Tiêu đề VIII -VECTORS.pdf ✅

6 VIII_OLYMPIAD_PHYSICS_VOL-1 MEMO GRAPH Physical Quantities Scalars With units Types of Vectors Without units Operations Vectors Multiplication Scalar Product Vector Product Addition Parallelogram Law Triangle Law Polygon Law Lami’s Theorem
7 VIII_OLYMPIAD_PHYSICS_VOL-1 WILHELM WIEN (1864 - 1928) Wurzbug University, Wurzburg, Germany “for his discoveries regarding the laws governing the radiation of heat’’ VECTORS PART-1 SYNOPSIS-1 1. Physical quantities are classified into scalars and vectors. 2. Scalar Quantities: Physical quantities having only magnitude are called scalars. Examples: Distance, speed, mass, time, temperature, density, work, energy, power,pressure etc., 3. Vector Quantities: Physical quantities having both magnitude and direction and which obey the laws of vector addition (discussed in the next synopsis) are called vectors. Examples : Displacement, velocity, acceleration, linear momentum, force etc., 4. Any directed line segment is a vector which has three characteristics viz; (namely)1) support (base) 2) length (magnitude) and 3) sense (direction). Moreover a vector should follow certain geometric laws. X Y A B Support The direction from A to B is denoted by AB  . The direction from B to A is denoted by BA  AB  implies modulus of the vector or length of the vector or magnitude of vector. (or) If a physical quantity has both magnitude and direction both then it does not always imply that it is a vector. For it to be a vector 1) it should be resolved into mutually perpendicular directions and 2) It should obey the certain geometric laws of vector addition. (which will be taught in course of time) 5. Representation of vector: A vector is represented by a directed line segment. Length of that line segment is proportional to the magnitude of the physical quan- tity which it represents and the arrow of that denotes the direction of vector. For example : If a force of 1 newton is represented by a vector of length 1 cm, then a force of 2 N is represented by a vector of length 2cm. 1 cm : 1 N 2 cm : 2 N 5 cm : 5 N
8 VIII_OLYMPIAD_PHYSICS_VOL-1 A  denotes magnitude of A  . 6. A vector remains unchanged when it is moved parallel to itself. Using this prin- ciple, any vector can be shifted in the same plane. 7. Certain pairs of physical quantities have same units or dimensions but one of those is a scalar and the other is vector. e.g : Speed and velocity, distance and displacement. Types of vectors 1. A  B  A  B  A  B  A  B  Equal vectors: If two vectors have same magnitude and direction they are said to be equal vectors. 2. A  2B   B  A  2B   B  Like vectors:-If two vectors have the same direction but different mag- nitudes, they are said to be like vectors. In the figure A  and B  are parallel vectors as both have same direction and magnitude of A  is twice of magnitude of B  . 3. A  2B   B  A  2B   B  Unlike vectors: If two vectors have opposite directions and different magnitudes, they are said to be unlike vectors. In the figure A  and B  are antiparallel vectors as both have opposite direction and magnitude of A  is twice of magnitude of B  . 4. Negative vector: If two vectors A  and B  have equal magnitude but opposite direc- tions, then each vector is negative vector of the other. i.e., A B     or B A     B  A   A  B  A   A  5. Unit vector: A vector having unit magnitude is called unit vector. If A  is given vector then unit vector in its direction is given by A aˆ A    ( A  or A is the magni- tude of A  )   A A aˆ   aˆ is the unit vector parallel to A  6. A vector parallel to A  and having magnitude same as that of B  is given by B a   .
9 VIII_OLYMPIAD_PHYSICS_VOL-1 7. Zero vector: A vector of zero magnitude is called zero vector or null vector. It is denoted as O  . The initial point and terminal point of a null vector coincide. So, direction of null vector is indeterminate. GUSTAF DALEN (1869 - 1937) Stockholm, Sweden “for his invention of automatic regulators for use in conjunction with gas accumulators for illuminating light houses and buoys’’ Examples: Velocity of a body projected vertically up at the highest point, velocity of bob of a simple pendulum at the extreme position. Properties of zero vector : 1) a o a      2) a b o a b          3) a a o      4) no o    8. Any vector of non zero magnitude is called proper vector. If A  is a proper vector then A 0   . 9. Co–planar vectors: Vectors, acting in the same plane are called co–planar vectors. A  B  C  A  B  C  In the diagram A,B   and C  are coplanar vectors. 10. Non Co–planar vectors: Vectors, acting in different planes are called non-copla- nar vectors. 11. Angle between two vectors: To find angle between two vectors, the two vectors from a point are drawn such that their arrow heads should be away from that point. The angle obtained in this way, is the angle between the vectors. A  B  120° A  B  60° the angle between A  and B  is not 120°. *Whenever angle between two vectors is to be taken we must make sure that either their heads coincide or their tails coincide. If heads coincide or tails coincide then internal angle is the angle between two vectors as in figure (1). If heads coincide with tail then external angle is the angle between the two vectors as in figure (2).