The discovery of the laws of motion was a dramatic moment in the history of science. Before Newton’s time, the motions of things like the planets were a mystery, but after Newton there was complete understanding.

The contribution of Newton was three laws:

Þ      The first law describes what happens when the body is left alone or is not disturbed.

Þ      The second law gives a specific way of determining how the velocity changes under different influences called forces.

Þ      The third law characterizes the basic nature of force.

OBJECTIVE

In this chapter we will study the three important laws of motion given by Newton. The study includes the dynamics of particle under influence of different forces.

After studying this chapter we will be able to answer the question like Why can’t we drive fast on icy road, in cricket how hard should a player hit the ball for a sixer or how does a small engine pull a large train and many other similar questions.

PREQUISITE

We know by experience that all bodies in nature interact in some way with one another.

Before actually going into details of the three Laws of motion we need to get familiar with  “force”. It was believed and has been accepted that “force” is an external or internal agent present to “influence” the natural state of motion of an object. So this is an influence (force) needed to change the natural state of body; that is of rest or of uniform motion.

Having this idea of force we can now readily move to the Newton’s first law of motion.

Newton’s first law of motion

It states that every object persists in its natural state of motion i.e. continues to be at rest or moves in a straight line with uniform (constant) velocity. (This is what is meant by natural state of motion); In the absence of a net external force acting (impressed) on it.

Mathematically, it is equivalent to say that for producing acceleration (that is for changing velocity) in a body, we need to have a net external force. (By net external force we mean vector sum of all the external forces acting on it)

It can be easily deduced from the statement of change in the state of motion. It is directly related to a frame of reference about which we have discussed earlier. To mark the point here, we can discover that by viewing objects from different frame of references the natural state of motion as perceived by different observers will be obviously different (can only be same if the frames are truly equivalent). Therefore, the change in state will also depend on the choice of reference frame. Finally, the amount of acceleration produced in a body (or change in velocity) will depend on our choice of reference frames.

A mass ‘M’ is lying (figure 3.1) on a table which is at rest (w.r.t. table). Explain its state with the help of Newton’s First Law of motion.

Since ‘M’ is lying on a table, there is no external force acting on it (forget about gravity just for the immediate discussion). As per Newton’s first law of motion it will keep on lying at rest with respect to table for infinite time.

Here, comes out a very important, intrinsic (that is inherent) property of a body which is that it retains its state of motionlessness (as well as of motion, if its in motion) which is termed as INERTIA of an object. This is present in all materialistic bodies in this universe.

Now we shall discuss another example of two-dimensional motion that is motion of a particle on a circular path. This type of motion is called circular motion.

Consider a particle P is moving on circle of radius r on X – Y plane with origin O as centre.

The position of the particle at a given instant may be described by angle q, called angular position of the particle, measured in radian. As the particle moves on the path, its angular position (q) changes. The rate of change of angular position is called angular velocity (w) measured in radian per second.

The rate of change of angular velocity is called angular acceleration, measured in rad/s². Thus, the angular acceleration is

RELATION BETWEEN DIFFERENT PARAMETERS OF CIRCULAR MOTION

ü        It is easy to derive the equations of rotational kinematics for the case of constant angular acceleration with fixed axis of rotation. These equations are of the same form as those for one-dimensional translational motion.

w = w₀ +αt                                                             …(i)

Φ = Φ₀ + w₀t + αt²/2                                 …(ii)

w² = w₀² + 2α (Φ- Φ₀)                             …(iii)

Φ= Φ₀ + (w₀ +w)/(2t)                               …(iv)

Here,Φ is the initial angle and w₀ is the initial angular speed.

Illustrations

1.     (a) What is the angular velocity of the minute and hour hands of a clock?

(b) Suppose the clock starts malfunctioning at 7 AM which decelerates the minute hand at the rate of 4p radians/day. How much time would the clock loose by 7 AM next day?

Sol. Angular speed of

minute hand : wmh = 2Π rad/hr

= 48Π rad/day = (Π/1800) rad/sec

hour hand : whh = (Π/6) rad/hr

= 4Π rad/day = (Π/21600) rad/sec

(b)     Assume at t = 0, Φ₀ = 0, when the clock begins to malfunction. Use equation (ii) to get the angle covered by the minute hand in one day.

Hence the minute hand complete 23 revolutions, so the clock losses 1 hour.

along the vertical y-axis with a uniform downward acceleration ‘g’ and

• along the horizontal x-axis with a uniform velocity forward.

Consider a particle projected with an initial velocity u at an angle α with the horizontal x-axis as shown in figure 2.17. Velocity and accelerations can be resolved into two components:

Velocity along x-axis = u_x = u cos α

Acceleration along x-axis a_x = 0

Velocity along y-axis = u_y = u sin α

Acceleration along y-axis a_y = -g

Here we use different equation of motions of one dimension derived earlier to get the different parameters.

v² = v₀² – 2g (y – y₀)                                …(C)

Total Time of flight

When body returns to the same horizontal level, the resultant displacement in vertical y-direction is zero. Use equation B.

Therefore,             0 = (u sin α) t – (½)gt²

or                   t = 2uSinα / g

(as t cannot equal to 0)

Horizontal Range

Horizontal Range (OA)  = Horizontal velocity × Time of flight

=        u cos α × 2u sin α/g

= u² sin 2α/g

=

Maximum Height

At the highest point of the trajectory, vertical component of velocity is zero.

Therefore              0 = (u sin α)² – 2g H_max

or,           H_max = u² sin² α/ 2g

Equation of trajectory

Assuming the point of projection as the origin of co-ordinates and horizontal direction as the x-axis and vertical direction as the y-axis. Let P (x, y) be the position of the particle at instant after t second.

Then x = u cos α.t                and         y = u sin α.t – 1/2 gt²

Eliminating  ‘t’ form the above equations, we get,

y = x tan α – gx²/ 2u² cos² α

This is the equation of trajectory which is a parabola  (y = ax + bx²).

Illustrations

1.      A gun moving at a speed of 30m/sec fires at an angle 300 with a velocity 150m/s relative to the gun. Find the distance between the gun and the projectile when projectile hits the ground . (g = 10 m/sec)

Sol. Vertical component of velocity = 150 sin 300 = 75 m/sec

Horizontal component of velocity relative to gun = 150 cos 300

=                       =   75 √3 m/sec

Horizontal component of velocity relative to ground

=                              = 75 √3 + 30 ≈ 160 m/sec

Time of flight = (2 * 75 )/g = 15 sec

Range of projectile              = 160 × 15 = 2400 m

Distance moved by the gun and projectile = 2400 – 450 = 1950 m.

Consider a particle projected horizontally with a velocity u  from a point O as shown in figure 2.18.

Assuming the point of projection O as the origin of coordinates and horizontal direction as the X-axis and vertical direction as Y-axis. Let P (x, y) be the position of the particle after t seconds.

x = horizontal distance covered in time t = ut. …(1)

y = vertical distance covered in time t  = ½gt² ….(2)

Eliminate t from equations  (1) and (2) then

We get, y = (1/2 ) (g/u²) x²

This is the equation of parabola passing through the origin, with its vertex at the origin O. Hence the trajectory is a parabola.

• The position of object can change on a straight line (like on x-axis with respect to origin) or on a plane with respect to some fixed point or frame. So we can define motion as follows:
• An object or a body is said to be in motion if its position continuously changes with time with reference to a fixed point (or fixed frame of reference)
• When the position of object changes on a straight line i.e. motion of object along straight line is called motion in one dimension.
• To understand the essential concepts of one dimensional motion we have to go through some basic definitions.
• One can see the platform from a running train, and it seems that all the objects placed on platform are continuously changing their position. But one, who is on platform, concludes that the objects on the platform are at rest. It means if we will take the train as reference frame the objects are not stationary and taking reference frame as platform the objects are stationary. So the study of motion is a combined property of the object under study and the observer. Hence there is a need to define a frame of reference under which we have to study the motion of an object. We can define the frame of reference as follows:
• A frame of reference is a set of coordinate axes which is fixed with respect to a space point (a body or an object can also be treated as a point mass and therefore it can become a site for fixing a reference frame), which we have arbitrarily chosen as per our observer’s requirement. The essential requirement for a frame of reference is that, it should be rigid.
• The position of an object is defined with respect to some frame of reference. As a convention, we define position of a point (essentially we treat body as a point mass) with the help of three co-ordinates X, Y and Z. Hence (X, Y, Z) is a set of coordinate axes representing a 3-dimensional space and each point in this space can be uniquely defined with the help of a set of X, Y and Z coordinate, all three axes being mutually perpendicular to each other. The line drawn from origin to the point represents the position vector of that point.

In the figure 2.1, the position of a point P is specified and

is called the position vector.

• Consider a case in which the position of an object changes with time. Suppose at certain instant ‘t’ the position of an object is x₁ along the x axis and some other instant ‘T’ the position is x ₂ then the displacement Δx is defined as

Δx = x₂ – x₁

• It can be seen in the figure 2.2 where x₁ and x₂ are instantaneous position of the object at time t and T respectively.
• Now consider the motion of a point A with respect to a reference point O. The motion of point A makes its radius vector vary in the general case both in magnitude and in direction as shown in figure 2.3. Suppose the point A travels from point 1 to point 2 in the time interval Δt.

Distance and Displacement

• To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O to height h.
• After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It’s because distance is a scalar quantity but displacement is a vector quantity.

Speed and Velocity

• Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.

• Imagine flying birds, moving planets, gushing air, flowing water and so on, all these phenomena are happening around us continuously. All the above mentioned phenomena can be summarized in one word “motion”.
• When something moves, there are several factors, which can be observed. If it moves, it will cover some distance, the distance covered may be different in different time periods. It may be moving slow or fast. Further its speed or velocity may be changing with time. All these factors depend on each other and we need to study their relationships.             Various kinds of motion can be systematically grouped under few broad categories. The point of distinction is made on grounds of the velocity vector. On these grounds motion can be:
• one dimensional, where only one dimension is required to describe the motion
• two-dimensional, where we require two dimensions to describe the motion of the particle
• three-dimensional, where three dimensions are required.

Pre-requisite

State of Rest and Motion

• If the position of a particle is changing with respect to its surroundings in a given time interval, we say the particle is in motion and on the other hand if the position particle is not changing with respect to its surroundings, we can say the particle is at rest.

Position

The position of a particle refers to its location in the space at a certain moment. A vector joining the moving particle with the origin can describe the position of that particle at a particular moment.

Displacement

It is the vector joining the initial position of the particle to its final position in a given time interval.

Velocity

The rate of change of position of a moving particle is known as its velocity.

Acceleration

The rate of change of velocity with time is called the acceleration.

Distance and Displacement

• To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O to height h.
• After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It’s because distance is a scalar quantity but displacement is a vector quantity.

Speed and Velocity

• Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.

• The unit of any derived quantity depends upon one or more fundamental units. This dependence can be expressed with the help of dimensions of that derived quantity. In other words, the dimensions of a physical quantity show how its unit is related to the fundamental units.
• To express dimensions, each fundamental unit is represented by a capital letter. Thus the unit of length is denoted by L, unit of mass by M, unit of time by T, unit of electric current by I, unit of temperature by K and unit of luminous intensity by C.
• Remember that speed will always remain distance covered per unit of time, whatever be the system of units, so the complex quantity speed can be expressed in terms of length L and time T. Now, we say that dimensional formula of speed is LT-1. We can relate the physical quantities to each other (usually we express complex quantities in terms of base quantities) by a system of dimensions.
• Dimension of a physical quantity are the powers to which the fundamental quantities must be raised to represent the given physical quantity.

Illustrations

1. Find the dimension of density.

Sol. Density of a substance is defined to be the mass contained in unit volume of the substance.
Hence, [density] =[Mass]/[volume]

= M/L³

So, the dimensions of density are 1 in mass, – 3 in length and 0 in time.
Hence the dimensional formula of density is written as

[ρ]=ML^-3T^0

Important

• Constants such as ½, Π, or trigonometric functions such as “sin Αt” have no units or dimensions because they are numbers, or ratios which are also numbers.
• Two physical quantities can be equated, added (or subtracted) if and only if they have the same dimension. Why? [Verify if two quantities which have different dimensions can be multiplied (or divided) or not.

Broadly speaking, dimension is the nature of a Physical quantity. Understanding of this nature helps us in many ways.

The order of magnitude of a number is the power of ten closest to the number.

Following table gives us some of the commonly used prefixes for power of ten.

Positive Powers of 10

Negative Powers of 10

Some Derived SI units and their symbols

The following conventions are adopted while writing a unit.

• Even if a unit is named after a person the unit is not written in capital letters. i.e. we write joules not Joules.

• For a unit named after a person the symbol is a capital letter e.g. ‘J’ for joules and the rest of them are in lowercase letters e.g. ’s’ for seconds.

• The symbols of units do not have plural form i.e. 70 m not 70 ms or 10 N not 10 Ns.

• Not more than one solid’s is used i.e. all units of numerator is written together before the ‘ / ‘ sign and all in the denominator are written after that.

i.e. It is 1 ms-2 or 1 m/s2 not 1m/s/s.

• Punctuation marks are not written after the unit e.g. 1 litre = 1000 cc not 1000 c.c.

Introduction

Physics is that branch which deals with the study of nature and natural phenomenon. The word physics comes from the Greek word ‘fusis’ meaning nature. In this unit we will discuss some of the important aspect of measurement in physics. We will also discuss why we need a unit to measure a physical quantity.

In the measurement of any physical quantity, we require some ‘reference standard’. This reference standard of measurement is called a unit. These are independent quantities i.e. they do not need any other quantity to represent them. Let us consider three physical quantities mass, length and time. These quantities are independent of each other. So, three separate units are required for the measurement of these quantities. Thus, it becomes important to establish a system of units.

Measurement in Physics

Fundamental Units

Measurement of a physical quantity involves:

• The standard or unit in which the quantity is being measured
• The numerical value representing the number of times the quantity contains that unit.

The physical quantities which do not depend upon other quantities are called fundamental quantities. In M.K.S. system the fundamental quantities are mass, length and time, while in more general Standard International (S.I.) system the Fundamental quantities are mass, length, time, temperature, luminous intensity, current and amount of substance. The units of fundamental quantities are called fundamental units and are discussed below.

Derived Units

The units of physical quantities which may be derived from fundamental units are called derived units, for example:

Unit of area:

area = length × breadth

unit of area =  unit of length × unit of breadth

= m × m = m²

Unit of Velocity:

velocity = Displacement/Time

unit of velocity =Unit of Displacement/Unit of Time

= m/s = ms^-1

Hence m2 and ms-1 are derived units.

Systems of Units :

There are following principal system of units:

1. C.G.S System :

length → centimetre (cm),
mass  → gram  (g)
time    → second (s).

2. F.P.S System :

length → foot (ft),
mass  → pound (lb),
time    → second (s).

3. M.K.S. System:

length  → metre (m),
mass   → kilogram (kg),
time     → second (s).

4. S.I. System :

It has SEVEN fundamental units.

Length                                     → metre (m),
Mass                                       → kilogram (kg),
Time                                     → second (s).
Temperature                          → kelvin (K),
Luminous intensity                 → candela (cd),
Electric current                     → ampere (A),
Amount of substance             → mole (mol).

In S.I. system there are two supplementary units.

P        Radian (rad) : Unit of plane angle

P        Steradian (st) : Unit of solid angle

General: Units and dimensions, dimensional analysis; least count, significant figures; Methods of measurement and error analysis for physical quantities pertaining to the following experiments: Experiments based on using Vernier calipers and screw gauge (micrometer), Determination of g using simple pendulum, Young’s modulus by Searle’s method, Specific heat of a liquid using calorimeter, focal length of a concave mirror and a convex lens using u-v method, Speed of sound using resonance column, Verification of Ohm’s law using voltmeter and ammeter, and specific resistance of the material of a wire using meter bridge and post office box.

Mechanics: Kinematics in one and two dimensions (Cartesian coordinates only), projectiles; Uniform Circular motion; Relative velocity.
Newton’s laws of motion; Inertial and uniformly accelerated frames of reference; Static and dynamic friction; Kinetic and potential energy; Work and power; Conservation of linear momentum and mechanical energy.
Systems of particles; Centre of mass and its motion; Impulse; Elastic and inelastic collisions. Law of gravitation; Gravitational potential and field; Acceleration due to gravity; Motion of planets and satellites in circular orbits; Escape velocity.

Rigid body, moment of inertia, parallel and perpendicular axes theorems, moment of inertia of uniform bodies with simple geometrical shapes; Angular momentum; Torque; Conservation of angular momentum; Dynamics of rigid bodies with fixed axis of rotation; Rolling without slipping of rings, cylinders and spheres; Equilibrium of rigid bodies; Collision of point masses with rigid bodies.

Linear and angular simple harmonic motions.
Hooke’s law, Young’s modulus.

Pressure in a fluid; Pascal’s law; Buoyancy; Surface energy and surface tension, capillary rise; Viscosity (Poiseuille’s equation excluded), Stoke’s law; Terminal velocity, Streamline flow, equation of continuity, Bernoulli’s theorem and its applications.
Wave motion (plane waves only), longitudinal and transverse waves, superposition of waves; Progressive and stationary waves; Vibration of strings and air columns;Resonance; Beats; Speed of sound in gases; Doppler effect (in sound).
Thermal physics: Thermal expansion of solids, liquids and gases; Calorimetry, latent heat; Heat conduction in one dimension; Elementary concepts of convection and radiation; Newton’s law of cooling; Ideal gas laws; Specific heats (Cv and Cp for monoatomic and diatomic gases); Isothermal and adiabatic processes, bulk modulus of gases; Equivalence of heat and work; First law of thermodynamics and its applications (only for ideal gases); Blackbody radiation: absorptive and emissive powers; Kirchhoff’s law; Wien’s displacement law, Stefan’s law.

Electricity and magnetism: Coulomb’s law; Electric field and potential; Electrical potential energy of a system of point charges and of electrical dipoles in a uniform electrostatic field; Electric field lines; Flux of electric field; Gauss’s law and its application in simple cases, such as, to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell.

Capacitance; Parallel plate capacitor with and without dielectrics; Capacitors in series and parallel; Energy stored in a capacitor.

Electric current; Ohm’s law; Series and parallel arrangements of resistances and cells; Kirchhoff’s laws and simple applications; Heating effect of current.

Biot Savart’s law and Ampere’s law; Magnetic field near a current-carrying straight wire, along the axis of a circular coil and inside a long straight solenoid; Force on a moving charge and on a current-carrying wire in a uniform magnetic field.

Magnetic moment of a current loop; Effect of a uniform magnetic field on a current loop; Moving coil galvanometer, voltmeter, ammeter and their conversions.

Electromagnetic induction: Faraday’s law, Lenz’s law; Self and mutual inductance; RC, LR and LC circuits with d.c. and a.c. sources.

Optics: Rectilinear propagation of light; Reflection and refraction at plane and spherical surfaces; Total internal reflection; Deviation and dispersion of light by a prism; Thin lenses; Combinations of mirrors and thin lenses; Magnification.

Wave nature of light: Huygen’s principle, interference limited to Young’s double-slit experiment.

Modern physics: Atomic nucleus; Alpha, beta and gamma radiations; Law of radioactive decay; Decay constant; Half-life and mean life; Binding energy and its calculation; Fission and fusion processes; Energy calculation in these processes.

Photoelectric effect; Bohr’s theory of hydrogen-like atoms; Characteristic and continuous X-rays, Moseley’s law; de Broglie wavelength of matter waves.