PHY 101: GENERAL PHYSICS I 2013/2014 HARMATTAN SEMESTER Dr. L.A. SUNMONU
COURSE OUTLINE 1 Measurement 2 Motion along a straight line 3 Vectors
1 MEASUREMENT
INTRODUCTION The goal of physics is to provide an understanding of the physical world by developing theories based on experiments. The basic laws of physics involve physical quantities as force, velocity, volume, and acceleration. The 3 most fundamental quantities are Length (L), Mass (M), and Time (T) In order to communicate the result of a measurement of a certain physical quantity, a unit for the quantity must be defined.
MEASUREMENT The physical quantities in the study of mechanics can be expressed in terms of 3 fundamental quantities: length, mass, and time, The SI units for the 3 are meters (m), kilograms (kg), and seconds (s), respectively. Some of the most frequently used metric prefixes representing powers of 10 are shown in Table 1
MEASUREMENT TABLE 1.2 Approximate Values of Some Masses Mass (kg) Observable 1 10 52 Universe Milky Way galaxy 7 10 41 Sun 2 10 30 Earth 6 10 24 Moon 7 10 22 Shark 1 10 2 Human 7 10 1 Frog Mosquito Bacterium Hydrogen atom Electron 1 10 1 1 10 5 1 10 15 2 10 27 9 10 31 TABLE 1.4 Some Prefixes for Powers of Ten Used with Metric (SI and cgs) Units Power Prefix Abbreviation 10 18 atto- a 10 15 femto- f 10 12 pico- p 10 9 nano- n 10 6 micro- 10 3 milli- m 10 2 centi- c 10 1 deci- d 10 1 deka- da 10 3 kilo- k 10 6 mega- M 10 9 giga- G 10 12 tera- T 10 15 peta- P 10 18 exa- E
MEASUREMENT TABLE 1.1 Approximate Values of Some Measured Lengths Length (m) Distance from Earth to most remote known quasar 1 10 26 Distance from Earth to most remote known normal galaxies 4 10 25 Distance from Earth to nearest large galaxy (M31, the Andromeda galaxy) 2 10 22 Distance from Earth to nearest star (Proxima Centauri) 4 10 16 One light year 9 10 15 Mean orbit radius of Earth about Sun 2 10 11 Mean distance from Earth to Moon 4 10 8 Mean radius of Earth 6 10 6 Typical altitude of satellite orbiting Earth 2 10 5 Length of football field 9 10 1 Length of housefly 5 10 3 Size of smallest dust particles 1 10 4 Size of cells in most living organisms 1 10 5 Diameter of hydrogen atom 1 10 10 Diameter of atomic nucleus 1 10 14 Diameter of proton 1 10 15
TABLE 1.3 MEASUREMENT Approximate Values of Some Time Intervals Time Interval (s) Age of Universe 5 10 17 Age of Earth 1 10 17 Average age of college student 6 10 8 One year 3 10 7 One day (time required for one revolution of Earth about its axis) 9 10 4 Time between normal heartbeats 8 10 1 Period a of audible sound waves 1 10 3 Period a of typical radio waves 1 10 6 Period a of vibration of atom in solid 1 10 13 Period a of visible light waves 2 10 15 Duration of nuclear collision 1 10 22 Time required for light to travel across a proton 3 10 24 a A period is defined as the time required for one complete vibration.
MEASUREMENT Meter is defined as the distance travelled by light in vacuum during the interval of 1/299 792 458 second. Kilogram is defined as the mass of a specific platinum-iridium alloy cylinder (kept at the International Bureau of Weights & Measures). Second is defined as 9 192 631 700 times the period of oscillation of radiation from the cesium atom
CONVERSION OF UNITS Often, we need to change the units in which a quantity is expressed. Chain-link conversion method, multiplying by a conversion factor, is used. Converting units is a matter of multiplying the given quantity by a fraction, with one unit in the numerator and its equivalent in the other units in the denominator, arranged so the unwanted units in the given quantity are cancelled out in favor of the desired units.
DIMENSIONAL ANALYSIS Dimension denotes the physical nature of a quantity. Symbols used to specify the dimensions of length, mass & time are [L], [M] & [T] respectively. Dimensional analysis can be used to check equations and to assist in deriving them.
UNCERTAINTY IN MEASUREMENT No physical quantity can be determined with complete accuracy. The concept of significant figures affords a basic method of handling these uncertainties. A significant figure is a reliably known digit, other than a zero, used to locate the decimal point. The two rules of significant figures are as follows: u When multiplying or dividing using two or more quantities, the result should have the same number of significant figures as the quantity having the fewest significant figures. u When quantities are added or subtracted, the number of decimal places in the result should be the same as in the quantity with the fewest decimal places.
PROBLEM-SOLVING STRATEGY Read Problem Draw Diagram Label physical quantities Identify principle(s); list data Choose Equation(s) Solve Equation(s) Substitute known values Check Answer
2 MOTION ALONG A STRAIGHT LINE
MOTION ALONG A STRAIGHT LINE
MOTION ALONG A STRAIGHT LINE 1. Displacement, Velocity, and Speed 2. Instantaneous Velocity and Speed 3. Acceleration 4. One-Dimensional Motion with Constant Acceleration 5. Freely Falling Objects
MOTION ALONG A STRAIGHT LINE As a first step in studying classical mechanics, we describe motion in terms of space and time while ignoring the agents that caused that motion. This portion of classical mechanics is called kinematics. We first define displacement, velocity, and acceleration. Then, using these concepts, we study the motion of objects traveling in one dimension with a constant acceleration.
MOTION ALONG A STRAIGHT LINE In physics we are concerned with three types of motion: translational, rotational, and vibrational. In this chapter, we are concerned only with translational motion. In our study of translational motion, we describe the moving object as a particle regardless of its size.
DISPLACEMENT, VELOCITY & SPEED The motion of a particle is completely known if the particle s position in space is known at all times. If a particle is moving, we can easily determine its change in position. The displacement of a particle is defined as its change in position. x x f x i 1
DISPLACEMENT, VELOCITY & SPEED The average velocity of a particle is defined as the particle s displacement Δx divided by the time interval Δt during which that displacement occurred t v x x t 2 u subscript x indicates motion along the x axis.
DISPLACEMENT, VELOCITY & SPEED The average speed of a particle, a scalar quantity, is defined as the total distance traveled divided by the total time it takes to travel that distance Average speed total distance total time 3 Unlike average velocity, average speed has no direction and hence carries no algebraic sign.
EXAMPLE 2.1 Estimate the average speed of the Apollo spacecraft in m/s, given that the craft took five days to reach the Moon from Earth. (The Moon is 3.8 x 10 8 m from Earth.)
INSTANTANEOUS VELOCITY & SPEED Often we need to know the velocity of a particle at a particular instant in time, rather than over a finite time interval. Instantaneous velocity V x equals the limiting value of the ratio Δx/Δt as Δt approaches zero x v x lim t:0 t x v x lim t:0 t dx dt
EXAMPLE 2.2 A particle moves along the x axis. Its x coordinate varies with time according to the expression x = -4t + 2t 2, where x is in meters and t is in seconds. (a) Determine the displacement of the particle in the time intervals t = 0 to t = 1 s and t = 1 s to t = 3 s. (b) Calculate the average velocity during these two time intervals. x(m) 10 8 6 Slope = 4 m/s 4 2 0 2 Slope = 2 m/s 4 0 1 2 3 4 t(s)
ACCELERATION When the velocity of a particle changes with time, the particle is said to be accelerating. The average acceleration of the particle is defined as the change in velocity v x divided by the time interval t during which that change occurred: t i t f x a x v x t v xf v xi t f t i v = v xi v = v xf (a) 4
EXAMPLE 2.3 The velocity of a particle moving along the x axis varies in time according to the expression: v x = (40-5t 2 ) m/s, where t is in seconds. (a) Find the average acceleration in the time interval t = 0 to t = 2.0 s (b) Determine the acceleration at t = 2.0s v x (m/s) 40 30 20 Slope = 20 m/s 2 10 0 t(s) 10 20 30 0 1 2 3 4
ACCELERATION Instantaneous acceleration equals the derivative of the velocity w.r.t. time It is the slope of the velocity time graph a x lim t:0 v x t dv x dt 5
ONE-DIMENSIONAL MOTION WITH CONSTANT ACCELERATION A very common and simple type of onedimensional motion is that in which the acceleration is constant. a x v x t v xf v xi t f t i v xf v xi a x t This powerful expression enables us to determine an object s velocity at any time t if we know the object s initial velocity and its (constant) acceleration. 6
ONE-DIMENSIONAL MOTION WITH CONSTANT ACCELERATION Because velocity at constant acceleration varies linearly in time, we can express the average velocity in any time interval as the arithmetic mean ial velocity v xi and the final velocity v x v x v xi v xf 2 (for constant a x ) on for average velocity applies only in x f x i v x t 1 2 (v xi v xf )t 7
ONE-DIMENSIONAL MOTION WITH CONSTANT ACCELERATION x f x i v x t 1 2 (v xi v xf )t v xf v xi a x t x f x i 1 2 (v xi v xi a x t)t x f x i 1 2 (v xi v xi a x t)t x f x i v xi t 1 2 a xt 2 8
ONE-DIMENSIONAL MOTION WITH CONSTANT ACCELERATION We can obtain an expression for the final velocity that does not contain a time interval by substituting the value of t from Equation 6 into Equation 7: v xf 2 v xi 2 2a x (x f x i ) 9
ONE-DIMENSIONAL MOTION WITH CONSTANT ACCELERATION v xf v xi a x t x f x i v xi t 1 2 a xt 2 v xf 2 v xi 2 2a x (x f x i )
ONE-DIMENSIONAL MOTION WITH CONSTANT ACCELERATION v xf v xi a x t 1 x f x i (v xi v xf )t 2 1 x f x i v xi t a x t 2 v xf 2 v xi 2 2a x (x f x i ) 2 Velocity as a function of time Displacement as a function of velocity and time Displacement as a function of time Velocity as a function of displacement Note: Motion is along the x axis.
EXAMPLE 2.4 A jet lands on an aircra9 carrier at 140 mi/h ( 63 m/s). (a) What is its acceleragon if it stops in 2.0 s? (b) What is the displacement of the plane while it is stopping?
FREELY FALLING OBJECTS In the absence of air resistance, all objects dropped near the Earth s surface fall toward the Earth with the same constant acceleration under the influence of the Earth s gravity. A freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion. Any freely falling object experiences an acceleration (g) directed downward, regardless of its initial motion. Free-fall acceleration g 9.80 m/s 2
EXAMPLES 3) A golf ball is released from rest at the top of a very tall building. Neglecting air resistance, calculate the position and velocity of the ball after 1.00 s, 2.00 s, and 3.00 s.
t = 2.04 s y max = 20.4 m v = 0 EXAMPLES A stone is thrown from the top of a building with an initial velocity of 20.0 m/s straight upward, at an initial height of 50.0 m above the ground. The stone just misses the edge of the roof on its way down, as shown in this Figure. Determine (a) the time needed for the stone to reach its maximum height, (b) the maximum height, (c) the time needed for the stone to return to the height from which it was thrown and the velocity of the stone at that instant; t = 0, y 0 = 0 v 0 = 20.0 m/s 50.0 m t = 4.08 s y = 0 v = 20.0 m/s t = 5.00 s y = 22.5 m v = 29.0 m/s t = 5.83 s y = 50.0 m v = 37.1 m/s
VECTORS: Outline Coordinate Systems Vector and Scalar Quantities Some Properties of Vectors Components of a Vector and Unit Vectors
3 VECTORS
VECTOR & SCALAR QUANTITIES A scalar quantity is specified by a single value with an appropriate unit and has no direction. Examples are volume, mass, temperature, time intervals, etc. The rules of ordinary arithmetic are used to manipulate scalar quantities. A vector quantity has both magnitude and direction. Examples are velocity, displacement, acceleration, etc.
VECTORS Physics deals with great many physical quantities that have both numerical and directional properties. This needs a special mathematical language VECTORS to describe them. It has magnitude as well as direction. It follows certain rule of combination
VECTORS The simplest VECTOR quantity is displacement. A V E C TO R t h a t r e p r e s e n t s d i s p l a c e m e n t i s c a l l e d DISPLACEMENT vector. DISPLACEMENT vector tells us nothing about path that the particle takes.
VECTOR ALGEBRA Following definitions are fundamental: EQUALITY: Two vectors A and B are equal if they have the same magnitude and the same direction. NEGATIVE: The negative of the vector A is defined as the vector that gives zero when added to A. ADDING VECTORS: Geometrically or Algebraically. SUBTRACTING VECTORS: Vector subtraction makes use of the definition of the negative of a vector. UNIT VECTOR: A unit vector is a dimensionless vector having a magnitude of exactly 1.
LAWS OF VECTOR ALGEBRA Commutative law è A+B = B+A Associative law è (A + B) + C = A + (B + C) Distributive law è m(a + B) = ma + mb Distributive law è (m + n)a = ma + na Commutative law for multiplication è ma = Am Associative law for multiplication è m(na) = (mn)a
VECTOR ADDITION: GEOMETRIC To add vector B to vector A geometrically, draw A on a piece of graph paper to some scale, such as 1cm = 1m, Specify its direction relative a coordinate system. Then draw vector B to the same scale with the tail of B starting at the tip of A. B The resultant vector R = A + B is the vector drawn from the tail of A to the tip of B. This procedure is known as the triangle method of addition. A R = A + B
VECTOR ADDITION: GEOMETRIC A geometric construction can also be used to add more than two vectors. R = A + B + C + D A The resultant Figure vector 3.8 R = A + B + C + D is the vector that completes the polygon. R is the vector drawn from the tail of the first vector to the tip of the last one. D B C
COMPONENTS OF VECTORS tically, One method of adding vectors makes use of the projections of a vector along the axes of a rectangular coordinate system. These projections are called components. : A : : A x A y : tan A y A x A x A cos A y A sin A y y tan θ A = A y A x A Ax 2 Ay 2 O θ A x x
VECTOR ADDITION: ALGEBRAIC Here, most of the time vectors are added algebraically in terms of their components. Suppose: of the time v : : : R A B. Then the components of the resultant vector R are given by: R x A x B x R y A y B y
EXAMPLE 3.2 Find the sum of two vectors A and B lying in the xy-plane and given by: A (2.0i 2.0j) m B (2.0i 4.0j) m
EXAMPLE 3.3 A particle undergoes three consecutive displacements: d 1 (15i 30j 12k) cm, ree consecutive displacement d 2 (23i 14j 5.0k) cm, (15i 30j 12k) cm, Find the components of the re d 3 ( 13i 15j) cm. Find the components of the resultant displacement and its magnitude. displacement and its ma
EXAMPLES: MISCELLANEOUS Cars A and B are travelling in adjacent lanes along a straight road (as shown below). At Gme, t = 0 their posigons and speeds are as shown in the diagram. If car A has a constant acceleragon of 0.6 m/s 2 and car B has a constant deceleragon of 0.46 m/s 2, determine when A will overtake B.
EXAMPLES: MISCELLANEOUS The length of a piece of paper is measured as 297±1 mm. Its width is measured as 209±1 mm. What is the area of one side of the piece of paper?
EXAMPLES: MISCELLANEOUS An aircra9 is trying to fly due north at a speed of 100 ms - 1 but is subject to a cross- wind blowing from the west to the east at 50 ms - 1. What is the actual velocity of the aircra9 relagve to the surface of the earth?
EXAMPLES: MISCELLANEOUS A pargcle starts from origin with a velocity of 15 cms- 1. Its acceleragon is always zero. What is its x- coordinate 3 seconds later?