 5 useful symbols sdispacement u initial velocity v final velocity t time taken aacceeration vuat lt quotgtt 122 u2 2as t atz s u 2 t atz 312 2 The first equation is derived from the definition of acceleration which is the rate of change of velocity over time v u a t atv u vuat The second equation is derived from the definition of velocity which is the rate of change of displacement over time 3 average velocity E vu average velocity Combine the two equations and svu t 2 ltvut 3 2 The third fourth and fifth equations are derived from both the first and the second equations v u t S T 2 vuat uatu sltgtt lt2uatt 3 2 at2 2ut 3 2 2 Zas 122 u2 122 u2 2as When an object is falling down in a vertical motion in a uniform gravitational field while ignoring the effect of air resistance that object is experiencing freefall In the absence of air resistance all falling objects have the same acceleration of freefall regardless of their mass Terminal velocity is the maximum velocity that an object can travel due to the system that the object is under A force is the cause of a velocity change The SI unit for the measurement of forces is the newtonN or kgms392 A force causes a change in velocity Since a change in velocity means there is an acceleration a force causes an acceleration IF there is no change in velocity or constant velocity a force is not necessary 0 Since only one force can act on one object the description of a force should include o The magnitude o The direction o The object that it acts o The object that exerts the force o The nature of the force 0 The following forces below are all the forces that pushes or pulls objects that exists in nature Gravitational force freefall Electrostatic force electrons electric shock Magnetic forces The ability of a positive nucleus to attract a negative electron Normal reaction Friction The forces that slows down an object in motion due to the surface drag Tension Hanging pictures and frames Compression Gas syringes Up thrust The initial force to start a motion of an object OOOOOOO o Lift Lifting a book directly up from a desk 0 Forces can be divided into two categories contact forces and force between the distance that separates the objects o Contact forces include but not limited to friction o Forces between the distance that separates the objects include but not limited to Electrostatic forces 0 The spring constant is the amount of force required to stretch a piece of string per m 0 t s SI unit is Nm391 0 A freebody diagram shows the all the forces that an object is receiving 0 It can only show one object 22 Newton39s three laws of motion First law 0 Newton39s first law states that an object continues in uniform motion or uniform velocity NO ACCELERATON in a straight line or at rest unless a resultant external forces acts 0 When the net force on an object is zero that object is said to be in an equilibrium 0 EON two things either an object is travelling at uniform velocity or at zero speed 0 The following table shows the formal descriptions of the 9 forces mentioned above Name of force Description Gravitational forces The force between objects due to its mass cause of weight Electrostatic forces The forces between objects due to their electric charges Magnetic forces The forces between magnets and electric currents Normal reactions The force between two surfaces that acts at right angles to the surface Friction The force that opposes the relative motion of surface Tension When a string is stretched it has equal yet opposite forces on its ends pulling outwards Compression When a rod is compressed it has equal yet opposite force on its pushing inward Up thrust The upward forces that act on an object when it is submerged in a fluid cause of buoyancy Lift The force exerted on an object when a fluid flows over it in an asymmetrical way enables airplanes to fly due to aerodynamics 23 Newton39s three laws of motion Second law 0 Newton39s second law enables us to measure force 0 In formal terms his second law is that the resultant force is proportional to the rate of change of momentum 0 In SI unit terms the resultant force is equal to the rate of change of momentum 0 Momentum is the product of mass of an object and it s velocity Amomentum 2 of force Atime 0 In terms of symbols A ZF p where p is momentum At Amv At mAv F Tt ma where a 13 acceleration 24 Newton39s three laws of motion Third law 0 Newton39s third law enables us to know that forces come in pairs 0 In formal terms his third law is when two bodies A and B interact the force that A exerts on B is equal to the opposite of the force that B exerts on A 0 In simpler terms For every action on one object there is an equal but opposite reaction on another object 0 In terms of symbols FABFBA 0 There are two important key points 0 The two opposing forces in the pair act on DIFFERENT objects in other words two equal yet opposite forces that act on the same object are NOT Newton39s third law force pairs 0 The two opposing forces in the pair must be equal opposite and must be the same type of force 25 Mass and Weight 0 Due to everyday languages mass and weight are often confused to be the same thing 0 Mass is the amount of matter contained in an object measured in kg 0 Weight of an object is essentially the mass of the object multiplied by the gravitational pull 0 For example in Jupiter your mass would still be the same however your weight will be about 25 times heavier due to the larger gravitational pull caused by Jupiter39s strong gravitational pull 0 Gravitational force mg 26 Momentum 0 Linear momentum is defined as the product of mass and velocity 0 Impulse is the change of momentum 0 According to Newton39s second law force is the change of momentum or final momentum subtracted by the initial momentum divided by time 0 The law of conservation of linear momentum states that The total linear momentum of a system of interacting particles remains constant assuming that are no resultant external forces 0 When two objects collide with each other two possible situations can occur o The first scenario is an elastic collision where no mechanical energy is lost during the collision I Elastic collisions occur rarely in real life the only situations where elastic collisions occur is the collision between gas molecules I Inelastic collision is where energy is lost can be lost in forms of sound and heat however momentum is still conserved 27 Work 0 Work is done when a force moves an objects in the direction of the force Work done F 3 C036 Work done F orce distance 0 When an object travels in a constant speed or does not move there is no acceleration therefore there is no force therefore the work done by the object is zero 0 In a force vs distance graph the total work can be determined by finding the area under the graph 0 To find out the work done when lifting an object vertically is to find out the product of mass gravitational force and the height 0 To find out the work done in compressing or extending a spring is kxz 28 Energy and power 0 The amount of energy transferred is equal to the work done 0 Energy is a measure of the amount of work done 0 It means that the units of energy and the units of work are the same joules 0 When energy is lost by one object it must be gained by another in other words energy cannot be destroyed 0 Due to the part that energy cannot be destroyed there are several statements about energy that is known as the principle of conservation of energy 0 The principle of conservation of energy can be stated in several ways o The total energy of any closed system must be constant 0 Energy is neither created or destroyed it just switches form for example in mechanics energy turns from gravitational potential energy into kinetic energy o Finally there is no change in the total energy in the Universe 0 There are 12 different types of energies that exists in nature Kinetic energy Gravitational potential energy Elastic potential energy Electrostatic potential energy Thermal energy Electrical energy Chemical potential energy Nuclear energy ONLY NUCLEAR FUSlON Internal energy Radiant energy Solar energy nuclear fusion o Light energy OOOOOOOOO 0 Kinetic energy can be defined as the product of mass and velocity squared divided by 2 mvz o Kinetic energy 0 Gravitational potential energy can be defined as the product of mass the Earth39s gravitational potential energy and the change in height o Gravitational potential energymgh 0 Elastic potential energy is the product of spring constant and the change in displacement squared divided by 2 kxz o Elastic potential energyT 0 Power is defined as the rate at which energy is transferred in other words the change of energy used over time 0 Power can also be stated as the rate at which work is done 0 The SI unit for power is joules per second Js391 or watts W 0 When something is moving at a constant velocity v against a constant frictional force f the power can be stated as the product of the frictional force and the velocity o Power F orceVelocity fv 0 Efficiency is the ratio of useful energy to the total energy transferred 0 Efficiency can be defined as useful work OUT useful energy OUT o Efficiency total energy transformed total energy transformed useful energy OUT 0 Efficiency Total energy IN useful power OUT 0 Efficiency total power IN 29 Uniform circular motion 0 Uniform circular motion is used to describe an object that is going around in a circular motion at a constant speed 0 The magnitude of the velocity or the speed is constant during the circular motion 0 However the direction of the velocity of the object is constantly changing 0 Since there are changes in direction the object is undergoing acceleration 0 The acceleration of an object in a circular motion is called centripetal acceleration 0 Since there is acceleration in order to drag a mass into a circular motion it requires a force 0 This force which causes centripetal acceleration is called the centripetal force 0 Centripetal acceleration is defined as the quotient of velocity squared and the radius of the circular path that the object makes during the actual circular motion 122 o C entripetal acceleration 7 where v instantaneous velocity of an object at a certain point in time and r is the radius 0 Centripetal force is defined as the product of the mass and the centripetal acceleration which can be redefined as the product of the mass and the velocity squared divided by the radius of the circular path that the object makes during the actual circular motion 2 o C entripetal force my 0 Centripetal force is the sum of all the forces that allows the object to move in a circular motion o In other words it is not a new force 0 Finally since there is no change in displacement made during a circular motion circular motion does not do any work o This statement is very IMPORTANT

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