Physical Quantities, Vectors, Fovce and newton's Laws
Group “A” Usman Abrar, Kamran Sharif,Muneeba Idrees, Hassan Amjad, Ali Asad
Chapter # 01, 02, 03 Measurement Motion in one Dimension & Force and Newton’s Laws
Chap#01 MeasurementPhysical Quantities:- “Quantities which can be measured are called Physical Quantities.” These require Magnitude, Unit and Sometime Direction for their complete description. Here we will discuss SI( System International) for measurements… Height of the girl, l = 1.55 m For example Symbol ofUnit of the the Numerical value physical physical quantity for the quantity magnitude of the physical quantity
System International:-Types of physical Quantities:- Base/ Basic Quantities . Derived Quantities.Basic Quantities:-“ Quantities which are chosen as a base and many other measuring quantities are derived from them are called basic quantities and their units are called base units…”They are :
Physical Quantities:-Derived quantities:- “All quantities other then “7” basic quantities are called Derived quantities because they are derived from Base quantities and their units are called derived units.”For Example :-
System internationalUnits:- “Units are symbol associated with every physical quantity whether its basic or Derived quantity….”Like, Meter(m) for length ; kilogram(kg) for Mass and second(s) for time etc…Standards:- “Physicals quantities must have a standard value so that every calculation is accurate and scientist in different place can calculate things without any ambiguity…”Some standards are discussed here...
System internationalThe Standard of time:-Anything which repeats its motion periodically can be set as timestandard like Oscillating Pendulum, A mass-Spring system, aQuartz crystal etc…But in SI time standard is defined as:“The second is the duration of 9,192,631,770 vibrations of (specific) radiation emitted by a Specific isotope Of the Cesium atom” Fig shows the current national frequancy standard, so-called Cesium foundation clock.
System internationalThe Standard of length:SI unit of length is Meter (m)… defined as: “The meter is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second”Length is also measured in cm, km, miles, feets etc…The Standard of Mass:SI unit Base unit for Mass is “Kg”“The SI standard of mass is a platinum-iridium cylinder kept at IBWM”
Precision And Significant FiguresPrecision:-“An indication of the scale on the measuring device that was used.”In other words, the more correct a measurement is, the moreaccurate it is. On the other hand, the smaller the scale on themeasuring instrument, the more precise the measurement.Fig illustrates differenceb/w Precision and accuracy.
Significant Figures“The significant figures (also known as significant digits, and often shortened to sig figs) of a number are those digits that carry meaning contributing to its precision. This includes all digits”except: leading and trailing zeros which are merely place holders to indicate the scale of the number. spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.
Identifying significant figuresThe rules for identifying significant figures when writing or interpreting numbers are asfollows: All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5). Zeros appearing anywhere between two non-zero digits are significant.Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2. Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2. Trailing zeros in a number containing a decimal point are significant. For example,12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only sixsignificant figures (the zeros before the 1 are not significant). In addition, 120.00 has fivesignificant figures since it has three trailing zeros. This convention clarifies the precision of suchnumbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23then it might be understood that only two decimal places of precision are available. Stating theresult as 12.2300 makes clear that it is precise to four decimal places (in this case, six significantfigures).
Identifying significant figures The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue: A bar may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten). The last significant figure of a number may be underlined; for example, "2000" has two significant figures. A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant. In the combination of a number and a unit of measurement the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as 1300 g is ambiguous, while in a mass of 13 hg or 1.3 kg it is not.
Chapter # 02 Motion in one dimensionKinematics:- “Kinematics is the branch of mechanics that describes the motionof objects without necessarily discussing what causes the motion.”By specifying the velocity, position and acceleration f a object, wecan describe how this object moves, including the direction of itsmotion. How that direction changes with time, whether the objectspeeds up or slows down and so forth…Position, velocity, acceleration etc… can be found using vectors.
VectorsVector:-“Vector is that quantity that requires magnitude, unit as well as Direction for its complete description…”For example: Displacement, velocity (v) etc… Vector is represented by straight line with an arrow head on its either side… And vector quantities are represented either in Bold or making an arrow on is symbol… A.
zProperties of Vectors Az θ A Representation of a Vector:- Aan a Ax d n d x Ax A cos sen θ A Ax i Ay j Az k Ay Asen sen θ A A Ax2 Ay Az2 2 Az A cos θ
Sum ofVectors A C B C A B Law of the polygon R
Sum of Vectors A C B C A B Law of the polygon RThe resulting vector is one that vector fromthe origin of the first vector to the end of the last
Vectors Properties A A A ˆ -A Opposite Null 0 = A + ()-A A Unit vector μ A
Properties of Commutative the sum of Law vectors R AB BA Difference Associative Law R A-B R A (B C) ( A B) C R A (-B) -B A R B A
Commutative(Methodparallelogram) A B B B The vectors A and B can be displaced parallel to find the vector sum
Multiplication of a vector by a scalar Given two vectors AyB Are said to be parallel if A B si 0 A B si 0 A B si 1 A B
Dot product of two vectors A B AB cos θ Projection of A on B A B A cosθ Projection of B on A B A B cosθ
i i 1 ˆ ˆ i ˆ0 ˆ j ˆ ˆ 1 j j ˆ ˆ i k 0 ˆ ˆ k k 1 j ˆ ˆk 0A i Ax ˆ A ˆ Ay j A B A XB X A YB Y A ZB Z ˆA k Az
Vector product of two vectors C AB C AB senθ ˆˆ 0 i i ˆˆ 0 j j ˆ ˆ k k 0 j ˆ iˆ ˆ k j ˆ ˆ k iˆ ˆ k iˆ ˆ j
Demonstrate: C A B ( A x ˆ A y ˆ A z k) ( B x ˆ B y ˆ B z k) i j ˆ i j ˆ C X AY BZ AZ BY C y Az Bx Ax Bz C z Ax B y Ay Bx
Distance vs DisplacementDistance ( d ) Total length of the path travelled Measured in meters scalar Displacement ( d ) Change in position (x) regardless of path x = xf – xi B Measured in meters vector Displacement Distance A
Finding displacement 1 v area l w bh v – vo = at 2 vo velocity 1 d vot t at 2 t time 1 d v0t 2 at 2
Average VelocityThe displacement divided by the elapsed time. Displaceme nt Average velocity Elapsed time x xo x v t to tSI units for velocity: meters per second (m/s)
Finding velocity x 8m Slope 4 m s t 2s
Instantaneous Velocity & SpeedThe instantaneous velocity indicates how fast the car moves and thedirection of motion at each instant of time. x v lim t 0 tThe instantaneous speed is the magnitude of the instantaneous velocity
AccelerationThe notion of acceleration emerges when a change in velocityis combined with the time during which the change occurs. The difference between the final and initial velocity divided by the elapsed time vv v a o t to t SI units for acceleration: meters per second per second (m/s2)
Finding acceleration v 12 m s Slope 6 m s 2 t 2s
ExampleAcceleration and Increasing Velocity Determine the average acceleration of the plane. vo 0 m s v 260 km h v vo a to 0 s t 29 s t to 260 km h 0 km h km h a 9 .0 29 s 0 s s
ExampleAcceleration and Decreasing Velocity v vo 13 m s 28 m s a t to 12 s 9 s a 5 .0 m s 2
Equations of Kinematics for Constant AccelerationIt is common to dispense with the use of boldface symbolsoverdrawn with arrows for the displacement, velocity, andacceleration vectors (AP does not show arrows on givenequations nor expect them on open-ended problems). Wewill, however, continue to convey the directions with a plusor minus sign. (AP calls elapsed time “t” where t = t – to) v vo v vo a a t to t
Equations of Kinematics forConstant Acceleration v vo a at v v o t AP Equation #1 v v o at
Equations of Kinematics for Constant AccelerationIf, a is constant: x x0 1 1 v 0 v v v o atv v o v t 2 2 x x0 v 1 t x x0 v 0 v t 1 2 x xo v o v o at t 2 AP Equation #2 x xo vot 1 2 2 at
Equations of Kinematics for Constant Acceleration 1 v vo v vo x x0 v 0 v t t a 2 t a v vo x xo 1 2 v o v AP Equation a #3 v vo 2 2x xo v v 2 a x x0 2 2 2a o
Free FallIn the absence of air resistance, it is found that all bodiesat the same location above the Earth fall vertically withthe same acceleration. If the distance of the fall is smallcompared to the radius of the Earth, then the accelerationremains essentially constant throughout the descent.This idealized motion is called free-fall and the accelerationof a freely falling body is called the acceleration due togravity. g 9 . 80 m s 2 2 or 32 . 2 ft s g 10 m s 2 2 or 30 ft s
Freefalling bodiesI could give a boring lectureon this and work throughsome examples, but I’drather make it more real…
Free fall problemsUse same kinematic equations just substitute g for aChoose +/- carefully to make problem as easy as possible
Force Two types of forces ◦ Contact force Force caused by physical contact ◦ Field force Force caused by gravitational attraction between two objects
Isaac Newton Born 1642 Went to University of Cambridge in England as a student and taught there as a professor after Never married Gave his attention mostly to physics and mathematics, but he also gave his attention to religion and alchemy Newton was the first to solve three mysteries that intrigued the scientists ◦ Laws of Motion ◦ Laws of Planetary Orbits ◦ Calculus
Three Laws of Motion Newton’s Laws of Motion are laws discovered by Physicist and mathematician, Isaac Newton, that explains the objects’ motions depending on forces acted on them ◦ Newton’s First Law: Law of Inertia ◦ Newton’s Second Law: Law of Resultant Force ◦ Newton’s Third Law: Law of Reciprocal Action
Newton’s First Law An Object at rest remains at rest, and an object in motion continues in motion with constant velocity (that is, constant speed in a straight line), unless it experiences a net external force. The tendency to resist change in motion is called inertia ◦ People believed that all moving objects would eventually stop before Newton came up with his laws
Friction A force that causes resistance to motion Arises from contact between two surfaces ◦ If the force applied is smaller than the friction, then the object will not move If the object is not moving, then ffriction=Fapplied ◦ The object eventually slips when the applied force is big enough
Friction Friction was discovered by Galileo Galilee when he rolled a ball down a slope and observed that the ball rolls up the opposite slope to about the same height, and concluded that the difference between the initial height and the final height is caused by friction. Galileo also noticed that the ball would roll almost forever on a flat surface so that the ball can elevate to the same height as where it started.
Two types of Friction Static Friction Kinetic Friction ◦ Friction that exists while ◦ The friction that exists the object is stationary when an object is in motion ◦ If the applied force on ◦ F – fkinetic produces an object becomes acceleration to the greater than the direction the object is maximum of static moving friction, then the object ◦ If F = fkinetic, then the object starts moving moves at constant speed ◦ Fstatic ≤ μstatic n with no acceleration ◦ fkinetic= μkineticn ◦ Kinetic friction and the coefficient of kinetic friction are smaller than static friction and the static coefficient
Newton’s First Law When there is no force exerted on an object, the motion of the object remains the same like described in the diagram ◦ Because the equation of Force is F=ma, the acceleration is 0m/s². So the equation is 0N=m*0m/s² ◦ Therefore, force is not needed to keep the object in motion, when ◦ The object is in equilibrium when it does not change its state of motion
The car is traveling rightwardand crashes into a brick wall.The brick wall acts as anunbalanced force and stopsthe car.
The truck stops when it But the ladder falls in frontcrashes into the red car. of the truck because the ladder was in motion with the truck but there is nothing stopping the ladder when the truck stops.
Inertial FramesAny reference frame that moves with constant velocity relative to aninertial frame is itself an inertial frameA reference frame that moves with constant velocity relative to thedistant stars is the best approximation of an inertial frame We can consider the Earth to be such an inertial frame, although it has a small centripetal acceleration associated with its motion
Newton’s First Law – Alternative StatementIn the absence of external forces, when viewed from aninertial reference frame, an object at rest remains atrest and an object in motion continues in motion witha constant velocity Newton’s First Law describes what happens in the absence of a force Does not describe zero net force Also tells us that when no force acts on an object, the acceleration of the object is zero
Inertia and Mass“The tendency of an object to resist any attempt to change its velocity is called inertia”“Mass is that property of an object that specifies how muchresistance an object exhibits to changes in its velocity”Masses can be defined in terms of the accelerationsproduced by a given force acting on them: m1 a2 m2 a1 The magnitude of the acceleration acting on an object is inversely proportional to its mass
Mass vs. Weight Mass and weight are two different quantities Weight is equal to the magnitude of the gravitational force exerted on the object Weight will vary with location Example: wearth = 180 lb; wmoon ~ 30 lb mearth = 2 kg; mmoon = 2 kg
Newton’s Second Law The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass Fnet Acceleration
Unbalanced Force and Acceleration Force is equal to acceleration multiplied by mass ◦ When an unbalanced force acts on an object, there is always an acceleration Acceleration differs depending on the net force The acceleration is inversely related to the mass of the object
Net Force Force is a vector ◦ Because it is a vector, the net force can be determined by subtracting the force that resists motion from the force applied to the object. ◦ If the force is applied at an angle, then trigonometry is used to find the force Fnet
Gravitational Force The force that exerts all objects toward the earth’s surface is called a gravitational force. ◦ The magnitude of the gravitational force is called weight The acceleration due to gravity is different in each location, but 9.80m/s² is most commonly used Calculated with formula w=mg
Newton’s Third Law If two objects interact, the force exerted on object 1 by object 2 is equal in magnitude but opposite in direction to the force exerted on object 2 by object 1 Forces always come in pair when two objects interact ◦ The forces are equal, but opposite in direction Fn Fg
Newton’s Third Law As the man jumps off the boat, he exerts the force on the boat and the boat exerts the reaction force on the man. The man leaps forward onto the pier, while the boat moves away from the pier.
Newton’s Third Law Foil deflected upEngine pushedforward Flow backwardpushed backward Flow Foil deflected down deflected Foil down
Applications of Newton’s LawAssumptions Objects can be modeled as particles Interested only in the external forces acting on the object can neglect reaction forces Initially dealing with frictionless surfaces Masses of strings or ropes are negligible When a rope attached to an object is pulling it, the magnitude of that force is the tension in the rope.