Ray optics: Prism and Optical Instruments

1. Refraction through a prism is show in figure. After suffering refraction at two faces of a prism, the emergent ray is always found to bend towards the base of the prism. It is observed, that \[\angle A = \angle r_1 + \angle r_2\]and \[\angle A + \angle \delta = \angle i + \angle e\]Angle between the incident ray and the emergent ray i.e., $ \angle \delta $ is known as the angle of deviation. Its value depends upon the angle of incidence, refractive index of prism material and the angle of prism.

2. When refracted ray passes symmetrically through a prism i.e., when $ r_1 $ = $ r_2 $ and i = e, the light rays undergoes minimum deviation Dm and in such an eventuality, \[n_{21} = \frac{sin\frac{A+D_m}{2}}{sin{\frac{A}{2}}}\]where $ n_{21} $ is the refractive index of prism material with respect to the medium outside.

3. For a prism of small angle (i.e., if  $ \angle  A $ is small enough), the angle of deviation is given by \[\delta , D = (n_{21}-1)A\]

4. Dispersion is the phenomenon of splitting of light into its component colours (or wavelengths) on passing through a dispersive medium. The pattern of colour components of light is called its spectrum. For sunlight, the spectrum consists of seven constituent colours given by the acronym VIBGYOR. In white light spectrum the violet ray is deviated the most and the red ray is deviated the most and the red ray is deviated the least.

5. Cause of dispersion in variation of refractive index with wavelength of light. In fact, \[n = A + \frac{B}{\lambda ^2}\]where A and B are two constants are a given material. As a result, the refractive index of prism and consequently the angle of deviation is maximum for violet colour ray and least for red colour ray. It results in dispersion.

6. Angular dispersion produced by a prism for white light is difference in the angles of deviation of two extreme colours i.e., violet and red colours. Mathematically, Angular dispersion = $ \delta_v - \delta_r = (n_v - n_r)A $.

7. The light, while passing through earth’s atmosphere, gets scattered by the atmospheric particles. According to Rayleigh’s law of scattering, for scattering from tiny scattering objects e.g., air molecules the intensity of the light corresponding to a wavelength in the scattered light varies inversely as the fourth power of the wavelength. Mathematically, Amount of scattering $ \propto \frac{1}{\lambda^4} $

8. Blue colour of sky, blue colour of ocean water, reddish appearance of Sun at sunrise or sunset are some common phenomenon based on Rayleigh’s scattering. Due to this very reason, red light is used in danger signals.

9. Rainbow is an example of dispersion of light, caused by tiny water droplets hanging in the atmosphere after the rains.

10. The human eye is one of the most valuable and sensitive sense organ, the human being have. Our eyes have a lens system which focus the light rays coming from an object on the retina. Retina contains rods and cone which sense light intensity and colour respectively. Retina transmits electrical signals via the optic nerve to the brain, which analyses the information received and perceives the object.

11. The eyelens has the power of accommodation t adjust it focal length so as to focus objects situated at different distance form eye at the retina.

12. The least distance of distinct vision or near point of an eye is the minimum distance from the eye at which object can be seen distinctly. For a young adult with normal vision near point is at 25 cm.

13. The farthest point up to which an eye can see objects clearly is called the far point of eye. For a normal vision, the far point of eye lies at infinity. In this situation, our eye is least strained.

14. There are four common defects of vision. These are (i) myopia or short-sightedness (ii) hypermetropia or long-sightedness (iii) presbyopia and (iv) astigmatism.

15. A myopic eye can see near objects clearly but cannot see far off objects clearly i.e., the far point of defective eye is not at infinity but has shifted nearer to the eye. This defect may arise either due to (a) excessive curvature of the cornea, or (b) elongation of eyeball. The defect can be corrected by use of a concave (diverging) lens of appropriate power.

16. In hypermetropia, a person can see distance objects clearly but cannot see nearby objects distinctly i.e., for defective eye the near point has shifted away from the eye. This defect arises either due to less curvature of cornea or contraction of the eyeball. The defect can be corrected by use of convex (converging) lens of appropriate power. With increase in age the ciliary muscles gradually weaken and power of accommodation of eye decreases. It is called presbyopia. It can be corrected by using a converging lens for reading .

17. In astigmatism, a person cannot focus simultaneously on both horizontal and vertical lines. It arises when the cornea is not spherical in shape. The problem can be rectified by using cylindrical lens of desired radius of curvature with an appropriately directed axis.

18. A microscope is used for observing magnified images of nearby tiny objects. A simple magnifier or microscope is a convex lens of small focal length held near the object such that $ u \leq  f $.

19. In a simple microscope if image is formed at near point, the angular magnification of image is $ m = (1 + \frac{D}{f} ) $. However, if image is formed at infinity then magnification $ m = \frac{D}{f} $.

20. A compound microscope consists of two convex lenses, an objective lens of very small focal length ($ f_0 $) and small aperture and an eye lens of small focal length ($ f_e $) and slightly greater aperture, placed coaxially at a suitable fixed distance of distinct vision (D = 25 cm) from the eye and is virtual, inverted and highly magnified.

21. The angular magnification of a microscope is defined as the ratio of the angle subtended by the final image at the eye to the angle subtended by the object at the eye when seen directly. Angular magnification of a compound microscope is given by : 

(a) If final image is formed at near point of eye, then \[m = m_0 \times m_e = -\frac{v_0}{u_0}(1+\frac{D}{f_e})=-\frac{L}{f_0}(1+\frac{D}{f_e})\]

(b) If final image in a microscope is formed at infinity, then \[m =-\frac{L}{f_0}\frac{D}{f_e}\]

22. The resolving power of a compound microscope is its ability to show as distinct (separate), the images of two point objects lying close to each other. The limit of resolution of a microscope is measured by the minimum distance d between two point objects, whose images in microscope are seen as just separate. It is found that \[d = \frac{1.22 \lambda}{2nsin\alpha} = \frac{0.16 \lambda}{N.A.}\]where n = refractive index of medium between the object and the objective lens, $ 2\alpha $ = angle subtended by the diameter of objective lens at the focus point and N.A. = $ n sin \alpha $ = numerical aperture of objective. Resolving power of a microscope is the reciprocal of its limit of resolution. For higher resolving power the numerical aperture the numerical aperture of objective lens of microscope should be large and wavelength of light used should be as small as possible.

23. An astronomical telescope is used to form magnified and distinct images of heavenly bodies like planets stars, moons, galaxies etc. A refracting type astronomical telescope consists of a convex objective lens of large focal length and large aperture and another convex eyepiece lens of small total length and small aperture. Final image formed is inverted, magnified and at infinitly in normal adjustment.

24. The angular magnification of a telescope is defined as the ratio of the angle subtended at the eye by the final image to the angle subtended at the eye by the object directly. It is found that in normal adjustment \[m = -\frac{f_0}{f_e}\]and length of telescope tube $ L = f_0 + f_e $. 

25. In a reflecting type telescope we use a concave mirror (generally parabolic) of large aperture and large focal length as the objective and a convex lens of small focal length and aperture as the aberrations, are cheap, easy to construct and handle.

26. The limit of resolution of a telescope is measured by the angle ($ \Delta \theta $) subtended at its objective, by those two distant objects whose images are just seen separate through the telescope.

Resolving power of telescope = $ \frac{1}{\Delta \theta}= \frac{A}{1.22 \lambda}    $, where A is the aperture size of the telescope objective.

Read More

Ray Optics: Reflection and Refraction

1. Light is that form of energy which causes sensation of sight of your eyes. In fact, light is a part of electromagnetic radiation spectrum having its wavelength ranging from about 400 nm to 700nm.

2. In vacuum light of any wavelength (or any frequency) travels with a speed c = $ 2.99792458 \times 10^8 $ m/s but for ordinary calculations this value may be considered as c = $ 3 \times 10^8 $ m/s. The speed of light in vacuum is the highest speed attainable in nature. Moreover, speed of light in vacuum is independent of the relative motion between the source and the observer. No physical signal or message can travel with a speed greater than c.

3. As the wavelength of light is very small compared to the size of ordinary objects, a light wave can be considered to travel from one point to another point along a straight line path. Such a path is called a ray of light of right and a bundle of rays constitutes a beam of light.

4. In reflection of light, the light rays return to the same medium after striking the surface of another medium (say a mirror). The wavelength and the speed of light remains the same.

5. There are two basic laws of reflection, which are followed for every short of reflection. According to these

(i)  The incident ray, reflected ray and the normal at  the point of incidence all lie in the same plane.

(ii)  Angle of incidence (i) = angle of reflection (r).

6. For reflection from a plane mirror, the image formed is always erect, virtual and laterally inverted. The image is of exactly the same size as the object and the image is formed as such behind the mirror as the object is in front of it.

7. For a spherical mirror in the mid-point of reflecting surface is called its pole. The line passing through pole and the center of curvature of mirror is called its principle axis.

8. The principle focus of a spherical mirror is a point on its principle axis, where a beam of light incident parallel to principle axis of mirror, after reflection, actually converges to (in case of a concave mirror) or appears to diverge from (in case of a convex mirror). Distance of principle focus from pole is called the local length of given mirror.

9. Focal length of a spherical mirror is half of its radius of curvature i.e., f = R/2.

10. As per Cartesian sign convention system for mirrors, the light ray is taken to travel from left to right. All distances are measured from the pole as origin. The distance measured in the same direction as the incident light are taken as positive and those measured in the opposite direction are taken as negative. Thus, the distances to the right of pole will be + ve but distances to the left of pole will be – ve. Again distances above the principle axis are taken as + ve but distances below it – ve.

11. A concave mirror may form either a real or a virtual image depending upon the position of the object relative to the mirror. A convex mirror forms only virtual images.

12. If an object is placed at a distance u from the pole of a mirror of focal length f and its image is formed at a distance v from the pole, then according to mirror formula, we have \[\frac{1}{u} + \frac{1}{v}=\frac{1}{f}=\frac{2}{R}\].

13. If a thin linear object of height h is situated normally on principle axis of mirror at a distance u and its image of height h’ is formed at a distance v from the pole, then the linear magnification m is defined as  \[M = \frac{h'}{h} = -\frac{v}{u} = \frac{f}{f-u} = \frac{f-v}{f}\]-ve magnification means inverted image and +ve magnification means erect image.

14. When a light ray travels obliquely from one transparent medium to another, it changes the direction of its path at the interface of the two media. This is called “refraction” of light.

15. There are two laws of refraction, which are as follows:

(i)  The incident ray, the refracted ray and the normal to the interface at the point of incidence, all lie in the same plane.

(ii)    The ratio of the sine of the angle of incidence in 1^st medium t sine the angle of refraction in second medium is a constant, knows as the refractive index of 2^nd medium with respect to the 1^st medium. Mathematically,  \[\frac{sin (i)}{sin (r)} = n_{21}\]Second law of refraction is known as Snell’s law.

16. Value of refractive index depends upon the pair of media and the wavelength of light but is independent of the angle of incidence.

17. When a light ray obliquely enters from an optically rarer medium to an optically denser medium, the light ray bends towards the normal. However, if a light ray travels from denser to rarer medium, it bends away from the normal.

18. Absolute refractive index of a transparent medium is defined as the ratio of the speed of light vacuum (c) to the speed of light in given medium (v) i.e., $ n = \frac{c}{v}$. It can be show that $ n_{21} = \frac{n_2}{n_1} = \frac{v_1}{ v_2} = \frac{\lambda_1}{\lambda_2}$. It is found that $n_{12} = \frac{1}{n_{21}} $.

19. When a light ray passes through a parallel sided slab of a transparent medium, the final emergent ray is parallel to the incident ray, but is laterally displaced by a distance d given by \[d = t \frac{sin (i – r)}{cos r}\]The value of lateral shift depends upon (a) thickness (t) of the transparent slab, (b) angle of incidence (i) and (c) refractive index of the material of slab.

20. When an object situated in medium number 2 is viewed from medium number 1, the apparent depth (height) of object appears to be different from its real depth 9height) and these are co-related as: \[\frac{d_{Real }}{ d_{Apparent}} = n_{12} = \frac{1}{n_{21}}\]

21. If an object situated in an optically denser medium is viewed by an observer situated in optically rarer medium, the apparent height is less than its real height. However, if an object situated in rarer medium is viewed by an observer situated in denser medium, then the apparent height is found to be more than its real height.

22. On account of atmospheric refraction the Sun is visible about 2 minutes before the actual sunrise and for 2 minutes even after the actual sunset. Thus, Sun also appear to be of oval shape at the time of sunrise or sunset on account of atmospheric refraction.

23. For a pair of media in contact, circuital angle is the angle of incidence in the denser medium corresponding to which angle of refraction in the rarer medium is 90 degree. If a light ray is incident on the surface of a rarer medium 2 from a denser medium 1, then \[Sin(i_c) = n_{21}\]Here $ n_{12}$ is the refractive index of 1^st (denser) medium with respect to the 2^nd (rarer) medium.

24. Total internal reflection is the phenomenon of complete reflection of light back into the denser medium, when a light ray coming from denser medium is incident on the surface of a rarer medium.Two essential conditions for total internal reflection are:

(i)  The light ray should travel in a denser medium towards a rarer medium.

(ii)  Angle of incidence in the denser medium should be greater than the critical angle for the pair of media in contact.

25. Values of critical angle of glass-air and water-air interfaces are 41.5 degree and 48.75 degree, respectively.

26. The brilliance of diamond, action of optical fibres and mirage etc., are the phenomena based on total internal reflection of light.

27. Prism make use for total internal reflection phenomenon to bend light by 90 degree or by 180 degree or to invert images without changing their size. Such prism have one angle 90 degree and the other two angles 45 degree each and are known as totally reflecting prisms or poroprisms.

28. For refraction at a single spherical surface, all distances are measured from the pole of the refracting surface. The distances measured in the direction fo incidence of light are taken as positive and the distances measured in the opposite direction are taken as negative. If object is considered to be situated on left side of pole,  then the sign convention agrees with the cartesian coordinate system. Accordingly, all distances on left side of pole are taken as negative and on right side of pole as positive. The height measured above the principle axis are taken as positive as heights measured downwards are taken as negative.

29. For refraction at a single spherical surface \[\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2 - n_1}{R}\]Where light beam is going from medium of refractive index n1 to medium of refractive index n2. The relation is true for concave as well as convex spherical surfaces and irrespective of the fact whether refraction is taking places from rarer medium to denser medium or vice-versa.

30. According of lens maker’s formula \[\frac{1}{f}=(n_{21}-1)(\frac{1}{R_1}-\frac{1}{R_2})\]Where $n_{21}$ is the refractive index of lens material w.r.t. the surrounding medium, $R_1$ and $R_2$ are the radii of curvature of two surfaces of lens and f its focal length.

31. For image formed by a thin lens, we have \[\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\]All the above relation are the true for convex as well as concave surfaces/lenses and for real as well as virtual images.

32. Linear magnification (m) produced by a lens is defined as the ratio of the linear (lateral) size of the image to that of the object. Thus, \[m = \frac{h'}{h}=\frac{v}{u}=\frac{f-v}{f}=\frac{f}{f+u}\]For erect and virtual image, m is positive but for an inverted and real image, m is negative.

33. Power of lens is a measure of a degree of convergence or divergence of light incident on it. Mathematically, the power(p) of a lens is defined as the tangent of the angle by which it converges/ diverges a beam of light falling at unit distance from the optical centre. For a thin lens power is found to be the reciprocal of its focal length (f) i.e., \[P = \frac{1}{f}\]SI unit of power is dioptre (D).

34. Power of a converging (convex) lens is taken to be positive but that of a diverging (concave) less is taken negative.

35. For a combination of two (or more) thin lenses in contact, the effective focal length of the combination is given by \[\frac{1}{f} = \frac{1}{f_1}+\frac{1}{f_2}+......\]And in term of power, we have $ P = P_1 + P_2 + ...... $. 

36. For a combination of two or more lenses, the effective magnification for the combination is given by \[m = m_1 \times m_2 \times m_3 .......\]

Read More

Electromagnetic Waves

1. A time-varying magnetic field gives rise to an electric field. Maxwell argued that a time-varying electric field should also give rise to a magnetic field. Maxwell thus tried to apply Ampere’s circuital law to find magnetic field outside a capacitor connected to a time-varying current. However, he noticed an inconsistency in Ampere’s circuital law.

2. To remove the inconsistency of Ampere’s circuital law, Maxwell suggested the existence of  "Displacement current”.

3. Displacement current ($ I_d$) is currents which come into play whenever the electric field and, consequently, the electric flux is changing with time. Mathematically, \[I_d = \epsilon_0 \frac{d\phi_E}{dt}\]

4. The sum of conduction current (I) and displacement current ($ I_d $) has the property of continuity along any closed path, although individually they may not be continuous. Thus, Maxwell modified Ampere’s circuital law as \[\oint \vec{B}.\vec{dl} = \mu_0 (I+I_d)\]With this modification the problem of inconsistency observed by Maxwell was rectified.

5. Maxwell was the first person who theoretically predicted the existence of electromagnetic waves, which are coupled with time-varying electric and magnetic fields propagating in space. The speed of these waves in free space is the same as that of light i.e. $ 3 \times 10^8 $ m/s.

6. Electromagnetic waves are produced by accelerated charges (or oscillating charge). An oscillating charge, which is an example of accelerating charge produces an oscillating electric field in space, which produces an oscillating magnetic field, which in turn is a source of oscillating electric field and so on. The oscillating electric filed and magnetic fields, thus, regenerate each other i.e., electromagnetic wave propagates through the space.

7. The frequency of the electromagnetic wave is same as the frequency of oscillation of the charge (electric field E) or the frequency of oscillating magnetic field (B).

8. Hertz was the first scientist to experimentally demonstrate the production of electromagnetic waves employing a crude form of an oscillatory LC circuit arrangement. Later on, Jagdish Chandra Bose produced electromagnetic waves of much shorter wavelengths. Marconi succeeded in transmitting electromagnetic waves over a distance of many kilometers.

9. Electromagnetic waves do not require any material medium for their propagation. In free space, their speed is given by \[c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8\]In a medium of absolute permittivity (), the speed of electromagnetic waves is given by \[c = \frac{1}{\sqrt{\mu \epsilon}} = \frac{c}{\sqrt{K\mu_r}}\]

10. In an electromagnetic wave and electric and magnetic fields are in phase with each other. They attain their peak values at the same instant.

11. Electromagnetic waves are transverse in nature. The oscillating electric and magnetic fields are perpendicular to each other as well as perpendicular to the direction of propagation of the wave. In fact, the direction of ($ \vec{E}\times \vec{B} $) gives the direction of propagation of e.m. waves.

12. If we consider an electromagnetic wave propagating along positive x-axis then oscillating electric and magnetic fields may be represented as:\[\vec{E_y} = E_0sin(kx-\omega t)\hat{j}\] and \[\vec{B_z} = B_0sin(kx-\omega t)\hat{k}\]Here $\omega = 2\pi \nu $ is the angular frequency and $k = (\frac{2\pi}{\lambda})$ propagation constant of given electromagnetic wave.

13. In an electromagnetic wave, Amplitudes $E_0$ and $B_0$ of electric and magnetic fields in free space are related as: \[\frac{E_0}{B_0} = c\]

14. The energy density i.e., energy per unit volume of an electromagnetic wave consists of electric and magnetic contributions. Thus, The mean energy density \[U_m = U_E + U_B = \frac{1}{2}\epsilon_0 E^2_{rms} + \frac{1}{2\mu_0}B^2_{rms}\] It is found that average values of $ U_E $ and $ U_B $ are equal. 

15. Intensity of the electromagnetic wave is defined as the mean amount of energy passing through a unit area normally in unit time. It can be shown that Intensity \[I = U_m c = \frac{1}{2}\epsilon_0 c E^2_0 = \frac{c}{2\mu_0}B^2_0\]

16. The electromagnetic wave carries momentum too. If U be the total energy transferred to a surface by an electromagnetic wave in time t, then momentum delivered to this surface, assuming the surface to be completely absorbent, is \[p = \frac{U}{c}\]The average force exerted by e.m. wave on the surface will be \[F= \frac{p}{t} = \frac{U}{ct}\]

17. The classification of electromagnetic radiation waves according to frequency is known as “electromagnetic spectrum”. There is no sharp division between one kind of wave and the next and the classification is based roughly on how the waves are produced/ detected.

18. Complete electromagnetic spectrum in ascending order of frequency (or in decreasing order of wavelength) broadly consists of seven parts namely 

(i) Radio waves, (ii) Microwaves (iii) Infrared waves, (iv) Visible light rays, (v) Ultraviolet rays, (vi) X-rays, and (vii) Gamma rays.

19. Radio waves are produced by accelerated motion of charges in conducting wires and are used in radio and TV communication. They are in the frequency range of 500kHz to about 1000 MHz (or 1 GHz). These are further subdivided as a medium band, short band, HF band, VHF band, UHF band, etc.

20. Microwaves are extremely short-wavelength radio waves having a frequency range of $ 10^9 $ Hz to about 10 11 Hz and are produced by special vacuum tubes e.g., klystrons, magnetrons, and Gunn diodes. These are used in radar, microwave telecommunication, microwave oven, etc.

21. Inferred waves are produced by hot bodies and molecules and are characterized by their heating property. Inferred radiation plays an important role in maintaining the earth’s warmth by the greenhouse effect. Inferred rays are widely used in the remote switches of household electronic systems such as TV sets, video recorders, hi-fi systems, etc.

22. Visible parts are that part of the electromagnetic spectrum which is detected by the human eye. It runs from about $ 4 \times 10^{14} $ Hz to $ 7 \times 10^{14} $ Hz. Visible light emitted or reflected from objects around us provides us information about the world.

23. Ultraviolet rays consist of radiation in the frequency range $ 7 \times 10^{14} $Hz to $ 5 \times  10^{17} $ Hz (or wavelength range from 400 nm to 0.6 nm). These are produced by the sun, special lamps like mercury lamp, hydrogen tube etc, and very hot bodies. Ultraviolet rays have various uses such as in  LASIK eye surgery, to kill germs in water purifiers, as a disinfectant in hospitals, etc. however, ultraviolet light in large quantities has harmful effects on humans.

24. Ozone layer present in the atmosphere at an altitude of about 40 – 50 km absorbs most of the ultraviolet rays coming from the sun and thus, form a protective ring around the earth.

25. X-rays cover wavelengths from about 1 nm to $10^{-3} $ nm. These are produced by bombarding high energy electrons on a metal target. X-rays are used as a diagnostic tool in medicine, as a treatment for certain forms of cancer, and for scientific research.

26. Gamma rays are the hardest electromagnetic waves having wavelengths even less than $ 10^{-3} $ nm. These are produced in nuclear reactions and are also emitted during radioactive decay of the nuclei. These are used in medicine for destroying cancer cells.

Electromagnetic Spectrum

Type	Wavelength range	Production	Detection
Radio	> 0.1 m	Rapid acceleration and decelerations of electrons in aerials	Reciever's aerial
Microwave	0.1 m to 1 mm	Klystron valve or magnetron valve	Point contact diode
Infrared	1 mm to 700 nm	Vibration of atoms and molecules	Thermopiles, Bolometer, Infrared photographic film
Light	700 nm to 400 nm	Electrons in atom emit light when they move from one energy level to a lower energy level	The eye Photocells, Photographic film
Ultraviolet	400 nm to 1 nm	Inner shell electrons in atoms moving from one energy level to lower level	Photocells, Photographic film
X-ray	1 nm to $10^{-3}$ nm	X-ray tubes or inner shell electrons	Photographic film, Geiger tubes, Ionisation chamber
Gamma Ray	< $ 10^{-3} $ nm	Radioactive decay of the nucleus	- do -

Read More

Newton's Laws and Force

1. Newton’s three laws of motion form the basis of mechanics. According to Ist law, A body continues to be in its state of rest or of uniform motion along a straight line, unless it is acted upon by some external force to change the state. This law defines force and is also called law of inertia.

According to second law, the rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place in the direction of the applied force. This law gives us a measure of force. i.e. $ F \propto \frac{d\vec{p}}{dt} $. 

According to third law, To every action, there is always an equal and opposite reaction. This law gives us the nature of force.

2. Inertia is the inability of a body to change by itself, its state of rest, or its state of uniform motion along the straight line. Inertia is of three types: (i) Inertia of rest (ii) Inertia of motion, (iii)Inertia of direction.

3. From Newton’s 2^nd law, we obtain $ \vec{F_{ext}} = m \vec{a} $ i.e. an external force is the product of mass and acceleration of the body.

4. The absolute unit of force on SI in newton (N) and on cgs system, it is dyne. 

5. According to the principle of conservation of linear momentum, the vector sum of linear momentum of all the bodies in an isolated system is conserved and is not affected due to their mutual action and reaction. An isolated system is that on which no external force is acting. In other words, If external forces acting on the system is zero then it's linear momentum is constant.  Flight of rockets, jet planes, recoiling of a gun, etc. are explained on the basis of this principle. Newton’s 3^rd law of motion can also be derived from this principle and vice-versa.

6. Apparent weight of a man in an elevator is given by $ W' = m(g \pm a) $ where mg is real weight of the man. Acceleration is (+ a), when the lift is accelerating upward and (-a) when the lift is accelerating downwards. When lift is moving uniformly (upwards/downwards). a = 0. W’ = m g = real weight. In free fall, a = g,  W' = m (g – g) = 0 i.e. apparent weight becomes zero.

7. When two bodies of masses m₁ and m₂ are tied at the ends of an inextensible string passing over a light frictionless pulley, acceleration of the system is given by, \[a = \frac{|m_1 - m_2|}{m_1+m_2}g\], Tension is given by, \[T = \frac{2m_1m_2}{m_1+m_2}g\]

8. Impulse \[\vec{I} = \vec{F_{av}} \times t = \vec{P_2}-\vec{P_1}\] where t is the time for which average force acts $ (\vec{P_2 } – \vec{P_1})$ is change in linear momentum of the body.

9. The force which are acting at a point are called concurrent forces. They are said to be in equilibrium when their resultant is zero.

FRICTION

10. Friction is the opposing force that comes into play when one body is actually moving over the surface of another body or one body is trying to move over the surface of the other. Two causes of friction are: the roughness of surfaces in contact; Force of adhesion between the molecules of the surfaces in contact.

11. Limiting friction is the maximum value of static friction. Dynamic/Kinetic friction is somewhat less than the force of limiting friction.

12. Static friction is a self adjusting force.

13. Rolling friction is less than sliding friction.

14. Laws of limiting friction are: 

(i) $ F \propto R$, where R is normal reaction and F is the friction force.

(ii) Direction of F is opposite to the direction of motion.

(iii) F does not depend upon the actual area of contact.

(iv) F depends upon the nature of material and nature of polish of the surfaces in contact.

15. Coefficient of friction is given by, $ \mu  = \frac{F}{R} $.

16. Angle of Repose ($ \theta $) is the minimum angle of inclination of a plane with the horizontal, such that a body placed on the plane just begins to slide down.

17. Acceleration of the body down a rough inclined plane, \[a = g(sin\theta - \mu cos\theta)\]

18. Work done in moving a body over a rough horizontal surface, \[W = \mu mgd \]Work done in moving a body over a rough inclined plane, \[W = mg(sin\theta + \mu cos\theta)d\]

19. Friction is a necessary evil. Some of the methods of reducing friction are polishing, lubrication; streamlining the shape etc.

20. Centripetal force is the force required to move a body uniformly in a circle. The magnitude of this force is $ F = \frac{mv^2}{r}=mr\omega^2 $. It acts along the radius and towards the centre of the circle.

21. Centrifugal force is a force that arises when a body is moving actually along a circular path, by virtue of tendency of the body to regain its natural straight line path. Centrifugal force can be treated as the reaction of centripetal force. The magnitude of centrifugal force is same as that of centripetal force. The direction of centripetal force is along the radius and away from the centre of the circle.

22. While rounding a level curved road, the necessary centripetal force is provided by the force f friction between the tyres and the road. The maximum velocity with which a vehicle can go round a level curve without skidding is $ v = \sqrt{\mu rg}$. To avoid dependence on friction, curved roads are usually banked i.e. outer edge of the curved road is raised suitably above the inner edge. If θ is the angle of banking, then $ tan\theta = \frac{v^2}{rg}$.

23. While rounding a banked curved road, the maximum permissible speed is given by \[v_{max} = \sqrt{\frac{rg(\mu_s + tan\theta)}{(1-\mu_s tan\theta)}}\]When frictional force is ignored, the optimum speed is, \[v_{max} = \sqrt{rg tan\theta }\].

24. Motion along a vertical circle is a non-uniform circular motion. Tension in the string at any position is $ T = \frac{mv^2}{r} + mgcos\theta $ where θ is the angle with vertical line through the lowest point of the circle.

1.                   For looping the vertical loop, the velocity of projection at lowest point L is $ v_L \geq \sqrt{5rg}$.

2.                   The value of velocity at the highest point H is $ v_H \geq \sqrt{rg}$.

3.                   Difference in tension in the string at lowest point and highest point of vertical circle is, $ T_L - T_H = 6mg $.

4.                   For oscillation over the arc of vertical circle $ 0 < v_L \leq \sqrt{2rg} $.

5.                   For leaving the vertical circle somewhat between $ 90^{\circ} < \theta < 180^{\circ} $, $ \sqrt{2rg} < v_L < \sqrt{5rg} $.

6.                   The minimum height h through which a motor cyclist has to descend to loop a vertical loop of radius r is, $ h = \frac{5}{2}r $.

Read More