1 Lecture 1 Standards and units Dimensional analysis
- Slides: 70
1 Lecture 1 Standards and units Dimensional analysis Scalars Vectors Operations with vectors and scalars
2 System of units In order to communicate the result of a measurement, one must give units. The units given to mass, length, and time, etc. , form the basis of different systems of units; other units are derived from them Two common systems of units you will encounter are International System SI (aka metric) system and the British System.
3 SI system There are seven (7) fundamental units in the SI system: meter (m) second (s) kilogram (kg) - distance or length (d, x, l) (dimension L) - time (t) (dimension T) - mass (m) (dimension M) Ampere (A) Kelvin (K) mole (mol) candela (cd) - electric current - temperature - amount of substance - intensity of light We will use the SI system in this class. You will have to know the conversion factors from the British System. Ex. : 1 inch = 2. 54 cm; 1 pound ~ 0. 454 kg; 1 mile ~ 1. 61 km
4 Other dimensions The dimensions for all other physical quantities are derived from the fundamental ones: Ex. : - Volume – has the dimension L x L – so the unit is m 3 - Area – has the dimension L x L – unit = m 2 - Velocity – has the dimension L/T – unit = m/s - Density – has the dimension M/Volume – unit = kg/m 3
5 Common multipliers hecto- = 102 centi - = 10 -2 kilo- = 103 milli- mega- = 106 micro- = 10 -6 giga- = 109 nano- = 10 -9 tera- = 1012 pico- = 10 -12 peta- = 1015 femto- = 10 -15 For time we also use: 1 minute = 60 s 1 hour = 60 minutes = 3600 s = 10 -3
6 Dimensional analysis Whenever you solve quantitatively a physics problems make sure you check that the equations yield the correct dimensions for the quantity. Good way to catch errors gross errors, but will not tell you if the quantitative result is correct. Ex. : - In a one-dimensional motion with constant acceleration a (dimension = L/T 2) , the distance traveled during a time interval t (dimension T) is given by the equation: x = v 0*t + ½ a*t 2 (v 0 is the velocity at the start of the time interval)
7 Dimensional analysis (continued) x = v 0*t + ½ a*t 2 - Since x is a distance and has the dimension L each term on the right side of the equation must have the dimension L. Check: v 0*t has dimension (L/T)*T = L - correct ½ a*t 2 has dimension (L/T 2)*T 2 = L - correct (the factor of ½ is a dimensionless constant)
8 To convert a velocity from units of km/h to units of m/s, you must: (A) multiply by 1000 and divide by 60 (B) multiply by 1000 and divide by 3600 (C) multiply by 60 and divide by 600 (D) multiply by 3600 and divide by 1000 (E) none of these is correct
9 If x and t represent position and time, respectively, then the constant A in the equation x = A*cos(B*t) must (A) have the dimensions L/T (B) have the dimensions 1/T (C) have the dimensions L (D) have the dimensions L 2/T 2 (E) be dimensionless
10 The dimensions of two quantities MUST be identical if you are either ______ or ______ the quantities. (A) adding; multiplying (B) subtracting; dividing (C) multiplying; dividing (D) adding; subtracting (E) all of these are correct Hint: - do not mix oranges and apples
11 Scalars Quantities described by a single number (magnitude or absolute value) + unit Ex. : temperature (T) time (t)
Vectors 12 Quantities described not only by magnitude but also by direction. Ex. : velocity ( ) displacement ( ) The arrows indicate that we deal with a vector
Vector properties 13 magnitude Negative of a vector (reciprocal) but Multiplication with a scalar – m and magnitude = has the same direction as
14 Geometrical addition of vectors Mathematically the sum of two vectors Geometrically using the graphical representation: - triangle method (head-to tail) - parallelogram method Note: it is OK to translate a vector parallel to itself
15 Properties of addition Vector addition is commutative: Vector addition is associative:
16 Vector subtraction – subtracting adding to. from is equivalent to
17 If you move east in a straight line 1 km and then north the same distance, how far will you find yourself from the starting point (in a straight line to the origin)? (A) 1 km (B) 3. 22 km (C) 1. 50 km (D) 2 km (E) 1. 41 km
18 VERY IMPORTANT: except if
19 Vector decomposition Less cumbersome technique for vector addition than the geometrical method. Step 1 Choose a system of rectangular coordinates (cartesian coordinates) Step 2 Resolve the vector by projecting it on the x, y (2 -dimensional case) axes, by drawing Perpendicular lines from the two ends of the vector to the axes.
20 Vector decomposition (continued) Step 3 Geometrically the components of the vector will then be: If we know the components of a vector and want to find its magnitude and direction then:
The magnitude of the vector 21 is 3 m and the angle q = 30 o. Which statement is correct: (A) points in the negative x direction; Ay = 1. 5 m (B) points in the positive y direction; Ax = 1. 5 m (C) points in the positive x direction; Ay = 1. 5 m (D) points in the positive x direction; Ax = 1. 5 m (E) none of these is correct y θ x
22 Unit Vectors We have discussed only the algebraic components of the projections ax and ay. However and are vectors , but with well defined orientations along the directions chosen for the system of coordinates. Define a pair of unit vectors parallel with the x and y axis and oriented in their positive direction. - dimensionless
Vector decomposition - Summary 23 Choose a set of orthogonal coordinates and project the vector in its components. Introduce a set of dimensionless unit vectors oriented in the positive direction of the axes.
Examples of vector decomposition 24
Examples of vector decomposition Translate 25
26 The magnitude and direction of a vector are given by: (A) (B) (C) (D) (E) none of the above
27 2 -chances Which diagram best describes vector a cartesian system of coordinates (x, y)? y y x (A) y x (B) y x (C) in y x x (D) (E)
28 Ex. : q = 30 o q = 210 o q = 150 o q =330 o
29 Important – remember this convention - Always measure the angles from the positive direction of the x axis in the counter –clockwise direction. - If you measure the angle clockwise you will have to add a negative sign in order not to lose the information regarding the direction of your vectors in the analysis.
Trigonometric functions Choose a vector of magnitude one: 30 y 1 tanq sinq Then: i. e. the x and y components of the unitary vector give the values of the sin and cos functions. q cosq 1 x
Radians and degrees 31 y q = 90 o q= q q = 180 o q = 0 o x q = 360 o q = 270 o y q q= q=0 x q= q= To convert from degrees to radians (rad) multiply with 2 p and divide by 360. Ex: - convert 60 o to radians
Vector addition using components 32 - Problem - find y y x Step 1 Decompose the vectors on a set of orthogonal axes. y x Step 2 Add algebraically the components on each axis to obtain the components of the sum vector. x Step 3 Construct the sum vector using its components.
33 Example: Calculate the sum of the following two: and First express with components: Then add the components on each axis:
34 Vectors and have the following components: x-component +5 units -6 units y-component -2 units +2 units What are the components of vector (A) Cx = +7 units, Cy = +8 units (B) Cx = +3 units, Cy = +4 units (C) Cx = +3 units, Cy = -4 units (D) Cx = -1 units, Cy = 0 units 2 -chances
35 Vectors and have the following components: x-component +5 units -6 units y-component -2 units +2 units What is the magnitude of vector (A) = -1 units (B) = +1 units (C) = -2. 65 units (D) = +5 units
36 2 -chances Two vector quantities, whose directions can be altered at will, can have a resultant whose length is between the limits 5 and 15. What could the magnitudes of these two vector quantities be? (A) 2 and 3 (B) 5 and 10 (C) 10 and 25 (D) 3 and 12
37 Problem 23 (page 54) Oasis B is 25 km due east of oasis A. Starting from oasis A, a camel walks 24 km in a direction 15 o south of east and then walks 8 km due north. How far is the camel then from oasis B? N y D B A C The displacement left to travel is: x
38 Sample Problem 3 -6 (page 47) Three vectors satisfy the relation. has a magnitude of 22. 0 units and is directed at an angle of -47 o (clockwise) from the positive direction on an x axis. has a magnitude of 17 units and is directed counterclockwise from the positive direction of the x axis by an angle f. is in the positive direction of the x axis. What is the magnitude of ? y f x But:
Sample Problem 3 -6 (page 47) (continued) 39 y x
Coordinate system equivalence The choice of a coordinate system is not unique. The system that we have been using so far is convenient because it looks “proper” (its axes are parallel with the paper or blackboard edges) However the coordinate systems are equivalent since the magnitude and orientation of a vector is not affected by the system in which it is analyzed. 40
Operations of multiplication with vectors 41 Multiplication of a vector with a scalar - the direction of is the same with that of and is opposite if m<0 Multiplication of a vector by a vector - the scalar product - the vector product if m > 0
42 Scalar product The scalar product (or “dot” product) of two vectors is defined as: q If we work with components We used: Can be generalized to 3 – dimensions (see also the textbook)
Scalar product properties The scalar product is commutative The scalar product is distributive Particular cases: 43
Vector product 44 The vector product (or “cross” product) of two vectors is defined as a vector: - with magnitude: The vector is perpendicular on the plane formed by the vectors and its direction is determined by the right hand rule.
Vector product properties The vector product is not commutative. Particular cases: 45
46 Vector product components - using the unit vector properties: - determinant notation We will review the vector product later. Use this for reference.
47 Motion along a straight line (one – dimensional) Definition: motion is defined as the change of an object’s position with time In this chapter we will impose a series of restrictions: - motion is constrained in one dimension, i. e. along a straight line (typically along the x or y axis). - the motion can be either in the positive or in the negative direction of the axis used. - for now, we will neglect the forces (pushes or pulls) that determine an object to move). - the moving object is either a particle or an object that moves like a particle (all its points move in the same direction at the same rate (speed).
Position of an object 48 For a one-dimensional motion the position of the object is specified by a single coordinate x. In order to describe the position of an object: - take a snapshot of the object at different times and record its position. - plot this position as a function of time x = x(t) - you might need to fit the plot to obtain the missing points.
Examples of time dependence of position x (m) t (s) 49 t (s)
50 Displacement A change from one position x 1 to another position x 2: Note: displacement is a vector even if in this chapter we will not specify it all the time. Consequently, make sure that the sign (i. e. direction) is not ignored. but
51 There are four pairs of initial and final positions. Which pair(s) gives a negative displacement: (a) – 3 m, 5 m (b) – 3 m, -7 m (c) 3 m, -3. 4 m (d) – 3 m, 0 m (A) (a) and (b) (B) (b) and (d) (C) (c) and (d) (D) only (b) (E) (b) and (c)
52 For a certain interval of time, is the magnitude of the displacement always equal to the distance traveled? (A) Yes (B) No
Average velocity 53 One characteristic of the motion is the rate of change of the object (particle) position. The initial and final position of the two objects are the same. Average velocity vavg is the ratio of the displacement and the time interval over which it has occurred. - Dimension is: [vavg]= L/T - Unit: m/s
Graphic interpretation of the velocity 54 The velocity is the slope of the straight line that connects the two particular points over which the displacement is calculated. x x 2 Slope of this line x 1 t 2 t
55 According to the following graph, when the two bodies (1) and (2) have the same velocity: x (m) (A) t = 0 s (1) (B) t=5 s (C) t = 10 s (D) Never (E) None of the above (2) t (s) 0 5 10
Average speed 56 Average speed is the ratio of the total distance traveled (Ex. number of meters moved over the time interval Because the time interval is always positive the average speed is always positive
57 Example: A car goes form city A to city B and then returns to city C (see figure). If it takes 10 minutes to drive from A to B, and 14 minutes from B to C, and the car was stationed in city B for half an hour, what are the average velocity and speed? C A B Note: - typically the value of the average velocity and speed are different
Instantaneous velocity 58 To obtain the velocity at any instance, the time interval Dt over which the average velocity is calculated is reduced - v is the derivative of x with respect to t. e = ty n i i s l loc i th ve f o ge e a op ver l S a x(t) - v is the slope of the tangent to the position versus time curve at the point of interest. Speed = the magnitude of instantaneous velocity. It does not contain any indication about the direction of motion.
59 The time dependence of the position of an object is shown in the figure. At which point is the object at rest (zero velocity)? B x (m) (A) D and F (B) A and C (C) B and E (D) B, D, E and F (E) A, C, E and F A C F t (s) D E
60 Acceleration – characterizes the rate of change in the velocity of an object (particle) v v 2 - Dimension is: [aavg]=L/T 2 - Unit: m/s 2 v 1 - aavg – is the slope of this line t 1 t 2 t
Instantaneous Acceleration 61 Instantaneous acceleration – derivative of velocity with respect to time
The acceleration’s sign 62 A negative sign for the acceleration does not necessarily means that the speed of an object is decreasing. Ex. : - an object starts from rest and increases its speed to (– 10 m/s) in 5 s. v x - the acceleration is negative even if the objects accelerates, because the motion is in the negative direction of the x axis Note: - if the sign of the velocity and acceleration of an object are the same, the speed of the object increases. If the signs are opposite the speed decreases.
63 Problem: The position of a car versus time is described by the following graph. a) Find the displacement and total distance traveled in 60 s. b) Plot the time dependence of the velocity c) Calculate the average velocity and average speed d) Calculate the average acceleration in the time interval 0 s to 60 s. e) Plot the time dependence of the acceleration (comment on the result)
64 a) b) c)
d) e) Note: - the measurement of the time dependence of the position was poor and it does not completely reflect the reality. 65
66 More real measurement:
67 What is vx at t = 1 s? (A) +3 m/s (B) +2 m/s (C) -2 m/s (D) -3 m/s (E) None of the above
68 What is vx at t = 3 s? (A) +3 m/s (B) +2 m/s (C) -2 m/s (D) -3 m/s (E) None of the above
2 chances 69 What is the average acceleration between t = 1 s and t = 8 s? (A) ~ 0. 43 m/s 2 (B) ~ 0. 21 m/s 2 (C) ~ - 0. 43 m/s 2 (D) ~ 0 m/s 2 (E) None of the above
Constant acceleration case 70 When the acceleration is constant the average acceleration and instantaneous acceleration are equal: Eq. (1)
- Circular motion is one dimensional or two dimensional
- 01:640:244 lecture notes - lecture 15: plat, idah, farad
- Mussd
- Botox units per ml
- Absorption costing income statement
- Customer service standards table
- Marzano teacher evaluation model
- Laws of similitude
- Dimensional analysis and its applications
- Scientific notation and dimensional analysis
- Pre ap chemistry worksheets
- 2-7 literal equations and dimensional analysis
- Metric conversion scientific notation
- Ap chemistry dimensional analysis worksheet
- How to determine drip rate
- Dimensional analysis dosage calculation
- Dimensional analysis formula
- 5 400 inches to miles dimensional analysis
- Dimensional analysis
- Conantucki island
- Dimensional analysis grams to moles
- What are two dimensional analysis of objectives
- Dimensional analysis metric system
- Dimensional analysis definition math
- Metric conversion dimensional analysis
- Dominos thor
- Dimensional analysis quiz
- Dimensional analysis steps
- Dimensional analysis factor label method
- Dimensional analysis king henry
- Dimensional analysis picket fence method
- Unit 2 measurements
- Dimensionless groups in fluid mechanics
- Dimensional analysis vocabulary
- Khdbdcmm
- Dimensional analysis game
- Dimensional analysis warm up
- Dimensional analysis definition chemistry
- Dimensional analysis definition chemistry
- Metric conversion dimensional analysis
- Dimensional analysis lab
- Factor label method steps
- Conversion problems with answers
- Dimensions in physics
- Dimensional analysis moles
- Dimensional analysis
- Lesson 5: introduction to generators
- Dimensional analysis fluid mechanics
- Dimensional analysis converting one unit to another
- Medical dosage calculations a dimensional analysis approach
- Dimensional analysis factor label method
- Exploratory data analysis lecture notes
- Sensitivity analysis lecture notes
- Factor analysis lecture notes
- Analysis of algorithms lecture notes
- Streak plate method
- Zline 667-36
- Units of analysis
- Unit of analysis examples
- Units of analysis
- 2d motion equations
- Two dimensional shapes alike and different
- In the statement "int *arr[4];",arr is
- Two-dimensional structure composed of rows and columns
- Two-dimensional structure composed of rows and columns
- Hse management standards
- Hse management standards
- Magnetism
- Power system dynamics and stability lecture notes
- Microbial physiology lecture notes
- Limits fits and tolerances