MOTION IN 2-DIMENSION (Projectile & Circular motion And Vectors)
|
|
- Edith Owens
- 5 years ago
- Views:
Transcription
1 MOTION IN -DIMENSION (Projectile & Circular motion nd Vectors) INTRODUCTION The motion of an object is called two dimensional, if two of the three co-ordinates required to specif the position of the object in space, change w.r.t time. In such a motion, the object moes in a plane. For eample, a billiard ball moing oer the billiard table, an insect crawling oer the floor of a room, earth reoling around the sun etc. Two special cases of motion in Two Dimension are. Projectile motion. Circular motion Projectile n object that is gien an initial elocit obliquel, and that subsequentl follows a path determined b the graitational force (and no other force) acting on it, is called a Projectile. Eamples of projectile motion : cricket ball hit b the batsman for a si bullet fired from a gun. packet dropped from a plane; but the motion of the aeroplane itself is not projectile motion because there are forces other than grait acting on it due to the thrust of its engine. ssumptions of Projectile Motion : We shall consider onl trajectories that are of sufficientl short range so that the graitational force can be considered constant in both magnitude and direction. ll effects of air resistance will be ignored. Earth is assumed to be flat. Projectile Motion : The motion of projectile is known as projectile motion. It is an eample of two dimensional motion with constant acceleration. Projectile motion is considered as combination of two simultaneous motions in mutuall perpendicular directions which are completel independent from each other i.e. horizontal motion and ertical motion. = + Parabolic path = ertical motion + horizontal motion. Tpes of Projectile Motion () Oblique projectile motion () Horizontal projectile motion (3) Projectile motion on an inclined plane. Oblique Projectile Motion Oblique Projection on a Horizontal Surface (a) Change in position ector r gt gt sin r t sin gt. tan cos o t = o r cos = cos P(, ) (b) erage Velocit
2 (c) (d) (e) (f) (g) (h) gt gt sin a tan sin gt cos Instantaneous Velocit gt gt sin sin gt tan tan cos tan {tan (gt / )sec } sin gt tan tan = tan cos Equation of Trajector g ( tan ) sec Time of Flight sin T g Maimum height Range H ma sin g sin R g R is maimum when sin is maimum = 45. sin g cos ngle of Projection of Gien Ratio of Range and Maimum Height ttained R H tan 4 / 4H R tan 76 (when H = R or = ) (i) Projectile Passing Through Two Different Points of same height at Time t and t gt t (j) Speed and ngle of Projection so that Projectile Passes Through Two Gien Points & (, ) ngle of projection ( ) tan. The speed of projection g tan where tan (k) Minimum Velocit of Projection Required to Pass Through a Gien Point
3 min g( ) (l) (m) Critical angle of projection to pass through a gien point. ( c ) tan Position, Time and Speed at n ngular Eleation cos cos sin( ) t gcos sin( )cos gcos (n) Radius of Curature at an Point on the Path of a Projectile r gcos tan sin g. cos. HORIZONTL PROJECTION FROM GIVEN HEIGHT (a) Displacement Y r g r and g tan g tan / g H r = gt X P(, ) (b) Velocit ˆ i-gtj ˆ (c) g g and tan Range R H g (d) Equation of trajector g Eample : bomb is fired from a cannon with a elocit of m/s making an angle of 3 o with the horizontal (g = 9.8 m/s ). (i) What is the time taken b the bomb to reach the highest point? (ii) what is the total time of its motion?
4 (iii) With what speed the bomb will hit the ground and what will be its direction of motion while hitting? (i) What is the maimum height attained b the bomb? () t what distance from the cannon the bomb will hit the ground? Solution : (i) Let u be t o its angle of projection. The time taken b the bomb to reach the highest point is gien b o usin o sin3 t = = 5 s g (ii) The total time of its motion is, T = t = 5 = s. (iii) The bomb will hit the ground with the same speed with which it was fired. Hence its speed of hitting = m/s. lso, the angle of hitting with respect to the horizontal is 3 o. (i) The maimum height attained b the bomb is u sin o h = g = o sin () Horizontal range is R = u sin o sin6 o = m g 9.8 =.7 4 m 3. PROJECTILE ON N INCLINED PLNE (a) (b) Time of flight Range sin( ) T. gcos {sin( ) sin } R gcos gsin g g cos Range is maimum when sin is maimum, that is equal to. R ma. (up the plane) g( sin ) (c) R' ma (down the plane) g( sin ) Condition for retracing the path of a projectile on an inclined plane cos T gsin, where = tan cot cot tan tan (3cot ) Eample : Solution : From a point high enough on an inclined plane, whose rise is 7 in 5, a shot is fired with a elocit of 9.6m/s at an angle of 3 o with the horizontal (a) up the plane, (b) down the plane. Find the range in each case. (g = m/s ). Let be the inclination of the plane. Then sin = 7 4, and cos = 5 5
5 u cos sin (a) Let R be the range up the plane. Then R = gcos Here u = 9.6 m/s and = 3 o Putting the alues we get R = 7.5 m (b) For motion down the plane, u = u cos (3 + ) and a = g sin u = u sin (3 + ) and a = -g cos Then using s = u t + a t and noting that s = we get time of flight T as o usin 3 T = gcos Let R be the range down the plane. Then using s = u t + a t we get u 3 o R = u cos3 sin 3 o o gcos = 5.6 m CIRCULR MOTION When a particle moes in a plane such that its distance from a fied (or moing) point remains constant, then its motion is known as circular motion with respect to that fied (or moing) point. The fied point is called centre, and the distance of particle from it is called radius. Uniform Circular Motion r S r sin (a) (b) erage elocit V a r r sin( / ) t t ngular and linear speeds t O r r r / t and r (c) Change in elocit ( cos ) sin. (d) Centripetal acceleration ( / ) ar t t Eample 3 : Solution : r r. bod of mass kg reoles in a circle of diameter.4 m, making reolutions per minute. Calculate its linear elocit and centripetal acceleration. Putting / t we obtain ar r If the bod makes n reolution per second, then its angular elocit is = n = = /3 rad/s 6 If the radius of the circle is r, then the linear elocit of the bod is = r =. (3) = /3 m/s The centripetal acceleration is a = r = r =. (3) = r r π 9 m/s
6 Non-uniform circular motion Radial acceleration, ar r a T Tangential acceleration, at r Resultant acceleration, a a a R T a R Non-Uniform Circular Motion with constant ngular cceleration t t t. nalog between translator and angular motions in terns of equations u at t S ut at u as t t Eample 4 : Solution : VECTORS car is moing with a speed of 3 m/s on a circular track of radius 5 m. Its speed is increasing at the rate of m/s. Determine the magnitude of its acceleration. The speed of the car moing on a circular track is increasing. Therefore, besides the centripetal acceleration a c, the car has a tangential acceleration a t also. a c and a \t are mutuall at right angles, 3 3 Here a c =.8 m/s r 5 and a t = m/s (gien) resultant acceleration a = a a.8. c =.7 m/s t Introduction The phsical quantities specified completel b their magnitude as well as direction are called ector quantities. The magnitude and direction alone cannot decide whether a phsical quantit is a ector. In addition to the aboe characteristics, a phsical quantit, which is a ector, should follow laws of ector addition. For eample, electric current has magnitude as well as direction, but does not follow laws of ector addition. Hence, it is not a ector. ector is represented b putting an arrow oer it. The length of the line drawn in a conenient scale represents the magnitude of the ector. The direction of the ector quantit is depicted b placing an arrow at the end of the line. For eample : If cm length is equal to km/hr, then ector represents 6 km/hr due east. SCLRS ND VECTORS The point is called initial point or tail and point is called terminal point or head. Scalars Scalars are phsical quantities which are completel described b their magnitude onl. For eample: mass, length, time, temperature energ etc. W N S 3 cm E
7 Vectors Vectors are those phsical quantities haing both magnitude as well as direction and the obes ector algebra (eg. parallelogram law or triangle law of ector addition). For eample: displacement, elocit, acceleration, force, momentum, impulse, electric field intensit etc. Tpes of Vector () Equal ectors : Two ectors and are said to be equal when the hae equal magnitudes and same direction. () Parallel ector : Two ectors and are said to be parallel when (i) oth hae same direction. (ii) One ector is scalar (positie) non-zero multiple of another ector. (3) nti-parallel ectors : Two ectors and are said to be anti-parallel when (i) oth hae opposite direction. (ii) One ector is scalar non-zero negatie multiple of another ector. (4) Collinear ectors : When the ectors under consideration can share the same support or hae a common support then the considered ectors are collinear. (5) Zero ector () : ector haing zero magnitude and arbitrar direction (not known to us) is a zero ector. (6) Unit ector : ector diided b its magnitude is a unit ector. Unit ector for is  (read as cap or hat). Since,  ˆ. Thus, we can sa that unit ector gies us the direction. (7) Orthogonal unit ectors : ˆˆ i, j and ˆk are called orthogonal unit ectors. These ectors must form a Right Handed Triad (It is a coordinate sstem such that when we Curl the fingers of right hand from to then we must get the direction of z along thumb). The ĵ z kˆ î z î, ĵ, ˆk z i ˆ, j ˆ, z zkˆ (8) Polar ectors : These hae starting point or point of application. Eample displacement and force etc. (9) ial Vectors : These represent rotational effects and are alwas along the ais of rotation in accordance with right hand screw rule. ngular elocit, torque and angular momentum, etc., are eample of phsical quantities of this tpe. () Coplanar ector : Three (or more) ectors are called coplanar ector if the lie in the same plane. Two (free) ectors are alwas coplanar.
8 ddition of Vectors (i) Geometrical Method Two ectors a and b ma be added geometricall b drawing them to a common scale Start and placing then head to tail. The ector connecting the tail of the first to the head of the second is the sum ector c. Vector addition is commutatie and obes the associatie law. Start a b a b b a Finish Vector addition is commutatie a b b a a a b c a b Finish ddition of two ectors a and b a a b c b a b b c Vector addition is associatie a b c a b c c Eample : If the position ector of point and are a and b respectiel. Find the position ector of middle point of. b M C Solution: O O OC O O OM a b OM OM a b O a (ii) naltical Method (Parallelogram law of ector addition) If the two ectors a and b are gien such that the resultant ector c of their ector addition is gien b ector a c = a b abcos b sin a b cos a c b b a Parallelogram Law of Vector ddition It is a common error to conclude that if c = a + b, the magnitude of c should be just equal to the magnitude of a plus the magnitude of b. In general, the conclusion is wrong; one can see that c < a + b. The magnitude of the ector sum a + b depends on the magnitudes of a and of b and on the angle between a and b. Onl in the special case in which a and b are parallel; the magnitude of c = a + b equal to the sum of the magnitudes of a and b. contrast, when the ectors are antiparallel the magnitude of c equals the difference of the magnitudes of a and b. Eample : Two forces of 6N and 8N acting at an angle of 6 with each other, pull an object. What single pull would replace the gien forces?
9 Solution : Two forces are drawn from a common origin O, making an angle of 6. O and OC represent the forces 6N and 8N respectiel. The diagonal O represents the resultant R. R = cos 6 = = 48 R =.7N 8 sin6 ngle is gien, tan = 6 8cos6 Which gies, = 34.7 O C Subtraction of ectors : Q P P Q P Q Eample 3 : If the sum of two unit ectors and is also equal to a unit ector, find the magnitude of the ector. Solution: Gien that = R Hence the angle between and is Now PS + cos =+ + () 3 PS = 3. P + Q S Relatie elocit V V V (i) (ii) V V V V V cos (iii) If relatie elocit makes an angle with V then, V sin tan V V cos V V Resolution of ectors The component of F in a direction making an angle is F cos. The other component of F at right angles to F cos is F sin. F F F F = F sin P O F F = F cos Q Eample 4 : force of 3 N is acting at an angle of 6 with the -ais. Determine the components of the forces along and -aes. Solution : F = F sin6 = 3 3 = 5 3 N Y F = 3 N F = F cos6 = 3 = 5 N 6 X
10 Direction cosines (d.c s) of a ector The components are a acos a az acos acos cos, cos, cos are called direction cosines of a ector The ectors will be parallel, if their direction cosines are same. z kˆ ĵ î Eample 5 : If ˆ ˆ a = 3 i + 4 j and ˆ ˆ b = 7 i + 4 j, find the ector haing the same magnitude as b and parallel to a. Solution: Magnitude of a = nd magnitude of b = a 3 4 = 5 b 7 4 = 5 Now a unit ector parallel to a = â 3i ˆ 4j ˆ 5 The ector haing the same magnitude as b and parallel to a = 5 â = 5i ˆ ˆj Scalar product or Dot product. cos a ˆ i a ˆ j a k ˆ if z b ˆi b ˆj b kˆ z. a.b a.b a z.b z Note: ˆi. ˆi ˆj. ˆj k. ˆ kˆ ˆi. ˆj ˆj. ˆi ˆj. kˆ k. ˆ ˆj k.i ˆ ˆ ˆi.kˆ Vector product or Cross product sin nˆ where ˆn = unit ector perpendicular to plane containing and. ˆi ˆj kˆ a a a ˆ i a b a b ˆj a b a b kˆ a b a b b b b z z z z z z Note: ˆi ˆj kˆ ˆj ˆi kˆ ˆj kˆ ˆi kˆ ˆj ˆi kˆ ˆi ˆj ˆi kˆ ˆj ˆi ˆi ˆj ˆj kˆ kˆ kˆ î ĵ Eample 6 : If a = î + 3 ĵ ; b = 4 î + ĵ Find c = a. b Solution : C = ( î + 3 ĵ ).(4 î + ĵ ) = ()(4) + (3)() = 4
11 Eample 7 : If a = î + 3 î ; b = 4 î + ĵ Find d = a b Solution : d = ( î + 3 ĵ ) (4 î + ĵ ) d = () (4) ( î î ) + () () ( î ĵ ) + (3) (4) ( ĵ î ) + (3) () ( ĵ ĵ ) Since î î = ; ĵ ĵ = ; î ĵ = ˆk ; ĵ î = ˆk d = 4 ˆk ˆk = 8 ˆk
Lesson 3: Free fall, Vectors, Motion in a plane (sections )
Lesson 3: Free fall, Vectors, Motion in a plane (sections.6-3.5) Last time we looked at position s. time and acceleration s. time graphs. Since the instantaneous elocit is lim t 0 t the (instantaneous)
More informationVectors Primer. M.C. Simani. July 7, 2007
Vectors Primer M.. Simani Jul 7, 2007 This note gives a short introduction to the concept of vector and summarizes the basic properties of vectors. Reference textbook: Universit Phsics, Young and Freedman,
More informationChapter 3 Motion in a Plane
Chapter 3 Motion in a Plane Introduce ectors and scalars. Vectors hae direction as well as magnitude. The are represented b arrows. The arrow points in the direction of the ector and its length is related
More informationUNDERSTAND MOTION IN ONE AND TWO DIMENSIONS
SUBAREA I. COMPETENCY 1.0 UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS MECHANICS Skill 1.1 Calculating displacement, aerage elocity, instantaneous elocity, and acceleration in a gien frame of reference
More informationPhysics 4A Solutions to Chapter 4 Homework
Physics 4A Solutions to Chapter 4 Homework Chapter 4 Questions: 4, 1, 1 Exercises & Problems: 5, 11, 3, 7, 8, 58, 67, 77, 87, 11 Answers to Questions: Q 4-4 (a) all tie (b) 1 and tie (the rocket is shot
More informationRutgers University Department of Physics & Astronomy. 01:750:271 Honors Physics I Fall Lecture 4. Home Page. Title Page. Page 1 of 35.
Rutgers Uniersit Department of Phsics & Astronom 01:750:271 Honors Phsics I Fall 2015 Lecture 4 Page 1 of 35 4. Motion in two and three dimensions Goals: To stud position, elocit, and acceleration ectors
More informationPhys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole
Phs 221 Chapter 3 Vectors adzubenko@csub.edu http://www.csub.edu/~adzubenko 2014. Dzubenko 2014 rooks/cole 1 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists
More informationISSUED BY K V - DOWNLOADED FROM KINEMATICS
KINEMATICS *rest and Motion are relative terms, nobody can exist in a state of absolute rest or of absolute motion. *One dimensional motion:- The motion of an object is said to be one dimensional motion
More informationPhysics 1: Mechanics
Physics 1: Mechanics Đào Ngọc Hạnh Tâm Office: A1.53, Email: dnhtam@hcmiu.edu.n HCMIU, Vietnam National Uniersity Acknowledgment: Most of these slides are supported by Prof. Phan Bao Ngoc credits (3 teaching
More informationVectors and 2D Kinematics. AIT AP Physics C
Vectors and 2D Kinematics Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels
More informationFigure 17.1 The center of mass of a thrown rigid rod follows a parabolic trajectory while the rod rotates about the center of mass.
17.1 Introduction A body is called a rigid body if the distance between any two points in the body does not change in time. Rigid bodies, unlike point masses, can have forces applied at different points
More informationVECTORS. Vectors OPTIONAL - I Vectors and three dimensional Geometry
Vectors OPTIONAL - I 32 VECTORS In day to day life situations, we deal with physical quantities such as distance, speed, temperature, volume etc. These quantities are sufficient to describe change of position,
More informationPhysics 101. Vectors. Lecture 2. h0r33fy. EMU Physics Department. Assist. Prof. Dr. Ali ÖVGÜN
Phsics 101 Lecture 2 Vectors ssist. Prof. Dr. li ÖVGÜN EMU Phsics Department h0r33f www.aovgun.com Coordinate Sstems qcartesian coordinate sstem qpolar coordinate sstem qfrom Cartesian to Polar coordinate
More informationIntroduction to vectors
Lecture 4 Introduction to vectors Course website: http://facult.uml.edu/andri_danlov/teaching/phsicsi Lecture Capture: http://echo360.uml.edu/danlov2013/phsics1fall.html 95.141, Fall 2013, Lecture 3 Outline
More informationCHAPTER 1 MEASUREMENTS AND VECTORS
CHPTER 1 MESUREMENTS ND VECTORS 1 CHPTER 1 MESUREMENTS ND VECTORS 1.1 UNITS ND STNDRDS n phsical quantit must have, besides its numerical value, a standard unit. It will be meaningless to sa that the distance
More informationGround Rules. PC1221 Fundamentals of Physics I. Coordinate Systems. Cartesian Coordinate System. Lectures 5 and 6 Vectors.
PC1221 Fundamentals of Phsics I Lectures 5 and 6 Vectors Dr Ta Seng Chuan 1 Ground ules Switch off our handphone and pager Switch off our laptop computer and keep it No talking while lecture is going on
More informationChapter 4 MOTION IN TWO AND THREE DIMENSIONS
Chapter 4 MTIN IN TW AND THREE DIMENSINS Section 4-5, 4-6 Projectile Motion Projectile Motion Analzed Important skills from this lecture: 1. Identif the projectile motion and its velocit and acceleration
More informationMOTION IN A PLANE. Chapter Four MCQ I. (a) 45 (b) 90 (c) 45 (d) 180
Chapter Four MOTION IN A PLANE MCQ I 4.1 The angle between A = ˆi + ˆj and B = ˆi ˆj is (a) 45 (b) 90 (c) 45 (d) 180 4.2 Which one of the following statements is true? (a) A scalar quantity is the one
More informationCHAPTER 3: Kinematics in Two Dimensions; Vectors
HAPTER 3: Kinematics in Two Dimensions; Vectors Solution Guide to WebAssign Problems 3.1 [] The truck has a displacement of 18 + (16) blocks north and 1 blocks east. The resultant has a magnitude of +
More informationPhysics 101 Lecture 2 Vectors Dr. Ali ÖVGÜN
Phsics 101 Lecture 2 Vectors Dr. Ali ÖVGÜN EMU Phsics Department www.aovgun.com Coordinate Sstems qcartesian coordinate sstem qpolar coordinate sstem Januar 21, 2015 qfrom Cartesian to Polar coordinate
More information( ) ( ) A i ˆj. What is the unit vector  that points in the direction of A? 1) The vector A is given by = ( 6.0m ) ˆ ( 8.0m ) Solution A D) 6 E) 6
A i ˆj. What is the unit vector  that points in the direction of A? 1) The vector A is given b ( 6.m ) ˆ ( 8.m ) A ˆ i ˆ ˆ j A ˆ i ˆ ˆ j C) A ˆ ( 1 ) ( i ˆ ˆ j) D) Aˆ.6 iˆ+.8 ˆj E) Aˆ.6 iˆ.8 ˆj A) (.6m
More informationMotion in Two and Three Dimensions
PH 1-A Fall 014 Motion in Two and Three Dimensions Lectures 4,5 Chapter 4 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 4 Motion in Two and Three Dimensions In this chapter
More informationMotion in Two and Three Dimensions
PH 1-1D Spring 013 Motion in Two and Three Dimensions Lectures 5,6,7 Chapter 4 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 4 Motion in Two and Three Dimensions In this chapter
More informationa by a factor of = 294 requires 1/T, so to increase 1.4 h 294 = h
IDENTIFY: If the centripetal acceleration matches g, no contact force is required to support an object on the spinning earth s surface. Calculate the centripetal (radial) acceleration /R using = πr/t to
More informationThe Dot Product Pg. 377 # 6ace, 7bdf, 9, 11, 14 Pg. 385 # 2, 3, 4, 6bd, 7, 9b, 10, 14 Sept. 25
UNIT 2 - APPLICATIONS OF VECTORS Date Lesson TOPIC Homework Sept. 19 2.1 (11) 7.1 Vectors as Forces Pg. 362 # 2, 5a, 6, 8, 10 13, 16, 17 Sept. 21 2.2 (12) 7.2 Velocity as Vectors Pg. 369 # 2,3, 4, 6, 7,
More informationChapter 9 Uniform Circular Motion
9.1 Introduction Chapter 9 Uniform Circular Motion Special cases often dominate our study of physics, and circular motion is certainly no exception. We see circular motion in many instances in the world;
More informationSOLUTIONS TO CONCEPTS CHAPTER 2
SOLUTIONS TO CONCPTS CHAPTR 1. As shown in the figure, The angle between A and B = 11 = 9 A = and B = 4m Resultant R = A B ABcos = 5 m Let be the angle between R and A 4 sin9 = tan 1 = tan 1 (4/) = 5 4cos9
More informationFeb 6, 2013 PHYSICS I Lecture 5
95.141 Feb 6, 213 PHYSICS I Lecture 5 Course website: faculty.uml.edu/pchowdhury/95.141/ www.masteringphysics.com Course: UML95141SPRING213 Lecture Capture h"p://echo36.uml.edu/chowdhury213/physics1spring.html
More informationChapter 3 Vectors. 3.1 Vector Analysis
Chapter 3 Vectors 3.1 Vector nalysis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Coordinate Systems... 6 3.2.1 Cartesian Coordinate System... 6 3.2.2 Cylindrical Coordinate
More informationModule 3: Cartesian Coordinates and Vectors
Module 3: Cartesian Coordinates and Vectors Philosophy is written in this grand book, the universe which stands continually open to our gaze. But the book cannot be understood unless one first learns to
More information27 ft 3 adequately describes the volume of a cube with side 3. ft F adequately describes the temperature of a person.
VECTORS The stud of ectors is closel related to the stud of such phsical properties as force, motion, elocit, and other related topics. Vectors allow us to model certain characteristics of these phenomena
More information(a) Taking the derivative of the position vector with respect to time, we have, in SI units (m/s),
Chapter 4 Student Solutions Manual. We apply Eq. 4- and Eq. 4-6. (a) Taking the deriatie of the position ector with respect to time, we hae, in SI units (m/s), d ˆ = (i + 4t ˆj + tk) ˆ = 8tˆj + k ˆ. dt
More information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More informationθ Vman V ship α φ V β
Answer, Key { Homework 3 { Rubin H Landau 1 This print-out should hae 9 uestions. Check that it is complete before leaing the printer. Also, multiple-choice uestions may continue on the next column or
More informationEVALUATE: If the angle 40 is replaces by (cable B is vertical), then T = mg and
58 IDENTIY: ppl Newton s 1st law to the wrecing ball Each cable eerts a force on the ball, directed along the cable SET UP: The force diagram for the wrecing ball is setched in igure 58 (a) T cos 40 mg
More informationdifferent formulas, depending on whether or not the vector is in two dimensions or three dimensions.
ectors The word ector comes from the Latin word ectus which means carried. It is best to think of a ector as the displacement from an initial point P to a terminal point Q. Such a ector is expressed as
More informationChapter 3 Vectors 3-1
Chapter 3 Vectors Chapter 3 Vectors... 2 3.1 Vector Analysis... 2 3.1.1 Introduction to Vectors... 2 3.1.2 Properties of Vectors... 2 3.2 Cartesian Coordinate System... 6 3.2.1 Cartesian Coordinates...
More informationPhysics Department Tutorial: Motion in a Circle (solutions)
JJ 014 H Physics (9646) o Solution Mark 1 (a) The radian is the angle subtended by an arc length equal to the radius of the circle. Angular elocity ω of a body is the rate of change of its angular displacement.
More informationCHAPTER 3 : VECTORS. Definition 3.1 A vector is a quantity that has both magnitude and direction.
EQT 101-Engineering Mathematics I Teaching Module CHAPTER 3 : VECTORS 3.1 Introduction Definition 3.1 A ector is a quantity that has both magnitude and direction. A ector is often represented by an arrow
More informationPearson Physics Level 20 Unit I Kinematics: Chapter 2 Solutions
Pearson Phsics Leel 0 Unit I Kinematics: Chapter Solutions Student Book page 71 Skills Practice Students answers will ar but ma consist of: (a) scale 1 cm : 1 m; ector will be 5 cm long scale 1 m forward
More informationCircular Motion Act. Centripetal Acceleration and. SPH3UW: Circular Motion, Pg 1 -> SPH3UW: Circular Motion, Pg 2. Page 1. Uniform Circular Motion
SPH3UW Centripetal Acceleration and Circular Motion Uniform Circular Motion What does it mean? How do we describe it? What can we learn about it? SPH3UW: Circular Motion, Pg 1 SPH3UW: Circular Motion,
More informationPhysics 2A Chapter 3 - Motion in Two Dimensions Fall 2017
These notes are seen pages. A quick summary: Projectile motion is simply horizontal motion at constant elocity with ertical motion at constant acceleration. An object moing in a circular path experiences
More informationMain points of today s lecture: Example: addition of velocities Trajectories of objects in 2 dimensions: Physic 231 Lecture 5 ( )
Main points of toda s lecture: Eample: addition of elocities Trajectories of objects in dimensions: Phsic 31 Lecture 5 ( ) t g gt t t gt o 1 1 downwards 9.8 m/s g Δ Δ Δ + Δ Motion under Earth s graitational
More informationDownloaded from 3. Motion in a straight line. Study of motion of objects along a straight line is known as rectilinear motion.
3. Motion in a straight line IMPORTANT POINTS Study of motion of objects along a straight line is known as rectilinear motion. If a body does not change its position with time it is said to be at rest.
More informationProjectile Motion and 2-D Dynamics
Projectile Motion and 2-D Dynamics Vector Notation Vectors vs. Scalars In Physics 11, you learned the difference between vectors and scalars. A vector is a quantity that includes both direction and magnitude
More informationONLINE: MATHEMATICS EXTENSION 2 Topic 6 MECHANICS 6.6 MOTION IN A CIRCLE
ONLINE: MAHEMAICS EXENSION opic 6 MECHANICS 6.6 MOION IN A CICLE When a particle moes along a circular path (or cured path) its elocity must change een if its speed is constant, hence the particle must
More informationModule 24: Angular Momentum of a Point Particle
24.1 Introduction Module 24: Angular Momentum of a Point Particle When we consider a system of objects, we have shown that the external force, acting at the center of mass of the system, is equal to the
More informationVector Basics. Lecture 1 Vector Basics
Lecture 1 Vector Basics Vector Basics We will be using vectors a lot in this course. Remember that vectors have both magnitude and direction e.g. a, You should know how to find the components of a vector
More informationKinematics - study of motion HIGHER PHYSICS 1A UNSW SESSION s o t See S&J , ,
1 Kinematics - study of motion HIGHER PHYSICS 1A UNSW SESSION 1 01 s Joe Wolfe s o t See S&J.1-.6, 3.1-3.4, 4.1-4.6 Is this straightforward, or are there subtleties? See Physclips Chs &3 and support pages
More information1-D and 2-D Motion Test Friday 9/8
1-D and -D Motion Test Frida 9/8 3-1 Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocit, force, momentum A scalar has onl a magnitude. Some scalar
More informationMotion Part 4: Projectile Motion
Motion Part 4: Projectile Motion Last modified: 28/03/2017 CONTENTS Projectile Motion Uniform Motion Equations Projectile Motion Equations Trajectory How to Approach Problems Example 1 Example 2 Example
More informationPHY 110 Handout I. Outline. Introduction
Introduction PHY 110 Handout I Phsics is inherentl a science of measurement. It s a science dedicated to the stud of all natural phenomena. Phsics is a science whose objective is to stud the components
More informationSPH4UIW The Circle Centripetal Acceleration and Circular Motion Round Round
SPH4UIW Centripetal Acceleration and Circular Motion The Circle Bablonian Numbers And ou thought our homework was difficult SPH4UIW: Circular Motion, Pg 1 SPH4UIW: Circular Motion, Pg ound ound SPH4UIW:
More information10. The dimensional formula for c) 6% d) 7%
UNIT. One of the combinations from the fundamental phsical constants is hc G. The unit of this epression is a) kg b) m 3 c) s - d) m. If the error in the measurement of radius is %, then the error in the
More informationQ.1. Which one of the following is scalar quantity? Displacement Option Electric field Acceleration Work Correct Answer 4 w = F.ds; it does not have any direction, it s a scalar quantity. Q.. Which one
More informationMethods of Integration
U96-b)! Use the substitution u = - to evaluate U95-b)! 4 Methods of Integration d. Evaluate 9 d using the substitution u = + 9. UNIT MATHEMATICS (HSC) METHODS OF INTEGRATION CSSA «8» U94-b)! Use the substitution
More informationvector of point will be and if the point P is in a space and its coordinates are (x, y, z) then position vector can be expressed as
2.1 Motion in One Dimension : Position Position of any point is completely expressed by two factors : Its distance from the observer and its direction with respect to observer. That is why position is
More informationPHYSICS 221, FALL 2010 EXAM #1 Solutions WEDNESDAY, SEPTEMBER 29, 2010
PHYSICS 1, FALL 010 EXAM 1 Solutions WEDNESDAY, SEPTEMBER 9, 010 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively. In
More informationOn my honor, I have neither given nor received unauthorized aid on this examination.
Instructor(s): Field/Furic PHYSICS DEPARTENT PHY 2053 Exam 1 October 5, 2011 Name (print, last first): Signature: On my honor, I hae neither gien nor receied unauthorized aid on this examination. YOUR
More informationLesson 6: Apparent weight, Radial acceleration (sections 4:9-5.2)
Beore we start the new material we will do another Newton s second law problem. A bloc is being pulled by a rope as shown in the picture. The coeicient o static riction is 0.7 and the coeicient o inetic
More informationPhysics 40 Chapter 3: Vectors
Physics 40 Chapter 3: Vectors Cartesian Coordinate System Also called rectangular coordinate system x-and y- axes intersect at the origin Points are labeled (x,y) Polar Coordinate System Origin and reference
More informationMotion in Two and Three Dimensions
chapter 4 Motion in Two and Three Dimensions Projectile motion (Section 4.3) 1. Which target got hit first? Contet of the tetbook: Before Eample 4. 2. Projectile range problem comparable to Eample 7, ecept
More informationMiscellaneous (dimension, angle, etc.) - black [pencil] Use different colors in diagrams. Body outline - blue [black] Vector
1. Sstems of orces & s 2142111 Statics, 2011/2 Department of Mechanical Engineering, Chulalongkorn Uniersit bjecties Students must be able to Course bjectie Analze a sstem of forces and moments Chapter
More informationMotion in 2- and 3-dimensions. Examples: non-linear motion (circles, planetary orbits, etc.) flight of projectiles (shells, golf balls, etc.
Motion in 2- and 3-dimensions Examples: HPTER 3 MOTION IN TWO & THREE DIMENSIONS General properties of vectors the displacement vector position and velocity vectors acceleration vector equations of motion
More informationVectors. Introduction. Prof Dr Ahmet ATAÇ
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both n u m e r i c a l a n d d i r e c t i o n a l properties Mathematical operations of vectors in this chapter A d d i t i o
More informationChapter 1: Kinematics of Particles
Chapter 1: Kinematics of Particles 1.1 INTRODUCTION Mechanics the state of rest of motion of bodies subjected to the action of forces Static equilibrium of a body that is either at rest or moes with constant
More informationA. unchanged increased B. unchanged unchanged C. increased increased D. increased unchanged
IB PHYSICS Name: DEVIL PHYSICS Period: Date: BADDEST CLASS ON CAMPUS CHAPTER B TEST REVIEW. A rocket is fired ertically. At its highest point, it explodes. Which one of the following describes what happens
More informationPHYSICS 221, FALL 2009 EXAM #1 SOLUTIONS WEDNESDAY, SEPTEMBER 30, 2009
PHYSICS 221, FALL 2009 EXAM #1 SOLUTIONS WEDNESDAY, SEPTEMBER 30, 2009 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively.
More informationAP Physics C Mechanics Vectors
1 AP Physics C Mechanics Vectors 2015 12 03 www.njctl.org 2 Scalar Versus Vector A scalar has only a physical quantity such as mass, speed, and time. A vector has both a magnitude and a direction associated
More informationKinematics in Two Dimensions; Vectors
Kinematics in Two Dimensions; Vectors Vectors & Scalars!! Scalars They are specified only by a number and units and have no direction associated with them, such as time, mass, and temperature.!! Vectors
More informationCHAPTER 10 VECTORS POINTS TO REMEMBER
For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two
More informationCIRCULAR MOTION EXERCISE 1 1. d = rate of change of angle
CICULA MOTION EXECISE. d = rate of change of angle as they both complete angle in same time.. c m mg N r m N mg r Since r A r B N A N B. a Force is always perpendicular to displacement work done = 0 4.
More informationChapter 4. Motion in Two Dimensions
Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail Will treat projectile motion and uniform circular motion
More informationVersion 001 HW#5 - Magnetism arts (00224) 1
Version 001 HW#5 - Magnetism arts (004) 1 This print-out should hae 11 questions. Multiple-choice questions may continue on the net column or page find all choices before answering. Charged Particle in
More informationChapter 3 MOTION IN A PLANE
Chapter 3 MOTION IN A PLANE Conceptual Questions 1. No; to be equal the must also hae the same direction. If the magnitudes are different, the cannot be equal.. (a) Yes, since the direction matters. One
More informationVECTORS IN THREE DIMENSIONS
1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude
More informationDO PHYSICS ONLINE. WEB activity: Use the web to find out more about: Aristotle, Copernicus, Kepler, Galileo and Newton.
DO PHYSICS ONLINE DISPLACEMENT VELOCITY ACCELERATION The objects that make up space are in motion, we moe, soccer balls moe, the Earth moes, electrons moe, - - -. Motion implies change. The study of the
More informationهکانیک تحلیلی 1 درس اول صحرایی گر ه فیسیک دانشگاه رازی.
هکانیک تحلیلی 1 درس اول صحرایی گر ه فیسیک دانشگاه رازی http://www.rai.ac.ir/sahraei References: هنابع: naltical Mechanics Grant R. Fowles هکانیک تحلیلی فا لس ترجوو دکتر جعفر قیصری هرکس نشر دانشگاىی چاپ
More informationNiraj Sir SOLUTIONS TO CONCEPTS CHAPTER 3
SOLUTIONS TO ONEPTS HPTER 3 1. a) Distance travelled = 50 + 40 + 0 = 110 m b) F = F = D = 50 0 = 30 M His displacement is D D = F DF 30 40 50m In ED tan = DE/E = 30/40 = 3/4 = tan 1 (3/4) His displacement
More informationUnit 11: Vectors in the Plane
135 Unit 11: Vectors in the Plane Vectors in the Plane The term ector is used to indicate a quantity (such as force or elocity) that has both length and direction. For instance, suppose a particle moes
More information2- Scalars and Vectors
2- Scalars and Vectors Scalars : have magnitude only : Length, time, mass, speed and volume is example of scalar. v Vectors : have magnitude and direction. v The magnitude of is written v v Position, displacement,
More information8.0 Definition and the concept of a vector:
Chapter 8: Vectors In this chapter, we will study: 80 Definition and the concept of a ector 81 Representation of ectors in two dimensions (2D) 82 Representation of ectors in three dimensions (3D) 83 Operations
More informationChapter 8 Vectors and Scalars
Chapter 8 193 Vectors and Scalars Chapter 8 Vectors and Scalars 8.1 Introduction: In this chapter we shall use the ideas of the plane to develop a new mathematical concept, vector. If you have studied
More informationPhysics Electricity and Magnetism Lecture 09 - Charges & Currents in Magnetic Fields Y&F Chapter 27, Sec. 1-8
Phsics 121 - Electricit and Magnetism Lecture 09 - Charges & Currents in Magnetic Fields Y&F Chapter 27, Sec. 1-8 What Produces Magnetic Field? Properties of Magnetic ersus Electric Fields Force on a Charge
More informationThe Magnetic Force. x x x x x x. x x x x x x. x x x x x x q. q F = 0. q F. Phys 122 Lecture 17. Comment: What just happened...?
The Magnetic orce Comment: I LOVE MAGNETISM q = qe + q q Comment: What just happened...? q = 0 Phys 122 Lecture 17 x x x x x x q G. Rybka Magnetic Phenomenon ar magnet... two poles: N and S Like poles
More informationQ1. The density of aluminum is 2700 kg/m 3. Find the mass of a uniform solid aluminum cylinder of radius cm and height cm.
Coordinator: W. Al-Basheer Sunday, June 28, 2015 Page: 1 Q1. The density of aluminum is 2700 kg/m 3. Find the mass of a uniform solid aluminum cylinder of radius 10.00 cm and height 30.48 cm. A) 25.85
More informationPhysics 1A. Lecture 3B. "More than anything else... any guy here would love to have a monkey. A pet monkey." -- Dane Cook
Physics 1A Lecture 3B "More than anything else... any guy here would love to have a monkey. A pet monkey." -- Dane Cook Trajectories Since there is no horizontal acceleration (a x = 0) the horizontal position,
More informationPhys101 First Major-111 Zero Version Monday, October 17, 2011 Page: 1
Monday, October 17, 011 Page: 1 Q1. 1 b The speed-time relation of a moving particle is given by: v = at +, where v is the speed, t t + c is the time and a, b, c are constants. The dimensional formulae
More informationKing Fahd University of Petroleum and Minerals Physics Department Physics 101 Recitation Term 131 Fall 013 Quiz # 4 Section 10 A 1.50-kg block slides down a frictionless 30.0 incline, starting from rest.
More informationMAGNETIC EFFECTS OF CURRENT-3
MAGNETIC EFFECTS OF CURRENT-3 [Motion of a charged particle in Magnetic field] Force On a Charged Particle in Magnetic Field If a particle carrying a positie charge q and moing with elocity enters a magnetic
More informationComponents of a Vector
Vectors (Ch. 1) A vector is a quantity that has a magnitude and a direction. Examples: velocity, displacement, force, acceleration, momentum Examples of scalars: speed, temperature, mass, length, time.
More informationChapter 17 Two Dimensional Rotational Dynamics
Chapter 17 Two Dimensional Rotational Dynamics 17.1 Introduction... 1 17.2 Vector Product (Cross Product)... 2 17.2.1 Right-hand Rule for the Direction of Vector Product... 3 17.2.2 Properties of the Vector
More informationPlease Visit us at:
IMPORTANT QUESTIONS WITH ANSWERS Q # 1. Differentiate among scalars and vectors. Scalars Vectors (i) The physical quantities that are completely (i) The physical quantities that are completely described
More information2-9. The plate is subjected to the forces acting on members A and B as shown. If θ = 60 o, determine the magnitude of the resultant of these forces
2-9. The plate is subjected to the forces acting on members A and B as shown. If θ 60 o, determine the magnitude of the resultant of these forces and its direction measured clockwise from the positie x
More informationVectors a vector is a quantity that has both a magnitude (size) and a direction
Vectors In physics, a vector is a quantity that has both a magnitude (size) and a direction. Familiar examples of vectors include velocity, force, and electric field. For any applications beyond one dimension,
More informationVector Analysis 1.1 VECTOR ANALYSIS. A= Aa A. Aa, A direction of the vector A.
1 Vector nalsis 1.1 VECTR NYSIS Introduction In general, electromagnetic field problem involves three space variables, as a result of which the solutions tend to become complex. This can be overcome b
More informationVectors. Slide 2 / 36. Slide 1 / 36. Slide 3 / 36. Slide 4 / 36. Slide 5 / 36. Slide 6 / 36. Scalar versus Vector. Determining magnitude and direction
Slide 1 / 3 Slide 2 / 3 Scalar versus Vector Vectors scalar has only a physical quantity such as mass, speed, and time. vector has both a magnitude and a direction associated with it, such as velocity
More informationGeometry review, part I
Geometr reie, part I Geometr reie I Vectors and points points and ectors Geometric s. coordinate-based (algebraic) approach operations on ectors and points Lines implicit and parametric equations intersections,
More informationMATHEMATICS. Time allowed : 3 hours Maximum Marks: 100
MATHEMATICS Time allowed : 3 hours Maimum Marks: 00 General Instructions:. All questions are compulsory.. This question paper contains 9 questions. 3. Questions 4 in Section A are very short-answer type
More informationMotion In Two Dimensions
Motion In Two Dimensions 1. Projectile motion is: (a) One dimensional (b) Two dimensional (c) Three dimensional (d) Multi-dimensional 2. For projectile motion: (a) A body must be thrown vertically (b)
More information