Tiêu đề Motion in Plane.pdf ✅

1 | P a g e Address: Gayatri Mandir Road, Sudna, Daltonganj (+91 8789806481 / +91 6562352180) By – Vishal Sir Chapter 3: Motion in Plane OXYGENATION TUTORIALS Motion in Plane Scalar and Vectors The physical quantities are broadly classified into two categories: i. Scalar quantities ii. Vector quantities Scalar quantities A physical quantity which has only magnitude but not direction is called scalar quantity. Eg: Mass, distance, speed, electric current, work, etc. Vector quantities A physical quantity which has magnitude as well as direction is called vector quantity. Eg: displacement, velocity, acceleration, force, etc. Magnitude The magnitude of the physical quantity tells us how big or small a physical quantity is. The magnitude of any physical quantity is given as, Q nu = . Where, Q is any physical quantity, n is the numerical value, and u is the unit. Representation of Vector A vector is represented by a line segment (indicating the magnitude) and the arrow a headed (indicating the direction). Types of vectors: i. Equal vectors: When two vectors having the same magnitude as well as same direction are known as equal vectors. • The angle between two parallel vectors are 0 . ii. Anti – parallel vectors: When two vectors having same magnitude but opposite direction. Then it is said to be anti – parallel vector. • The angle between anti – parallel vector is 180 . iii. Zero or null or empty vectors: The vector whose magnitude is zero is called zero vector, or null vector, or empty vector. • It is denoted by 0 . Properties of zero vector: • When a vector is added with zero vector, the resultant remains the same, i.e., A A + = 0 . • When zero vector is subtracted with any vector, the resultant will be the same, i.e., A A − = 0 . • When zero vector is multiplied with any vector, the resultant will zero vector, i.e., A = 0 0 . • When a vector is added with negative of itself, the resultant will zero vector, i.e., A A − = ( ) 0. General representation of a vector A vector is generally represented as, 1 2 3 A a i a j a k ˆ ˆ ˆ = + +
Chapter 3: Motion in Plane 2 | P a g e Address: Gayatri Mandir Road, Sudna, Daltonganj (+91 8789806481 / +91 6562352180) By – Vishal Sir Here, • 1 a is coordinate and ˆ i is the unit vector along x − axis. • 2 a is coordinate and ˆ j is the unit vector along y − axis. • 3 a is coordinate and ˆ k is the unit vector along z − axis. Magnitude of Vector: The magnitude of the vector 1 2 3 A a i a j a k ˆ ˆ ˆ = + + is given as, 222 A A a a a = = + + 1 2 3 Q. Two vectors A i j = + 3 4 ˆ ˆ and B i j = + 4 3 ˆ ˆ . Then find the magnitude of: i. A ii. B iii. A B+ iv. A B− Q. If ˆ ˆ ˆ A i j k = − − 3 4 5 , ˆ ˆ ˆ B i j k = − + 2 3 and C i j = + ˆ ˆ . Find: i. C ii. A B C + + iii. A B C + − iv. A B C + + 2 v. A B C + + 2 3 Q. If ˆ ˆ ˆ A i j ck = + + 0.4 0.8 represent a vector of magnitude 1.2 . Then find the value of c . Unit Vector A vector whose magnitude is unity is called unit vector, i.e., ˆ ˆ ˆ i j k = = =1 In other words, the ratio of any vector with its magnitude is called unit vector. Mathematically, ˆ A A A = Q. If vector A i j = + 3 4 ˆ ˆ . Find the unit vector A ˆ . Q. If vector ˆ ˆ ˆ A i j k = + + 3 4 12 . Find the unit vector A ˆ . Triangle law of vector addition When two vectors are taken along two sides of a triangle, then the third side of it always be in reverse order. The third side is said to be resultant vector (R). The resultant vector R is equal to the vector sum of another two vectors. R A B = + Magnitude and Direction of Resultant Vector Consider a OPQ in which two sides OP and PQ represents two vectors A and B , respectively. Draw a perpendicular QM and PM . According to Pythagoras Theorem, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 1 OQ OM QM OQ OP PM QM R A PM QM R A PM QM = +  = + +  = + +  = + + In PQM , cos cos cos PM PM PM B PQ B    =  =  = .
3 | P a g e Address: Gayatri Mandir Road, Sudna, Daltonganj (+91 8789806481 / +91 6562352180) By – Vishal Sir Chapter 3: Motion in Plane And, sin sin sin QM QM QM B QP B    =  =  = . From equation (1), ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 cos sin cos 2 cos sin cos sin 2 cos 2 cos R A B B R A B AB B R A B AB R A B AB          = + +  = + + +  = + + +  = + + Direction: sin tan tan tan cos QM QM B OM OP PM A B      =  =  = + + Case i: When both vectors are in same direction When A and B are in same direction, i.e.,  = 0 . ( ) 2 2 2 2 2 R A B AB R A B AB R A B R A B = + +   = + +   = +  = + 2 cos0 2 1 Direction, sin 0 0 tan tan tan 0 0 cos0 1 B B A B A B       =  =  =  = +  +  Case ii: When both vectors are in opposite direction When A and B are in opposite direction, i.e.,  =  180 . ( ) ( ) 2 2 2 2 2 R A B AB R A B AB R A B R A B = + +   = + +  −  = −  = − 2 cos180 2 1 Direction, ( ) sin180 0 tan tan tan 0 0 or 180° cos180 1 B B A B A B       =  =  =  = +  +  − Case iii: When both vectors are perpendicular When A and B are perpendicular, i.e.,  =  90 . 2 2 2 2 2 2 R A B AB R A B AB R A B = + +   = + +   = + 2 cos90 2 0 Direction, sin90 1 tan tan tan cos90 0 B B B A B A B A      =  =  = +  +  Q. If two vectors having magnitude 3 and 4 gives the resultant 7 . Then find the angle between them. Q. If two vectors having magnitude 2 and 3 , respectively and the angle between them is 60 . Find the magnitude of the resultant vector. Q. At what angle of force p q + and p q − acts so that the resultant is 2 2 3p q + . Properties of Vector Addition • Vector addition obeys commutative law: A B B A + = + . • Vector addition obeys associative law: ( A B C A B C + + = + + ) ( ). • Vector addition obeys distributive law:    ( A B A B + = + ) ; here  is a constant. Law of parallelogram If two vectors are represented by two adjacent sides of a parallelogram drawn from a common point, then the diagonal of the parallelogram passing through the common point acts as the resultant vector.
Chapter 3: Motion in Plane 4 | P a g e Address: Gayatri Mandir Road, Sudna, Daltonganj (+91 8789806481 / +91 6562352180) By – Vishal Sir R A B = + Polygon law of vector addition If a number of vectors are represented by the sides of an open polygon taken in the same order, then their resultant is represented by the closing side of the polygon taken in opposite direction. R A B C D = + + + Resolution of vector The process of splitting of splitting a vector into two or more vectors in such a way that their combined effect is same as that of the given vector. The vectors into which the given vector is splitter are called component of vectors. The vector A is given as, ˆ ˆ A A i A j = + x y Here, A x is the component of A along x − axis and A y is the component of A along y − axis. Rectangular Component When a vector is resolved along two mutually perpendicular direction, the component so obtained are called rectangular component of the given vector. Rectangular component of a vector in a plane Let us consider a vector A which makes an angle  with x − axis. The component of vector A along with x − axis cos A A x =  . The component of vector A along with y − axis sin A A y =  . Therefore, the resolution of vector is given as, ˆ ˆ ˆ ˆ cos sin A A i A j A i A j = + = + x y  