Motion in a plane || motion in 2-D || 2-D motion || Vectors || Chapter 4 Physics || Notes ||

             1. Introduction

In order to describe motion of an

object in two dimensions (a plane) or three dimensions (space), we need to use vectors to describe the above-mentioned physical quantities. Therefore, it is first necessary to learn the language of vectors. As a simple case of motion in a plane, we shall discuss motion with constant acceleration and treat in detail the projectile motion. Circular motion is a familiar class of motion that has a special significance in daily-life situations.

       2. Scalars and vectors

We can classify quantities as scalars or

vectors. Basically, the difference is that a direction is associated with a vector but not with a scalar.

 A scalar quantity is a quantity with magnitude only. It is specified completely by a single number, along with the proper unit. Examples are : the distance between two points,mass of an object, the temperature of a body and the time at which a certain event happened. The rules for combining scalars are the rules of ordinary algebra.

Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers. 

A vector quantity is a quantity that has both a magnitude and a direction and obeys the triangle law of addition or equivalently the parallelogram law of addition.

2.1 Position and Displacement Vectors

It is important to note that displacement

vector is the straight line joining the initial and final positions and does not depend on the actual path undertaken by the object between the two positions. For example, given the initial and final positions as P and Q, the displacement vector is the same PQ for different paths of journey, say PABCQ, PDQ, and PBEFQ. Therefore, the magnitude of 

displacement is either less or equal to the path length of an object between two points. 

 This fact was emphasised in the previous chapter also while discussing motion along a straight line.

2.2 Equality of Vectors

Two vectors A and B are said to be equal if, and only if, they have the same magnitude and the same direction.

  3. MULTIPLICATION OF VECTORS BY 

                   Real NUMBERS

Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A :

 λ A  = λ  A    ,  if λ > 0.

For example, if A is multiplied by 2, the resultant vector 2A is in the same direction as A and has a magnitude twice of |A|.

Multiplying a vector A by a negative number −λ gives another vector whose direction is opposite to the direction of A and whose magnitude is λ times |A|.

The factor λ by which a vector A is multiplied could be a scalar having its own physical dimension. Then, the dimension of λ A is the product of the dimensions of λ and A. For example, if we multiply a constant velocity vector by duration (of time), we get a displacement vector.

     4. ADDITION AND SUBTRACTION OF

                           VECTORS 

4.1 Addition of vectors

The lengths of the line segments representing these vectors are proportional to the magnitude of the vectors. To find the sum A + B, we place vector B so that its tail is at the head of the vector A. Then, we join the tail of A to the head of B. This line OQ represents a vector R, that is, the sum of the vectors A and B. Since, in this procedure of vector addition, vectors are arranged head to tail, this graphical method is called the head-to-tail method. The two vectors and their resultant form three sides of a triangle, so this method is also known as triangle method of vector addition. If we find the resultant of B + A, the same vector R is obtained. Thus, vector addition is commutative:

               A + B = B + A 

The addition of vectors also obeys the associative law as illustrated in Fig. 4.4(d). The result of adding vectors A and B first and then adding vector C is the same as the result of adding B and C first and then adding vector A :

            (A + B) + C = A + (B + C) 

4.2 Subtraction of vectors

Subtraction of vectors can be defined in terms of addition of vectors. We define the difference of two vectors A and B as the sum of two vectors A and –B :

                         A – B = A + (–B)

The vector –B is added to vector A to get R2 = (A – B). The vector R1 = A + B is also shown in the same figure for comparison. We can also use the parallelogram method to

find the sum of two vectors. Suppose we have two vectors A and B. To add these vectors, we bring their tails to a common origin O. Then we draw a line from the head of A parallel to B and another line from the head of B parallel to A to complete a parallelogram OQSP. Now we join the point of the intersection of these two lines to the origin O. The resultant vector R is directed from the common origin O along the diagonal (OS) of the parallelogram . The triangle law is used to obtain the resultant of A and B and we see that the two methods yield the same result. Thus, the two methods are

equivalent.

      5. VECTOR ADDITION – ANALYTICAL

                              METHOD

Although the graphical method of adding vectors helps us in visualising the vectors and the resultant vector, it is sometimes tedious and has limited accuracy. It is much easier to add vectors by combining their respective components. Consider two vectors A and B in x-y plane with components Ax

, Ay

 and Bx

, By :

A î j= + A A x y

B î j= + B B x y

Let R be their sum. We have

R = A + B

 =+++ ( AA BB xy xy ) 

WORK, POWER and ENERGY

LAWS OF MOTION

MOTION IN A STRAIGHT LINE

HIMANSHU CHAUDHARY