Time has come for intensive study. May be at the rate of one chapter each week. It means solving lots of problems.
Week ending 25 May 2008 - Mathematics and Physics - Vector algebra and Calculus
Saturday, May 24, 2008
Tuesday, May 13, 2008
IIT JEE Physics Useful books - Download
Elementary Mechanics and Thermodynamics by John Norburyhttp://www.scribd.com/doc/90229/Elementary-mechanics-and-thermodynamics
Solutions manual
http://www.scribd.com/doc/90222/Solutions-manual-for-mechanics-and-thermodynamics
Quantum Theory - a Very short Introduction by John Polkinghornehttp://www.scribd.com/doc/2676192/Quantum-Theory
Read Albret Einstein's note on Relativity for people with Matriculate Physics Knowledge
http://www.scribd.com/doc/3408/ebook-PDF-Science-Albert-Einstein-Relativity
History of Physics
http://www.scribd.com/doc/322968/A-History-of-Physics-in-Its-Elementary-Branches
Solutions manual
http://www.scribd.com/doc/90222/Solutions-manual-for-mechanics-and-thermodynamics
Quantum Theory - a Very short Introduction by John Polkinghornehttp://www.scribd.com/doc/2676192/Quantum-Theory
Read Albret Einstein's note on Relativity for people with Matriculate Physics Knowledge
http://www.scribd.com/doc/3408/ebook-PDF-Science-Albert-Einstein-Relativity
History of Physics
http://www.scribd.com/doc/322968/A-History-of-Physics-in-Its-Elementary-Branches
Monday, May 12, 2008
Past JEE Objective Questions on Magnetism
Objective questions on magnetism, from the past JEE papers of 1982-88 are posted in
http://iit-jee-physics-ps.blogspot.com/2008/05/past-jee-objective-questions-magnetism.html
http://iit-jee-physics-ps.blogspot.com/2008/05/past-jee-objective-questions-magnetism.html
Thursday, May 8, 2008
IIT JEE Physics Concept Review - An-Az
Angular momentum
(from Rotational Mechanics chapter)
Angular momentum of a particle about a point O is defined as
l = r×p
l, r, and p are vectors.
l = angular momentum of a particle
r = positition vector of the particle from O
p = linear momentum of the particle
Angular momentum of a particle about a line say AB is the component of angular momentum about a point O on the line AB along the line AB. It means to find angular momentum of a particle about a line AB, we have to first find angular momentum about a point O on AB and then find its component along the line AB.
(from Rotational Mechanics chapter)
Angular momentum of a particle about a point O is defined as
l = r×p
l, r, and p are vectors.
l = angular momentum of a particle
r = positition vector of the particle from O
p = linear momentum of the particle
Angular momentum of a particle about a line say AB is the component of angular momentum about a point O on the line AB along the line AB. It means to find angular momentum of a particle about a line AB, we have to first find angular momentum about a point O on AB and then find its component along the line AB.
IIT JEE Physics Concept Review - Ca-Cm
Centrifugal Force:
Newton's laws are not valid if one is working from a noninertial frame. If the frame of reference rotates at a constant angular velocity ω with respect to an inertial frame, we use the pseudo force centrifugal force.
Centrifugal force is assumed act because we describe the particle from a rotating frame which is noninertial and still use Newton's laws.
Example: An observer sitting in a rotating cabin observed that a box is at rest. If the box is at rest, the resultant force on the box has to be zero. The forces on the box are, its weight and the normal contact force between the cabin floor and the box, and the friction force beween the cabin floor and the box. The friction force f = mω²r acts on the box towards the origin. As the weight and the normal contact force cancel each other, to make the resultant zero, a pseudo force with magnitude mω²r is to be assumed which acts on the box away from the centre. This is centrifugal force.
Newton's laws are not valid if one is working from a noninertial frame. If the frame of reference rotates at a constant angular velocity ω with respect to an inertial frame, we use the pseudo force centrifugal force.
Centrifugal force is assumed act because we describe the particle from a rotating frame which is noninertial and still use Newton's laws.
Example: An observer sitting in a rotating cabin observed that a box is at rest. If the box is at rest, the resultant force on the box has to be zero. The forces on the box are, its weight and the normal contact force between the cabin floor and the box, and the friction force beween the cabin floor and the box. The friction force f = mω²r acts on the box towards the origin. As the weight and the normal contact force cancel each other, to make the resultant zero, a pseudo force with magnitude mω²r is to be assumed which acts on the box away from the centre. This is centrifugal force.
IIT JEE Physics Concept Review - Fa-Fm
Flux of an electric field
If in a plane surface area of ∆s, a uniform electric field E exists, and makes an angle θ with the normal to the surface area (positive normal - you can arbitrarily decide which direction is positive), the quantity
∆Φ = E ∆s cos θ
is called the flux of the electric field through the chosen surface.
If ∆s is represented as a vector
∆Φ = E.∆s
Where E and ∆s are vectors and ∆Φ is a scalar quantity.
(Chapter: Gauss's Law)
If in a plane surface area of ∆s, a uniform electric field E exists, and makes an angle θ with the normal to the surface area (positive normal - you can arbitrarily decide which direction is positive), the quantity
∆Φ = E ∆s cos θ
is called the flux of the electric field through the chosen surface.
If ∆s is represented as a vector
∆Φ = E.∆s
Where E and ∆s are vectors and ∆Φ is a scalar quantity.
(Chapter: Gauss's Law)
IIT JEE 2010 Study Plan for Physics
Chapters to be covered during XI class (2008-09) from H C Verma
Physics Class XI Chapters (Maharashtra Board Syllabus)
1. Measurement
2. Scalars and vectors
3. Projectile motion
4. Force
5. Friction in solids and liquids
6. Sound waves
7. Thermal expansion
8. Refraction of light
9. Lens
10. Electrostatics
11. Current electricity
12. Magnetic effect of electric current
13. Magnetism
14. Electromagnetic waves
Chapters to be covered during XI class (2008-09) from H C Verma
1. Introduction to physics
2. Physics and mathematics
3. Rest and motion
4. The Forces
5. Newton’s laws of motion
6. Friction
15. Wave motion
16. Sound waves
18. Geometrical optics
19. Optical instruments
29. Electric field and potential
32. Electric current in conductors
34. Magnetic field
35. Magnetic effect of electric current
36. Permanent magnets
37. Electromagnetic waves
Physics Class XII Chapters (Maharashtra Board Syllabus)
1. Circular motion
2. Gravitation
3. Rotational motion
4. Oscillations
5. Elasticity
6. Surface tension
7. Wave motion
8. Stationary waves
9. Kinetic theory of gases
10. Radiation
11. Wave theory of light
12. Interference and diffraction
13. Electrostatics
14. Current electricity
15. Magnetic effect of current
16. Magnetism
17. Electromagnetic induction
18. electrons and photons
19. Atoms, molecules, nuclei
20. Semiconductors
21. Communication
Physics Class XI Chapters (Maharashtra Board Syllabus)
1. Measurement
2. Scalars and vectors
3. Projectile motion
4. Force
5. Friction in solids and liquids
6. Sound waves
7. Thermal expansion
8. Refraction of light
9. Lens
10. Electrostatics
11. Current electricity
12. Magnetic effect of electric current
13. Magnetism
14. Electromagnetic waves
Chapters to be covered during XI class (2008-09) from H C Verma
1. Introduction to physics
2. Physics and mathematics
3. Rest and motion
4. The Forces
5. Newton’s laws of motion
6. Friction
15. Wave motion
16. Sound waves
18. Geometrical optics
19. Optical instruments
29. Electric field and potential
32. Electric current in conductors
34. Magnetic field
35. Magnetic effect of electric current
36. Permanent magnets
37. Electromagnetic waves
Physics Class XII Chapters (Maharashtra Board Syllabus)
1. Circular motion
2. Gravitation
3. Rotational motion
4. Oscillations
5. Elasticity
6. Surface tension
7. Wave motion
8. Stationary waves
9. Kinetic theory of gases
10. Radiation
11. Wave theory of light
12. Interference and diffraction
13. Electrostatics
14. Current electricity
15. Magnetic effect of current
16. Magnetism
17. Electromagnetic induction
18. electrons and photons
19. Atoms, molecules, nuclei
20. Semiconductors
21. Communication
Wednesday, May 7, 2008
Concept Review - Chapter 1 Introduction
Physics is the study of nature and its laws.
The nature around us is like a big chess game played by Nature. Various events that happen are like the moves made by a chess players. We are allowed to watch the events that happen, and guess the rules, and then play to derive benefits that we want from nature. We may across new events which do not follow old rules that we have formulated and we need to guess the new rules.
Great scientists or scientists in general guess the rules from the observations available at that time and prove the usefulness of those rules by further experiments or events that happen subsequently. These rules may require modification subsequently if they are not able to explain some events observed that happen subsequently.
The description of nature becomes easy if we have the freedom to use mathematics.
Mathematics is the language of physics.
Units: Fundamental and derived
While there are a large number of physical quantities to be measured, only seven fundamental quantities are found to be sufficient. All other quantities can be measured using these seven fundamental quantities.
The set of fundamental quantities must have the following properties.
a. The fundamental quantities should be independent of each other; and
b. All other quantities may be expressed in terms of the fundamental quantities.
Fundamental quantities are also referred to as base quantities.
Who decides the units?
A body named Conference Generale des Poids or CGPM (General Conference on Weight and Measures in English) has been gibven the authority by international agreement.
Definition of base units
Metre
Kilogram
Second
Ampere
Kelvin
Mole
Candela
Metre
The distance travelled by light in vacuum in 1/[299,792,458] second is called 1 m.
Kilogram
the mass of cylinder made of planitum-iridium alloy kept at International Bureau of Weights and Measures is defined as 1 kg.
Second
The time duration in 9,192,631,770 time periods of the selected transition of radiation of Cesium-133 atom is defined as 1 s. (To understand this idea see Bohr model chapter)
Ampere
To under stand the definition of this unit, you have to read the chapter on electric field.
Two long straight wires with negligible cross section are to be placed parallel to each other at a separation of 1 m and electric current in the same amout is sent through them in the same direction. If the attractive force between the two wires adjusted, the current in each wire (one of the wires) that gives a force between them of 2*10^-7 newton per metre of the wires, is defined as 1 A of current.
Kelvin
The fraction 1/9273.16) of the thermodynamic temperature of triple point of water is called 1 K.
Mole
The amount of a substance that contain as many elementary entities (molecules or atoms if the substance is monatomic) as there are number of atoms in 0.012 kg of carbon-12 is called a mole.
Candela
It is the luminous intensity of a blackbody of surface area 1/[600,000] m² placed at the temperature of freezing platinum and at a pressure of 101,325 N/m², in the direction perpendicular to its surface.
Dimensions of physical quantities
When a physical quantity is expressed in terms of the base quantities, it is written as a product of different powers of the base quantities.
[Force] = MLT-2
M,L, and T are base quantities – mass, length and time
The exponent of a base quantity that enters into the expression (MLT-2) is called the dimension of the quantity in that base.
The dimensions or force are 1 in mass, 1 in length and -2 in time. The dimensions of all other base quantities are zero.
The base quantities are denoted as follows in writing expressions. The symbols used are M for mass, L of length, T for time, I for current, K for temperature, mol for mole and cd for candela.
The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets to inform that it is expressed in dimensions of base quantities. Such expression of a physical quantity in terms of dimensions of base quantities is called the dimensional formula.
Order of magnitude
Convert the number into 1*10^c form.
First convert the number into a*10^b form in this case 1≤a<10 an="" and="" b="" br="" integer.="" is="">If a is less than or equal to 5 assume it is one and if a is greater than 5 assume it is 10 and convert the number into 1*10^c form.
Then c is the order of magnitude of the number.
The structure of the world
For the recent revised version of this chapter, visit
http://iit-jee-physics.blogspot.com/2008/07/chapter-1-introduction-july-dec.html
The nature around us is like a big chess game played by Nature. Various events that happen are like the moves made by a chess players. We are allowed to watch the events that happen, and guess the rules, and then play to derive benefits that we want from nature. We may across new events which do not follow old rules that we have formulated and we need to guess the new rules.
Great scientists or scientists in general guess the rules from the observations available at that time and prove the usefulness of those rules by further experiments or events that happen subsequently. These rules may require modification subsequently if they are not able to explain some events observed that happen subsequently.
Mathematics is the language of physics.
Units: Fundamental and derived
While there are a large number of physical quantities to be measured, only seven fundamental quantities are found to be sufficient. All other quantities can be measured using these seven fundamental quantities.
The set of fundamental quantities must have the following properties.
a. The fundamental quantities should be independent of each other; and
b. All other quantities may be expressed in terms of the fundamental quantities.
Fundamental quantities are also referred to as base quantities.
Who decides the units?
A body named Conference Generale des Poids or CGPM (General Conference on Weight and Measures in English) has been gibven the authority by international agreement.
Definition of base units
Metre
Kilogram
Second
Ampere
Kelvin
Mole
Candela
Metre
The distance travelled by light in vacuum in 1/[299,792,458] second is called 1 m.
Kilogram
the mass of cylinder made of planitum-iridium alloy kept at International Bureau of Weights and Measures is defined as 1 kg.
Second
The time duration in 9,192,631,770 time periods of the selected transition of radiation of Cesium-133 atom is defined as 1 s. (To understand this idea see Bohr model chapter)
Ampere
To under stand the definition of this unit, you have to read the chapter on electric field.
Two long straight wires with negligible cross section are to be placed parallel to each other at a separation of 1 m and electric current in the same amout is sent through them in the same direction. If the attractive force between the two wires adjusted, the current in each wire (one of the wires) that gives a force between them of 2*10^-7 newton per metre of the wires, is defined as 1 A of current.
Kelvin
The fraction 1/9273.16) of the thermodynamic temperature of triple point of water is called 1 K.
Mole
The amount of a substance that contain as many elementary entities (molecules or atoms if the substance is monatomic) as there are number of atoms in 0.012 kg of carbon-12 is called a mole.
Candela
It is the luminous intensity of a blackbody of surface area 1/[600,000] m² placed at the temperature of freezing platinum and at a pressure of 101,325 N/m², in the direction perpendicular to its surface.
Dimensions of physical quantities
When a physical quantity is expressed in terms of the base quantities, it is written as a product of different powers of the base quantities.
[Force] = MLT-2
M,L, and T are base quantities – mass, length and time
The exponent of a base quantity that enters into the expression (MLT-2) is called the dimension of the quantity in that base.
The dimensions or force are 1 in mass, 1 in length and -2 in time. The dimensions of all other base quantities are zero.
The base quantities are denoted as follows in writing expressions. The symbols used are M for mass, L of length, T for time, I for current, K for temperature, mol for mole and cd for candela.
The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets to inform that it is expressed in dimensions of base quantities. Such expression of a physical quantity in terms of dimensions of base quantities is called the dimensional formula.
Order of magnitude
Convert the number into 1*10^c form.
First convert the number into a*10^b form in this case 1≤a<10 an="" and="" b="" br="" integer.="" is="">If a is less than or equal to 5 assume it is one and if a is greater than 5 assume it is 10 and convert the number into 1*10^c form.
Then c is the order of magnitude of the number.
The structure of the world
For the recent revised version of this chapter, visit
http://iit-jee-physics.blogspot.com/2008/07/chapter-1-introduction-july-dec.html
Concept Review - Chapter 3 Rest and Motion: Kinematics
Rest, Motion and Reference Frame
Motion is a combined property of the object under study and the observer. There is no meaning of rest or motion without the viewer.
For example, a book placed on the table remains on the table and we say that it is not moving, it is at rest. However, if we station ourselves on the moon, the whole earth is changing its position, and os the room, the table and the book are all continuously changing their positions.
To locate the position of a particle we need a frame of reference. We can fix up three mutually perpendicular axes, name them X,Y and Z and then specify the position of the particle with respect to that frame. Then if the coordinates of the particle change with respect to time, we can say that the body is moving with respect to this frame.
Many times the choice of frame is clear from the context.
Displacement
The magnitude of the displacement is the length of the straight line joining the initial and final position.
Displacement has magnitude as well as direction (initial position and final position).
It is a vector quantity.
Displacements add according to triangle rule of vectors.
For example, if a book kept on table is displaced and the table is also displaced. The net displacement of the book is obtained by the vector addition of the two displacements.
Distance
Average speed
Instantaneous speed
Average velocity
Average speed and average velocity of a body over a specified time interval may not turnout to be same.
Example See the worked out example 2 of HC Verma's book.
The teacher made 10 rounds back and forth in the room and the total distance moved is 800 feet (10 rounds back and forth of 40 ft room). As the time taken is 50 minutes, average speed is 800/50 = 16ft/min.
But because he went out of the same door that he has entered, displacement is zero and hence average velocity is zero.
Instantaneous velocity
Average acceleration
Instantaneous acceleration
Motion in a straight line
Choose the line as the X-axis
Position of the particle at time t is given by x.
Velocity is v = dx/dt
acceleration is a = dv/dt = d²x/dt²
If accelaration is constant dv/dt = a (constant)
initial velocity = u (at time t =0)
final velocity = v (at time t)
The v = u+at
x = distance moved in time t = ut+½at²
Also
v² = u²+2ax
Motion in a plane
Motion in plane is described by x coordinate and y coordinate.
The x-coordinate, the x component of velocity, and the x component of acceleration are related by equations of straight line motion along X axis.
Similarly y components.
Projectile
Projectile motion is an important example of motion in a plane.
Vertical motion of the projectile is the motion along Y axis and horizontal motion is motion along X axis.
Terms used in describing projectile motion
Point of projection
Angle of projection
Horizontal range
Time of flight
Maximum height reached
The motion of projectile can be discussed separately for the horizontal and vertical parts.
The origin is taken as the point of projection.
The instant the particle is projected is taken as t = 0.
X-Y plane is the plane of motion.
The horizontal line OX is taken as the X axis.
Vertical line OY is the Y axis.
Vertically upward direction is taken as positive direction of Y
Initial velocity of the particle = u
Angle between the velocity and horizontal axis = θ
ux – x-component of velocity = u cos θ
ax – x component of acceleration = 0
uy – y component of velocity = u sin θ
ay = y component of acceleration = -g
Horizontal motion – Equations of motion
ux = u cos θ
ax = 0
vx = ux +axt = ux = u cos θ (as ax = 0)
Hence x component of the velocity remains constant.
Displacement in horizontal direction = x = uxt+1/2ax t²
As ax = 0, x = ux t = ut cos θ
Vertical motion – Equations of motion
uy = u sin θ
ay = -g
vy = uy – gt
Displacement in y direction = y = uyt – ½ gt²
vy² = uy² - 2gy
Time of flight of projectile
A projectile is projected from the ground at point O, and after some travel in the space, it reaches the ground at point B. The time taken for this travel is called time of flight of the projectile. Over this time, the displacement in y direction becomes zero.
Hence we can write y = uyt – ½ gt² = u sin θ*t - ½ gt²
solving we get T = (2u sin θ)/g
Time of flight of the projectile = (2u sin θ)/g
Range of the projectile:
The distance OB travelled by the projetile in the horizontal direction is the range.
OB = (u²sin 2θ)/g
Maximum height reached
When the projectile reaches the highest level in vertical direction, the vertical component of the velocity becomes zero.
The time taken for the vertical component of velocity to become zero is
vy = 0 = uy - gt = u sin θ - gt
So t = (u sin θ)/g
So time taken for vertical component of velocity to become zero is (u sin θ)/g.
Note that this time is half of Time of flight.
so maximum height reached in t = (u sin θ)/g = (u² sin²θ)/2g
Expressing velocity w.r.t. one Frame w.r.t. to a different frame
If XOY is one frame called S and X'O'Y' is another frame called S' we can express velocity of a body w.r.t. S as a combination of velocity of body w.r.t. to S' and velocity of S' w.r.t to S.
V(B,S) = V(B,S')+V(S',S)
Where
V(B,S) = velocity of body wrt to S)
V(B,S') = velocity of body wrt to S')
V(S',S) = velocity of S' wrt to S)
we can rewrite above equation as
V(B,S') = V(B,S)- V(S',S)
We can interpret the above equation in terms of two bodies. Assume S', and B are two bodies. If we know velocities of two bodies with respect to a common frame (in this case S)we can find the velocity of one body with respect to the other body (V(B,S')
Motion is a combined property of the object under study and the observer. There is no meaning of rest or motion without the viewer.
For example, a book placed on the table remains on the table and we say that it is not moving, it is at rest. However, if we station ourselves on the moon, the whole earth is changing its position, and os the room, the table and the book are all continuously changing their positions.
To locate the position of a particle we need a frame of reference. We can fix up three mutually perpendicular axes, name them X,Y and Z and then specify the position of the particle with respect to that frame. Then if the coordinates of the particle change with respect to time, we can say that the body is moving with respect to this frame.
Many times the choice of frame is clear from the context.
Displacement
The magnitude of the displacement is the length of the straight line joining the initial and final position.
Displacement has magnitude as well as direction (initial position and final position).
It is a vector quantity.
Displacements add according to triangle rule of vectors.
For example, if a book kept on table is displaced and the table is also displaced. The net displacement of the book is obtained by the vector addition of the two displacements.
Distance
Average speed
Instantaneous speed
Average velocity
Average speed and average velocity of a body over a specified time interval may not turnout to be same.
Example See the worked out example 2 of HC Verma's book.
The teacher made 10 rounds back and forth in the room and the total distance moved is 800 feet (10 rounds back and forth of 40 ft room). As the time taken is 50 minutes, average speed is 800/50 = 16ft/min.
But because he went out of the same door that he has entered, displacement is zero and hence average velocity is zero.
Instantaneous velocity
Average acceleration
Instantaneous acceleration
Motion in a straight line
Choose the line as the X-axis
Position of the particle at time t is given by x.
Velocity is v = dx/dt
acceleration is a = dv/dt = d²x/dt²
If accelaration is constant dv/dt = a (constant)
initial velocity = u (at time t =0)
final velocity = v (at time t)
The v = u+at
x = distance moved in time t = ut+½at²
Also
v² = u²+2ax
Motion in a plane
Motion in plane is described by x coordinate and y coordinate.
The x-coordinate, the x component of velocity, and the x component of acceleration are related by equations of straight line motion along X axis.
Similarly y components.
Projectile
Projectile motion is an important example of motion in a plane.
Vertical motion of the projectile is the motion along Y axis and horizontal motion is motion along X axis.
Terms used in describing projectile motion
Point of projection
Angle of projection
Horizontal range
Time of flight
Maximum height reached
The motion of projectile can be discussed separately for the horizontal and vertical parts.
The origin is taken as the point of projection.
The instant the particle is projected is taken as t = 0.
X-Y plane is the plane of motion.
The horizontal line OX is taken as the X axis.
Vertical line OY is the Y axis.
Vertically upward direction is taken as positive direction of Y
Initial velocity of the particle = u
Angle between the velocity and horizontal axis = θ
ux – x-component of velocity = u cos θ
ax – x component of acceleration = 0
uy – y component of velocity = u sin θ
ay = y component of acceleration = -g
Horizontal motion – Equations of motion
ux = u cos θ
ax = 0
vx = ux +axt = ux = u cos θ (as ax = 0)
Hence x component of the velocity remains constant.
Displacement in horizontal direction = x = uxt+1/2ax t²
As ax = 0, x = ux t = ut cos θ
Vertical motion – Equations of motion
uy = u sin θ
ay = -g
vy = uy – gt
Displacement in y direction = y = uyt – ½ gt²
vy² = uy² - 2gy
Time of flight of projectile
A projectile is projected from the ground at point O, and after some travel in the space, it reaches the ground at point B. The time taken for this travel is called time of flight of the projectile. Over this time, the displacement in y direction becomes zero.
Hence we can write y = uyt – ½ gt² = u sin θ*t - ½ gt²
solving we get T = (2u sin θ)/g
Time of flight of the projectile = (2u sin θ)/g
Range of the projectile:
The distance OB travelled by the projetile in the horizontal direction is the range.
OB = (u²sin 2θ)/g
Maximum height reached
When the projectile reaches the highest level in vertical direction, the vertical component of the velocity becomes zero.
The time taken for the vertical component of velocity to become zero is
vy = 0 = uy - gt = u sin θ - gt
So t = (u sin θ)/g
So time taken for vertical component of velocity to become zero is (u sin θ)/g.
Note that this time is half of Time of flight.
so maximum height reached in t = (u sin θ)/g = (u² sin²θ)/2g
Expressing velocity w.r.t. one Frame w.r.t. to a different frame
If XOY is one frame called S and X'O'Y' is another frame called S' we can express velocity of a body w.r.t. S as a combination of velocity of body w.r.t. to S' and velocity of S' w.r.t to S.
V(B,S) = V(B,S')+V(S',S)
Where
V(B,S) = velocity of body wrt to S)
V(B,S') = velocity of body wrt to S')
V(S',S) = velocity of S' wrt to S)
we can rewrite above equation as
V(B,S') = V(B,S)- V(S',S)
We can interpret the above equation in terms of two bodies. Assume S', and B are two bodies. If we know velocities of two bodies with respect to a common frame (in this case S)we can find the velocity of one body with respect to the other body (V(B,S')
Concept Review - Chapter 4 The Forces
Force
Force is an interaction between two objects.
Force is exerted by an object A on another object B.
Force is a vector quantity. Hence if two or more forces act on a particle, we can find the resultant force using laws of vector addition.
The SI unit for measuring the force is called a newton.
Newton's third law of motion
If a body A exerts a force F on another body B, then B exerts a force -F on A,the two forces acting along the same line.
Gravitational force
Any two bodies attract each other by virtue of their masses.
The force of attraction between two point masses is
F = Gm1m2/r²
where m1 and m2 are the masses of the particles and r is the distance between them.
G is a universal constant having the value 6.67 x 10^-11 N-m²/kg²
The above rule was given for point masses. But it is analytically found that the gravitational force exerted by a spherically symmetric body of mass m1 on another such body of mass m2 kept outside the first body is Gm1m2/r² where r is the distance between teh centres of such bodies.
Thus, for the calculation of gravitational force between two spherically symmetric bodies, they can be treated as point masses placed at their centres.
Gravitational force on small bodies by the earth
For earth, the value of radius R and mass M are 6400 km and 6 x 10^24 kg respectively. Hence, the force exerted by earth on a particle of mass m kept at its surface is, F = GMm/R². The direction of this force is towards the centre of the earth.
The quantity GM/R² is a constant and has the dimensions of acceleration.
It is called acceleration due to gravity and is denoted by letter g.
Hence, g and G are different.
Value of g is approximately 9.8 m/s².
In calculations, we often use 10 m/s².
Now we know that force exerted on a small body of mass m, kept near the earth's surface is mg in the vertically downward direction.
Gravitational constant is so small that the gravitational force becomes appreciable only when one of the masses has a very large mass.
HC Verma gives the example of Force exerted by a body of 10 kg on another body of 10 kg when they are separted by a distance of 0.5 m. The force comes out to be 2.7*10^-8 N which can hold only 3 microgram. Such forces can be neglected in practice.
Hence we consider only gravitational force exerted by earth
Electromagnetic force
Apart from gravitational force between any two bodies, the particles may exert upon each other electromagnetic forces.
If two particles having charges q1 and q2 are at rest with respect to the observer, the force between them has a magnitude
F = (1/4πε0)(q1q2/r^2)
Where ε0 = permittivity of air or vacuum = 8.8549 x 10^-12 C² /N-m²
The quantity (1/4πε0) = 9.0 x 10^9 N-m² /C²
q1, q2 = charges
r distance between q1 and q2
This is called coulomb force and it acts along the line joining the particles.
Atoms are composed of electrons, protons and neutrons.
Each electron has 1.6*10^-19 coulomb of negative charge. Each proton has an equal amount of positive charge.
In atoms, the electrons are bound by the electromagnetic force exerted on them by charge on protons. Even the combination of atoms in molecules are brought about by electromagnetic forces only. A lot of atomic and molecular phenomena result from electromagnetic forces between subatomic particles (for example, theory is put forward that charged mesons are responsible for the stability of nucleus).
Examples of electromagnetic force:
1. Bodies in contact: The contact force between bodies in contact arises out of electromagnetic forces acting between the atoms and molecules of the surfaces of the two bodies. The contact force may have a components parallel to the contact surface. This component is known as friction.
2. Tension in a string: Tension in the string is due to electromagnetic forces between atoms or electrons and protons (free electrons and nucleus in metals).
3. Force due to spring: If a spring has natural length x0 and if it is extended to x, it will exert a force
F = k|x-x0| = k|∆x|
k, the proportionality constant is called the spring constant. This force comes into picture due to the electromagnetic forces between the atoms of the material.
Nuclear forces
The alpha particles is a bare nucleus of Helium. It contains two protons and two neutrons. It is a stable object and once created it can remain intact until it is not made to interact with other objects.
The protons in the nucleus will repel each other due to coulomb force and try to break the nucleus. Why does the Coulomb force fail to break the nucleus.
There are forces called nucluear forces and they are exerted only if the interacting particles are protons or neutrons or both. They are largely atractive, but with a short range. They are weaker than the Coulomb force if the separation between particles is more than 10^-14 m. For separation smaller than this the nuclear force is stronger than the Coulomb force and it holds the nucleus stable.
Radioactivity, nuclear energy (fission, fusion) etc. result from nuclear force.
Weak forces
A neutron can change into proton and simulataneously emit an electron and a particle called antinutrino.
a proton can also change into neutron and simulataneously emit a positron (and a neutrino). The forces responsible for these changes are called weak forces. The effect of this force is experienced inside protons and neutrons only.
Scope of classical physics
Physics based on Newton's Laws of motion, Newton's law of gravitation, Maxwell's electromagetism, laws of thermodynamics and the Lorentz force is called classical physics. The behaviour of all the bodies of linear sizes greater than 10^-6 m are adequately described by classical physics. Grains of sands and rain drops fall into this range as well as heavenly bodies.
But sub atomic particles like atoms, nuclei, and electrons have sizes smaller than 10^-6 m and they are explained by quantum physics.
The mechanics of particles moving at velocity equal to light are explained by relativistic mechanics formulated by Einstein in 1905.
Force is an interaction between two objects.
Force is exerted by an object A on another object B.
Force is a vector quantity. Hence if two or more forces act on a particle, we can find the resultant force using laws of vector addition.
The SI unit for measuring the force is called a newton.
Newton's third law of motion
If a body A exerts a force F on another body B, then B exerts a force -F on A,the two forces acting along the same line.
Gravitational force
Any two bodies attract each other by virtue of their masses.
The force of attraction between two point masses is
F = Gm1m2/r²
where m1 and m2 are the masses of the particles and r is the distance between them.
G is a universal constant having the value 6.67 x 10^-11 N-m²/kg²
The above rule was given for point masses. But it is analytically found that the gravitational force exerted by a spherically symmetric body of mass m1 on another such body of mass m2 kept outside the first body is Gm1m2/r² where r is the distance between teh centres of such bodies.
Thus, for the calculation of gravitational force between two spherically symmetric bodies, they can be treated as point masses placed at their centres.
Gravitational force on small bodies by the earth
For earth, the value of radius R and mass M are 6400 km and 6 x 10^24 kg respectively. Hence, the force exerted by earth on a particle of mass m kept at its surface is, F = GMm/R². The direction of this force is towards the centre of the earth.
The quantity GM/R² is a constant and has the dimensions of acceleration.
It is called acceleration due to gravity and is denoted by letter g.
Hence, g and G are different.
Value of g is approximately 9.8 m/s².
In calculations, we often use 10 m/s².
Now we know that force exerted on a small body of mass m, kept near the earth's surface is mg in the vertically downward direction.
Gravitational constant is so small that the gravitational force becomes appreciable only when one of the masses has a very large mass.
HC Verma gives the example of Force exerted by a body of 10 kg on another body of 10 kg when they are separted by a distance of 0.5 m. The force comes out to be 2.7*10^-8 N which can hold only 3 microgram. Such forces can be neglected in practice.
Hence we consider only gravitational force exerted by earth
Electromagnetic force
Apart from gravitational force between any two bodies, the particles may exert upon each other electromagnetic forces.
If two particles having charges q1 and q2 are at rest with respect to the observer, the force between them has a magnitude
F = (1/4πε0)(q1q2/r^2)
Where ε0 = permittivity of air or vacuum = 8.8549 x 10^-12 C² /N-m²
The quantity (1/4πε0) = 9.0 x 10^9 N-m² /C²
q1, q2 = charges
r distance between q1 and q2
This is called coulomb force and it acts along the line joining the particles.
Atoms are composed of electrons, protons and neutrons.
Each electron has 1.6*10^-19 coulomb of negative charge. Each proton has an equal amount of positive charge.
In atoms, the electrons are bound by the electromagnetic force exerted on them by charge on protons. Even the combination of atoms in molecules are brought about by electromagnetic forces only. A lot of atomic and molecular phenomena result from electromagnetic forces between subatomic particles (for example, theory is put forward that charged mesons are responsible for the stability of nucleus).
Examples of electromagnetic force:
1. Bodies in contact: The contact force between bodies in contact arises out of electromagnetic forces acting between the atoms and molecules of the surfaces of the two bodies. The contact force may have a components parallel to the contact surface. This component is known as friction.
2. Tension in a string: Tension in the string is due to electromagnetic forces between atoms or electrons and protons (free electrons and nucleus in metals).
3. Force due to spring: If a spring has natural length x0 and if it is extended to x, it will exert a force
F = k|x-x0| = k|∆x|
k, the proportionality constant is called the spring constant. This force comes into picture due to the electromagnetic forces between the atoms of the material.
Nuclear forces
The alpha particles is a bare nucleus of Helium. It contains two protons and two neutrons. It is a stable object and once created it can remain intact until it is not made to interact with other objects.
The protons in the nucleus will repel each other due to coulomb force and try to break the nucleus. Why does the Coulomb force fail to break the nucleus.
There are forces called nucluear forces and they are exerted only if the interacting particles are protons or neutrons or both. They are largely atractive, but with a short range. They are weaker than the Coulomb force if the separation between particles is more than 10^-14 m. For separation smaller than this the nuclear force is stronger than the Coulomb force and it holds the nucleus stable.
Radioactivity, nuclear energy (fission, fusion) etc. result from nuclear force.
Weak forces
A neutron can change into proton and simulataneously emit an electron and a particle called antinutrino.
a proton can also change into neutron and simulataneously emit a positron (and a neutrino). The forces responsible for these changes are called weak forces. The effect of this force is experienced inside protons and neutrons only.
Scope of classical physics
Physics based on Newton's Laws of motion, Newton's law of gravitation, Maxwell's electromagetism, laws of thermodynamics and the Lorentz force is called classical physics. The behaviour of all the bodies of linear sizes greater than 10^-6 m are adequately described by classical physics. Grains of sands and rain drops fall into this range as well as heavenly bodies.
But sub atomic particles like atoms, nuclei, and electrons have sizes smaller than 10^-6 m and they are explained by quantum physics.
The mechanics of particles moving at velocity equal to light are explained by relativistic mechanics formulated by Einstein in 1905.
Concept Review - Chapter 5 Newton's Laws of Motion
(This revision material will be meaningful and useful only when you have read HC Verma's Chaper)
First law of motion
If the (vector) sum of all the forces acting on a particle is zero then and only then the particle remains unaccelerated (i.e., remains at rest or moves with constant velocity).
We can say in vector notation
a = 0 if and only if resultant force F = 0
A frame of reference in which Newton's first law is valid is called an inertial frame of reference.
A frame of reference in whch Newton's first law is not valid is called a noninertial frame of reference. (Example: lamp in an elevator cabin whose cable had broken)
Example of lamp in an elevator who cable had broken:
In the cabin when on measures with reference to the cabin, the lamp hanging from the ceiling has no acceleration. Hence the forces acting on the lamp, its weight (W) and the tension in the string supporting it are balancing each other W = T.
But for an observer on the ground, lamp is accelerating with acceleration g, when he considers the forces acting on the lamp as w and T once again, W is not equal to T as lamp is accelarating. Both cannot be right at the same time, and it means in once of the frames Newton's first law is not applicable.
Interial frame
All frames moving uniformly with respect to an inertial frame are themselves inertial.
Examples: A train moving with uniform velocity with respect to ground, a plane flying with uniform velocity with respect to a high etc. The sum of forces acting on a suit case kept on the shelves of them with turnout to be zero.
Second law of motion
The acceleration of a particle as measured from an inertial frame is given by the (vector) sum of all the forces acting on the particle divided by its mass.
a = F/m or F = ma
Acceleration and force are measured at the same instant. If force becomes zero at an instant, acceleation also becomes zero at the same instant.
Third law of motion
Psuedo forces
Inertia
First law of motion
If the (vector) sum of all the forces acting on a particle is zero then and only then the particle remains unaccelerated (i.e., remains at rest or moves with constant velocity).
We can say in vector notation
a = 0 if and only if resultant force F = 0
A frame of reference in which Newton's first law is valid is called an inertial frame of reference.
A frame of reference in whch Newton's first law is not valid is called a noninertial frame of reference. (Example: lamp in an elevator cabin whose cable had broken)
Example of lamp in an elevator who cable had broken:
In the cabin when on measures with reference to the cabin, the lamp hanging from the ceiling has no acceleration. Hence the forces acting on the lamp, its weight (W) and the tension in the string supporting it are balancing each other W = T.
But for an observer on the ground, lamp is accelerating with acceleration g, when he considers the forces acting on the lamp as w and T once again, W is not equal to T as lamp is accelarating. Both cannot be right at the same time, and it means in once of the frames Newton's first law is not applicable.
Interial frame
All frames moving uniformly with respect to an inertial frame are themselves inertial.
Examples: A train moving with uniform velocity with respect to ground, a plane flying with uniform velocity with respect to a high etc. The sum of forces acting on a suit case kept on the shelves of them with turnout to be zero.
Second law of motion
The acceleration of a particle as measured from an inertial frame is given by the (vector) sum of all the forces acting on the particle divided by its mass.
a = F/m or F = ma
Acceleration and force are measured at the same instant. If force becomes zero at an instant, acceleation also becomes zero at the same instant.
Third law of motion
Psuedo forces
Inertia
Concept Review - Chapter 6 Friction
Contact force
When two bodies are kept in contact, electromagnetic forces act between the charged particles at the surfaces of the bodies. As a result, each body exerts a contact force on the other.
Friction
The perpendicular component of the contact force is called the normal contact force and the parallel component is called friction
Kinetic friction
Static friction
Laws of friction
1. If the bodies slip over each other, the force of friction is given by
fk = μ kN
Where
fk is the force of friction
N is the normal contact force
μ k is the coefficient of friction between the surfaces
2. The direction of kinetic friction on a body is opposite to the velocity of this body with respect to the body applying the force of friction.
3. If the bodies do not slip over each other, the force of friction is given by
fs = μ sN
where
fs = static force of friction
μ s = coefficient of static friction between the bodies
N = normal force between them
The direction and magnitude of static friction are such that the condition of no slipping between the bodies is ensured.
If the applied force is small, the friction force will be equal to it. As the applied force is increased friction force also increases to a maximum limit and if additional force is applied the bodies start slipping.
4. The frictional force fk or fs does not depend on the areas of contact. It depend on normal contact force only.
Friction coefficient
When two bodies are kept in contact, electromagnetic forces act between the charged particles at the surfaces of the bodies. As a result, each body exerts a contact force on the other.
Friction
The perpendicular component of the contact force is called the normal contact force and the parallel component is called friction
Kinetic friction
Static friction
Laws of friction
1. If the bodies slip over each other, the force of friction is given by
fk = μ kN
Where
fk is the force of friction
N is the normal contact force
μ k is the coefficient of friction between the surfaces
2. The direction of kinetic friction on a body is opposite to the velocity of this body with respect to the body applying the force of friction.
3. If the bodies do not slip over each other, the force of friction is given by
fs = μ sN
where
fs = static force of friction
μ s = coefficient of static friction between the bodies
N = normal force between them
The direction and magnitude of static friction are such that the condition of no slipping between the bodies is ensured.
If the applied force is small, the friction force will be equal to it. As the applied force is increased friction force also increases to a maximum limit and if additional force is applied the bodies start slipping.
4. The frictional force fk or fs does not depend on the areas of contact. It depend on normal contact force only.
Friction coefficient
Concept Review - Chapter 7 Circular Motion
Angular position of the particle
If you visualize a particle moving in a radius of r with O as the centre of the circle. That means OP is equal to r. With O as origin mark OX on the X axis.
The position of P can be described by angle θ between Op and OX.
Angle θ is called the angular position of the particle
Angular velocity
The rate of change of angular position is called angular velocity.
ω = dθ/dt
Angular acceleration
The rate of change of angular velocity is angular acceleration.
Tangential accelartion
Radial unit vector
Tangential unit vector
Centripetal acceleration
Centripetal force
Centrifugal force
Colatitude
7.5 Circular turnings and banking of roads
When vehicles go through turnings, they travel along a nearly circular arc. That means there is centrepetal accelaration. What forces cause this acceleration? Friction fs can act towards the centre. However this may not be sufficient and the vehicle may skid.
To take care of it, the roads are banked t the turn so that outer part of the road is somewhat lifted up as compared to the inner part. therefore the normal force makes an angle θ wit hte vertical. The horizontal component of the normal force helps in providing the accelaration required.
The θ required for a speed of the vehicle of v is given by
tanθ = v²/rg
7.6 Centrifugal force
What psuedo force is required if the frame of reference rotates at a constant angular velocity ω with respect to an inertial frame?
Cetrifugal force
Coriolis force
Note: It is a common minsconception among the beginners that centrifugal force acts on a particle because the particle goes on a circle. Centrigual force acts (or is assumed to act) becasue we describe the particle from a rotating frame which is noninertial and still use Newton's laws.
7.7 Effect of earth's rotation on apprarent weight
A plumb line stays in a direction which is different from true vertical to earth at that point. The walls of building are built by making them parallel to the plumb line and not to the true vertical.
The weight of a body is mg' and not mg and g' is less than g.
g' = g only at the poles as the poles themselved do not rotate and hence the effect of earth's rotation is not felt there.
If you visualize a particle moving in a radius of r with O as the centre of the circle. That means OP is equal to r. With O as origin mark OX on the X axis.
The position of P can be described by angle θ between Op and OX.
Angle θ is called the angular position of the particle
Angular velocity
The rate of change of angular position is called angular velocity.
ω = dθ/dt
Angular acceleration
The rate of change of angular velocity is angular acceleration.
Tangential accelartion
Radial unit vector
Tangential unit vector
Centripetal acceleration
Centripetal force
Centrifugal force
Colatitude
7.5 Circular turnings and banking of roads
When vehicles go through turnings, they travel along a nearly circular arc. That means there is centrepetal accelaration. What forces cause this acceleration? Friction fs can act towards the centre. However this may not be sufficient and the vehicle may skid.
To take care of it, the roads are banked t the turn so that outer part of the road is somewhat lifted up as compared to the inner part. therefore the normal force makes an angle θ wit hte vertical. The horizontal component of the normal force helps in providing the accelaration required.
The θ required for a speed of the vehicle of v is given by
tanθ = v²/rg
7.6 Centrifugal force
What psuedo force is required if the frame of reference rotates at a constant angular velocity ω with respect to an inertial frame?
Cetrifugal force
Coriolis force
Note: It is a common minsconception among the beginners that centrifugal force acts on a particle because the particle goes on a circle. Centrigual force acts (or is assumed to act) becasue we describe the particle from a rotating frame which is noninertial and still use Newton's laws.
7.7 Effect of earth's rotation on apprarent weight
A plumb line stays in a direction which is different from true vertical to earth at that point. The walls of building are built by making them parallel to the plumb line and not to the true vertical.
The weight of a body is mg' and not mg and g' is less than g.
g' = g only at the poles as the poles themselved do not rotate and hence the effect of earth's rotation is not felt there.
Concept Review - Chapter 11 Gravitation
Universal law of gravitation
F = GMm/r²
Universal constant of gravitation (G)
G = 6.67*10^-11 N-m² /kg²
Gravitational potential energy
Gravitational potential
Gravitational potential at a point is equal to the change in potential energy per unit mass, as the mass is brought from the reference point to the given point.
Gravitational field
It is assumed that a body say A, creates a gravitational field in the space around it. The field has its own existence and has energy and momentum. When another body B is placed in gravitational field of a body, this field exerts a force on it. The direction and intensity of the field is defined in terms of the force it exerts on a body placed in it.
The intensity of gravitational field vector E at a point is defined by the equation
vector E = force vector F/mass
where F is the force vector exerted by the field on a body of mass m placed in the field. The intensity of gravitational field is abbreviated as gravitational field. Its SI unit is N/kg.
By the way they are defined, intensity of gravitational field and acceleration due to gravity have equal magnitudes and directions, but they are two separate physical quantities.
Acceleration due to gravity (g)
Variation in the value of g
Kepler's laws
Time period of a planet
Escape velocity
gravitational binding energy
Black holes
Inertial and gravitational mass
Geostationery satellite
F = GMm/r²
Universal constant of gravitation (G)
G = 6.67*10^-11 N-m² /kg²
Gravitational potential energy
Gravitational potential
Gravitational potential at a point is equal to the change in potential energy per unit mass, as the mass is brought from the reference point to the given point.
Gravitational field
It is assumed that a body say A, creates a gravitational field in the space around it. The field has its own existence and has energy and momentum. When another body B is placed in gravitational field of a body, this field exerts a force on it. The direction and intensity of the field is defined in terms of the force it exerts on a body placed in it.
The intensity of gravitational field vector E at a point is defined by the equation
vector E = force vector F/mass
where F is the force vector exerted by the field on a body of mass m placed in the field. The intensity of gravitational field is abbreviated as gravitational field. Its SI unit is N/kg.
By the way they are defined, intensity of gravitational field and acceleration due to gravity have equal magnitudes and directions, but they are two separate physical quantities.
Acceleration due to gravity (g)
Variation in the value of g
Kepler's laws
Time period of a planet
Escape velocity
gravitational binding energy
Black holes
Inertial and gravitational mass
Geostationery satellite
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