What are the Different Types of Dimensional Motion?

Physics Topics such as mechanics, thermodynamics, and electromagnetism are fundamental to many other scientific fields.

Reference Frame : Cartesian Frame and Polar Frame

Introduction

Mechanics is the branch of science which deals with the motion of bodies and the effect of force on stationary or moving bodies. The branches of mechanics are as follows:

1. Dynamics: A branch of mechanics in which motion of a body and the effect of force on the motion of the body are discussed.

Kinematics: It is restricted to the study of motion and not the causes of motion.

Kinetics: It incorporates the study and analysis of the motion of a body together with the causes of motion. Also, the mass of a body and effect of force on the mass are studied here.

2. Statics: A branch of mechanics which deals with the equilibrium of a body under the action of a number of forces and studies the conditions of equilibrium.

One-dimensional, two-dimensional and three-dimensional motions:

We will be discussing only one-dimensional motion in this chapter. The motion which is confined to a straight line is called one-dimensional motion or rectilinear motion. This type of motion can be explained by one-dimensional, two-dimensional or three-dimensional coordinates (see section 1.3). The motion of a car moving in a straight line or the motion of a body under gravity are examples of one-dimensional motion.

The motion which is confined to a plane is called two-dimensional motion or planar motion. This type of motion can only be described by two-dimensional or three-dimensional coordinates. The motion of a planet around the sun, a body revolving in a circle, a billiard ball moving over the billiard table etc. are the examples of two-dimensional motion.

The motion which is not confined to a plane is called three-dimensional motion. This type of motion can only be explained by three-dimensional coordinates. The examples of three dimensional motion are the spiral motion of a particle or the motion of an aeroplane.

Rest And Motion

Rest: When a body does not change its position with time, the body is said to be at rest. For example buildings, roads, trees, etc. appear to be in rest.

Motion: When a body changes its position with time, the body is said to be in motion. A moving car, an aeroplane flying in the air, the earth revolving around the sun, etc. are examples of moving objects.

Absolute rest: The earth revolves around the sun and also simultaneously spins on its own axis. It is therefore classified as a moving object. So the plants and buildings on earth which seem to be at rest are actually in states of motion, with respect to the sun and other heavenly bodies. The sun is also in motion with respect to other stars in our galaxy. Different galaxies are also in motion with respect to one another. Hence, nothing can be identified to be in absolute rest in the universe. Rest is an apparent state of an object.

Absolute motion: Motion of an object with respect to other bodies or objects at absolute rest (if it exists) can be defined as absolute motion. But, a body in absolute rest is yet to be known; hence absolute motion too, of an object, cannot be identified.

Reference Frame

We just realised that the concepts of absolute rest and absolute motion are physically meaningless. As a consequence, the states of rest and motion of a body are always studied with respect to sonic other body in the surrounding. This ‘some other body’ provides the frame of reference.

Suitable reference frame: For the study of the position or motion of different objects in nature, a number of different reference frames are available. In every individual case, only one of those reference frames is chosen and that choice is entirely determined by convenience and simplicity of the study of rest and motion. A few examples are

i) Motion of the earth: The sun is the most convenient reference frame. Here, the sun is considered to be at rest, and the motion of the earth and the other planets is studied with reference to the sun.

ii) Motion of objects on the earth: Here, the earth provides the most convenient stationary frame of reference. It then becomes easy to study the rest and motion of almost everything around us that we observe in our daily life. Actually, we instinctively take the earth as stationary and use it as the most advantageous reference frame in our day-to-day observations. When we say that the car is at rest’ or The train is moving’, the reference frame is obviously the earth; we even feel it unnecessary to specify the reference frame at all.

iii) Motion inside a running train: The states of passengers sitting or moving inside a train compartment can be best described by taking the train itself as a stationary frame of reference.

It may be noted that, for the study of rest and motion of any object, the choice of any one of different reference frames is physically allowed. But obviously, every reference frame is not equally convenient. For example, the sun may be chosen as the reference frame to study the motion of a car on the earth. But then, the study would he highly complicated and therefore the choice would be impractical.

Any state of rest or motion is relative. A body at rest in one reference frame may be in motion with respect to another frame of reference. This is in accordance with the concept that, all motions in the universe are realtive, as there exists no absolute rest or absolute motion.

Once a reference frame is conveniently chosen, we come to the task of measurements. For example, we may have to measure the position, velocity, acceleration etc. of a moving body at any instant of time. For this purpose, the reference frame should initially be assigned with an origin, some reference axes and some reference coordinates.

This assignment is different in different types of use; accordingly, the measuring systems are classified as cartesian, plane polar, spherical polar, cylindrical etc. Loosely, the systems are called cartesian reference frame, polar reference frame etc.; but it is important to note that there are often different ways of representing the same frame of reference.

Cartesian Frame of Reference

The reference frame invented by the French mathematician Descartes is called cartesian reference frame or cartesian coordinate system.

One-dimensional reference frame: The instantaneous position of a particle moving in a straight line, can be specified conveniently and sufficiently by the distance x of the particle P from a fixed point O along the straight path OX [Fig.].

The motion in such case is one-dimensional motion and the line OX is a one-dimensional frame of reference.

Two-dimesional reference frame: To denote the position of a particle on a plane, two mutually perpendicular axes OX and OY are taken [Fig.]. In this case, the position of the particle P is uniquely expressed by the coordinates (x, y). The reference frame, constituted of X and Y axes, is called a two-dimensional frame of reference.

If r is the linear distance of the particle P from the origin O at any instant,
r = OP = \(\sqrt{O A^2+A P^2}\) = \(\sqrt{x^2+y^2}\)

Motion of a particle on a fixed plane is called two-dimensional motion. The two-dimensional motion of a particle cannot be described by less than two coordinates, x and y.

Three-dimensional reference frame: This is obtained by drawing three perpendicular lines OX, OY and OZ from a chosen origin O [Fig.]. The position of a particle P in space is completely expressed by the coordinates (x, y, z). This system is known as a three-dimensional reference frame. The linear distance of a particle P from the origin O, is given by r = \(\sqrt{x^2+y^2+z^2}\).

Motion of a particle in a three dimensional space is called three-dimensional motion. The three-dimensional motion of a particle cannot be described by less than three coordinates, x, y and z.

Polar Frame of Reference

This is an alternative choice of frame for two-dimensional motion of particles.

In this reference frame, the position of the particle P is determined in terms of

1. its linear Y distance r, from origin (or pole) O, [Fig.] and

2. the polar angle θ, that the line joining OP subtends with the polar axis OX. The coordinates of P are taken as (r, θ). If the coordinates of P in cartesian frame of reference are (x, y), then from Fig. we can write,
x = rcosθ, y = rsinθ
∴ r = \(\sqrt{x^2+y^2}\) and tanθ = \(\frac{y}{x}\)
The polar coordinates of any point are related to the cartesian coordinates by the above equations.

Spherical Frame of Reference

In this frame of reference, which is three dimensional, the position of the particle P is denoted by

1. the linear distance r of the particle from the origin or pole O,
2. the angle θ between the line OP and the Z-axis and
3. the angle ϕ between OP’ (projection of OP on the XY plane) and X-axis [Fig.]. Hence the coordinates of P are denoted by (r, θ, ϕ), and they are called the spherical polar coordinates.

Note that while r is a linear coordinate, θ and ϕ are angular coordinates. θ and ϕ are called polar angle and azimuthal angle respectively. If it is necessary to define a unique set of spherical coordinates for each point, one must restrict their ranges. The ranges of the three coordinates are -r ≥ 0, 0° ≤ θ ≤ 180° (π radian), 0° ≤ ϕ ≤ 360° (2π radian).

Example: The latitude and longitude of a place on the earth are the angular coordinates of that place. Actually, if the sphere of Fig. is the earth and the axis OZ passes through the north pole, then for a place P on the earth’s surface, latitude = 90° – θ and longitude = ϕ. The direction of the axis OX is defined in such a way that the longitude of Greenwich, London is ϕ = 0.