MOTION IN A PLANE

M OTION IN A P LANE . INTRODUCTION In the last chapter we developed the concepts of position, displacement, velocity and acceleration that are needed to describe the motion of an object along a straight line. We found that the directional aspect of these quantities can be taken care of by + and – signs, as in one dimension only two directions are possible. But 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. What is a vector? How to add, subtract and multiply vectors ? What is the result of multiplying a vector by a real number ?

We shall learn this to enable us to use vectors for defining velocity and acceleration in a plane. We then discuss motion of an object in a plane. 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.

We shall discuss uniform circular motion in some detail. The equations developed in this chapter for motion in a plane can be easily extended to the case of three dimensions. . SCALARS AND VECTORS In physics, 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 . Introduction . Scalars and vectors .

Multiplication of vectors by real numbers . Addition and subtraction of vectors — graphical method . Resolution of vectors . Vector addition — analytical method .

Motion in a plane . Motion in a plane with constant acceleration . Projectile motion . Uniform circular motion Summary Points to ponder Exercises just as the ordinary numbers * .

For example, if the length and breadth of a rectangle are . m and . m respectively, then its perimeter is the sum of the lengths of the four sides, . m + .

The length of each side is a scalar and the perimeter is also a scalar. Take another example: the maximum and minimum temperatures on a particular day are . °C and . °C respectively.

Then, the difference between the two temperatures is . °C. Similarly, if a uniform solid cube of aluminium of side cm has a mass of . kg, then its volume is – m (a scalar) and its density is .

× kg m – (a scalar). 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 . So, a vector is specified by giving its magnitude by a number and its direction. Some physical quantities that are represented by vectors are displacement, velocity, acceleration and force.

To represent a vector, we use a bold face type in this book. Thus, a velocity vector can be represented by a symbol v . Since bold face is difficult to produce, when written by hand, a vector is often represented by an arrow placed over a letter, say r v . Thus, both v and r v represent the velocity vector.

The magnitude of a vector is often called its absolute value, indicated by | v | = v . Thus, a vector is represented by a bold face, e.g. by A, a, p, q, r, ... x, y , with respective magnitudes denoted by light face A, a, p, q, r, ...

x, y . . . Position and Displacement Vectors To describe the position of an object moving in a plane, we need to choose a convenient point, say O as origin.

Let P and P ′ be the positions of the object at time t and t ′ , respectively [Fig. . (a)]. We join O and P by a straight line.

Then, OP is the position vector of the object at time t . An arrow is marked at the head of this line. It is represented by a symbol r , i.e. OP = r .

Point P ′ is represented by another position vector, OP ′ denoted by r ′ . The length of the vector r represents the magnitude of the vector and its direction is the direction in which P lies as seen from O. If the object moves from P to P ′ , the vector PP ′ (with tail at P and tip at P ′ ) is called the displacement vector corresponding to motion from point P (at time t ) to point P ′ (at time t ′ ). Fig.

. (a) Position and displacement vectors. (b) Displacement vector P Q and different courses of motion. 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, in Fig. . (b), given the initial and final positions as P and Q , the displacement vector is the same P Q for different paths of journey, say PABC Q , PD Q , and PBEF Q . 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. . . Equality of Vectors Two vectors A and B are said to be equal if, and only if, they have the same magnitude and the