Motion in a Straight Line: Point Object

Concept of point object (particle) in physics

A small piece of matter with no dimension but only a position is called particle.  This is the mathematical or ideal concept of a particle.

But the concept of a point object in physics is different.  In physical system, we can even consider big objects like earth, moon etc as point objects when distance involved is quite large and thus, size is of no importance.

"Thus, an object can be considered as a point object if during its motion in a given time, it covers distances much greater than its own size."

Thus, whether we can treat a real object as point object, it depends on the circumstances.  Let us consider some examples.

Examples

1. While studying the revolution of earth around the sun, earth can be regarded as a point object as diameter of earth is very small as compared to length of its orbit around the sun.
2. A car travelling a few hundred km. may be treated as a point object.
3. A kite flying in the sky

Conclusion:  In physical system, it is not the actual size of the object but the ratio of its size to the distance traveled by it during the motion which dtermines whether or not the object should be treated as point object

In the study of one, two and three dimensional motion, we treat the object as a particle unless stated.

Representation of position of an object in physics

The space is three dimensional. So, we define the position of an object or a point in space by three numbers.  For this, we choose a convenient reference point O and draw a set of three mutually perpendicular axes OX, OY and OZ through O.  This is known as rectangular (Cartesian) coordinate system.  It is clear that point of intersection of three axes called origin (O) serves as the reference point.  The three numbers (x,y,z) - the distances along OX axis, OY axis and OZ axis - will describe the position of the point P in space and we say that co-ordinate of point P are x, y and z.  This coordinate system constitutes a frame of reference.  If all the three coordinate x,y and z of a particle do not change with time, the particle is said to be at rest w.r.t the coordinate system.  But if any one or more of the coordinate change with time, the particle is said to be in motion w.r.t. the coordinate system.

Motion in Straight Line: Rest & Motion

Meaning of Rest and Motion

Rest and motion are the two states of a body.

Rest: A body is said to be rest if it does not change its position with time w.r.t. its surroundings.  Ex. a book lying in air, a man sitting on desk etc.

Motion: A body is said to be in motion if it changes its position with time w.r.t. its surroundings. Ex. a bird flying in air, a man walking on road etc.

The world, and everything in it, moves.  Even seemingly stationary things, such as roadway, move with Earth's rotation, Earth's orbit around the Sun, the sun's orbit around the centre of the Milky Way galaxy, the galaxy's migration relative to other galaxies.

Rest and motion are relative

It means that an object can be rest w.r.t. one object and at the same time can be in motion w.r.t. another object

Example (i) When we are sitting in a moving train compartment, we are at rest w.r.t. the other passenger sitting in it.  But to an observer outside the compartment, we appear to be moving.

(ii) A person sitting in his house is at rest w.r.t. earth but is in motion w.r.t. another planet.

From the above two examples, it is clear that whether the position if of rest or of uniform motion, it is only relative i.e. depends only upon the position of the observer.  Thus, there is no meaning of rest or motion without observer.

Meaning of Absolute Rest and Absolute Motion

Absolute rest is complete absence of motion.  To determine this, we require some 'fixed reference object'.  If a body is at rest w.r.t. this fixed reference object, it will have absolute rest.  But there is no fixed object in this universe.  All heavenly bodies are moving w.r.t. one another.  Thus, absolute rest is beyond our imagination.

Similarly, absolute motion is also practically not possible as we do not have any reference point which is absolutely fixed in space.

Motion in Straight Line: Mechanics

Mechanics

The branch of physics which deals with the study of state of rest and state of motion of material objects is called Mechanics.  It is broadly classified into two categories:

1. Statics
2. Dynamics

Statics

It is the study of object in motion.  It is further subdivided into

1. Kinematics
2. Dynamics proper

Kinematics: It is the study of motion of objects without taking into account the cause of motion.  In kinematics, we do not consider the size, shape, mass etc. of the body.  We deal only with the properties of motion i.e. speed, velocity, accerlation etc.

The term kinematics is derived from Greek word 'Kinema' meaning motion.  Motion is the continuous change of position.

Dynamics proper: It is the study of motion of objects considering the cause of motion.  Since force is the cause of motion, therefore, the study of dynamics involves the study of force.

The word dynamics comes from the Greek word 'dynamics' meaning power.

Importance of Mechanics

By learning a few basic principles of mechanics, we can describe and explain the motion of  planets, build bridges, fly aeroplanes and put satellites into orbits.  Motion of objects along a straight line is also called rectilinear motion.

Units and Measurements: Applications

1. To check the correctness of a physical relation

Example: Show that the expression of the timer period T of a simple pendulum of l given by
T = 2π√l/g is dimensionally correct

2. To convert one system of units to another

This is based on the fact that the product of the numerical value (n) and its corresponding unit (u) is a constant.  Thus n[u] = constant or n₁[u₁] = n₂[u₂]

Let us consider a physical quantity whose dimensional formula is [M^aL^b T^c].  Let n₁ & n₂ be its numerical value in two systems having different units u₁ & u₂

n₁u₁ = n₂u₂

(In 1st system of units)   (In 2nd system of units)

u₁ = [M₁^aL₁^b T₁^c] and u₂ = [M₂^aL₂^b T₂^c]

Thus, n₁ [M₁^aL₁^b T₁^c] = n₂  [M₂^aL₂^b T₂^c]

∴ n_{2 =} n₁ [M₁/M₂]^a[L₁/L₂]^b[T₁/T₂]^c

Thus, if we know n₁ we can calculate n₂. 

3. To find the dimensions of constant in a given physical relation

For this, we again use the principle of homogeneity of dimensions i.e. only like quantities can added or subtracted

4. To derive the relationship between different physical quantities

If we know the factors on which a given physical quantity depends, we can find the relation between the physical quanity and those factors on which it depends by using the principle of homogeneity of dimensions.

4. To derive the forgotten formula

Suppose a person is sure of physical quantities involved in a relation but is doubtful about the powers or a person is reading a book in which an equation has been written whose power are not legible.  In such situations, we can use the principle of homogeneity of dimensions to obtain the correct relation.

Units and Measurement: Principle of homogeneity of dimensions

Principle of homogeneity of dimensions

"A given physical relation is dimensionally correct if the dimensions of the each term on either side of the relation are the same."

This principle is based on the fact that only quantities of the same kind can be added or subtracted.  For example, velocity cannot be added to the force.

This principle is very helpful in checking the correctness of an equation.  If the dimensional formula of each term of the equation is not the same, the equation is wrong.

Example:  Consider the physical relation
s = ut + ½ at²  In this relation, dimensional formula of each term is same i.e.

Dimensional formula for s = L, ut = LT^-1T, ½ at²= LT^-2T² = L

Thus, we find that the given relation is dimensionally correct as each term has the same dimensional formula.  But we must note that dimensional consistency does not guarantee correct equation.

Reason: In dimensional analysis, it does not consider dimensionaless quantities, dimensionaless function (e.g. trigonometric functions, logarithmic functions, exponential functions etc.) pure numer, ratio of two similar quantities etc.

Conclusion: If an equation fails the consistency test, it is wrong.  But if it passes, it is not proved right.  Thus, a dimensionally correct equation may or may not be actually correct, but a dimensionally wrong equation must be wrong.

Uses of dimensional equation 

1. To check the correctness of a physical relation (using principle of homogenity of dimension)
2. To convert one system of units to another
3. To find the dimensions of constants in a given relation
4. To derive relationships between different physical quantities
5. To write a forgotten formula

Units & Measurement: Dimensional Formula

How to obtain dimensional equation

It can be obtained by the relations between base quantities & derived quantities.  Table below gives the dimensional formulas of a few physical quantities:

S. No.	Physical quantity	Relation with other physical quantities	Dimensional Formula
1	Area	Length x breadth	M0L2 T0
2	Volume	Length x breadth x height	M0L3 T0
3	Density	Mass/Volume	M1L-3 T0
4	Speed/Velocity	Distance/time	M0L1 T-1
5	Acceleration	Velocity/time	M0L1 T-2
6	Force	Mass x acceleration	M1L1 T-2
7	Momentum	Mass x velocity	M1L1 T-1
8	Work	Force x distance	M1L2 T-2
9	Power	Work / Time	M1L2 T-3
10	Pressure	Force/Area	M1L-1 T-2
11	Kinetic Energy	½ Mass x Velocity2	M1L2 T-2
12	Potential Energy	Mass x Gravity x Height	M1L2 T-2
13	Impulse	Force x time	M1L1 T-1
14	Torque	Force x distance	M1L2 T-2
15	Stress	Force / Area	M1L-1 T-2
16	Strain	Change in Length / Original Length	M0L0 T0

Units & Measurement: Dimensional formula

Dimensional Formula

When a physical quantity is expressed in terms of fundamental quantities (using M, L, T, A, K, mol, cd), the expression obtained is called dimensional formula of that physical quantity.

Thus, "the dimensional formula of a physical quantity is an expression which shows on which of the fundamental quantities and how, the given physical quantity depends."

Example: The dimensional formulae of density and acceleration are ML^-3 and  MLT^-2 

From dimensional formula of density, we come to know that density depends on 1 & -3 powers of M & L respectively.  It does not depend on time.

Dimensional Equation

"An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity"

Example: Dimensional equations of force (F) and Density (D) are as follows;

[F] = [MLT^-2]

[D] = [MLT^-3]

Note: When we represent a physical quantity in terms of M, L, T etc. (i.e., in terms of base quantitative), the magnitudes are not considered.  We consider only the quality or the type of physical quantity.

Thus, all the following physical quantities will have same dimensional formula [LT^-1]:

• a change in velocity
• initial velocity
• average velocity
• final velocity
• speed
• change in speed etc.

Units and Measurement: Dimensions

Dimensions and Dimensional Analysis

"The dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity"

Example: Force = mass x acceleration = mass x velocity / time = mass x displacement / time x time = mass x length / time²

Thus, force has one dimension in mass, one in length and minus two in time

Similarly, Density = mass / volume = mass / length³ = mass x length^-3

Thus, density has one dimension in mass and minus three in length.

Thus, all physical quantities can be expressed in terms of fundamental quantities.  But for convenience, the fundamental quantities, instead of writing as such, are represented as follows:

Mass by M, length by L, time by T while electric current, temperature, the amount of substance and luminous intensity are denoted by the symbols of their units, A, K, mol and cd.

Thus, all physical quantities can be expressed in terms of M, L, T, A, K, mol, cd.

In mechanics, all physical quantities can be expressed in terms of only three fundamental quantities - M, L and T

Density = mass / length³ = ML^-3

Force = mass x acceleration = mass x length / time² = MLT^-2

ML^-3 and  MLT^-2 are called dimensional formulae of density and force.

Units and Measurement: Physical Quantities

Physical Quantities

Those quantities in terms of which physics is studies and which can be measured directly or indirectly are called physical quantities

Ex.: mass, force, speed, acceleration etc.

Fundamental and derived physical quantities

Physical quantities are often divided into fundamental and derived quantities.  In physics, there are seven fundamental or base quantities.  These are : Mass, length, time, temperature, electric current, luminous intensity and amount of substance.  These are called fundamental quantities because;

1. These are the simplest physical quantities and cannot be broken into more simple physical quantities.
2. All other quantities can be obtained only from these.

Thus, it is clear that fundamental quantities are not defined in terms of other physical quantities and these are independent of other physical quantities

All other quantities (except fundamental quantities) which can be expressed in terms of some suitable combination of seven fundamental quantities are called derived physical quantities.   As all physical quantities are expressed in terms of fundamental quantities, therefore, fundamental quantities or base quantities are called seven dimensions of the physical world.