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24. Photons, X-Rays and Dual Nature of Matter 767–794
25. Atoms and Nuclei 795–830
26. Electronic Devices
27. Units and Measurements
Preface
As you all know that book market is flooded with many books for the purpose of NEET and other medical entrance examinations. But, in most of the books, the theory is given in contemporary way with only few illustrations and a set of unsolved problems at the end of chapter but there is a lack of practice problems.
I have great pleasure in presenting this book Objective Physics for NEET and Other Medical Examinations. This book is written to build a strong practice of fundamental principles of physics among the medical aspirants. This book is mainly designed for NEET/AIPMT and all other medical entrance examinations like AIIMS, JIPMER and state-level exams. In this book, each chapter starts with key concepts followed by a large number of MCQs. Hints and explanations to all the problems are given at the end each chapter. Questions from previous AIPMT and AIIMS exams with genuine solution have also been included. As in AIPMT, NCERT-based questions plays an important role hence these questions have been included with proper explanation in the Board-Text Drills section. Reason-Assertion type questions are also arranged systematically. I sincerely wish that this book will fulfill all the requirements, regarding physics, of medical aspirants and propel them to success in exams.
Although utmost care has been taken to make the book error-free, but some errors inadvertently may have crept in. I would be grateful to the students and teachers for bringing errors and mistakes to my notice, so that those may be rectified and incorporated in the subsequent editions. Please feel free to contact me at kumar.abhayk@gmail.com
—Abhay Kumar
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Acknowledgements
I express my deepest gratitude to my teachers and parents, without their support this book would not have been completed. I am also grateful to my wife, Awani; and my little daughter, Meethee. I extend my sincere thanks to the editorial team of Pearson Education for their constant support and sincere suggestions.
—Abhay Kumar
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C hapter 1 Vector and Scalar
Key ConCepts
☞ The quantities which can be measured are called physical quantities.
Physical Quantities:
Scalar Quantity: ☞ A physical quantity which is completely known by its magnitude only, i.e., a physical quantity which has only magnitude and has no direction, is called a scalar quantity or simply a scalar.
Forexample: Mass, length, volume, density, time, temperature, pressure, speed, work, etc.
Vector Quantity: ☞ A vector is that physical quantity which is completely known only when its magnitude and direction are known and obeys the laws for vectors.
Forexample: Force, acceleration, displacement, momentum, etc.
Localized Vector: ☞ A vector is said to be a localized vector, if it passes through a fixed point in space. Thus, a localized vector cannot be shifted parallel to itself.
Free Vector: ☞ A vector is said to be free vector, if it is not localized. Thus, a free vector can be taken anywhere in space.
Unless otherwise stated, all vectors will be considered as free vectors.
Vector Addition of Two Vectors: ☞ Law of parallelogram of vector addition or triangle law of vector addition:
Vector Addition of More Than Two Vectors: ☞ The above method can be applied for only two vectors, and the component method or polygon law of vector addition can be applied for resultant of two or more than two vectors.
Vector addition is commutative, i.e., if a and b be any two vectors, then
C hapter o utline
Vector addition is associative, i.e., if ab , and c be any three vectors, then abc ++() = () abc ++ .
Vector addition is distributive, i.e., if a and b be any two vectors, then mab() + = mamb +
Vector Subtraction: ☞ Vector subtraction is not a new kind of vector operation, but it is also the resultant of first vector and reverse of second vector.
If S = AB and S = || S , then S = AB +−() , S = ABAB 22 2 +− cos. θ
Null Vector: ☞ It is a vector which has zero magnitude and an arbitrary direction. It is represented by 0 and, is also known as zero vector:
(i) makes vector algebra complete.
(ii) represents physical quantities in a number of situations.
Physical Meaning of Zero Vector: ☞
(i) It represents the position vector of the origin.
(ii) It represents the displacement vector of a stationary particle.
(iii) It represents the acceleration vector of a particle moving with uniform velocity.
Rotation of a Vector: ☞
(i) If the frame of reference is rotated or translated, the given vector does not change. The components of the vector may, however, change.
(ii) If a vector is rotated through an angle θ, which is not an integral multiple of 2π, the vector changes.
Dot Product of Two Vectors: ☞ It is the multiplication of two vectors, such that the field is a scalar quantity, and it is
AB ⋅ = AB cosθ, where θ is the angle between A and B AB ⋅ = A x
4. Ordinary algebraic laws are true for a dot product.
5. If θ is acute, dot product is positive. If θ is obtuse, dot product is negative and; if θ is 90°, dot product is zero. Hence, dot product of two perpendicular vectors is zero.
6. The scalar product of two identical vectors AAA ⋅= 2 .
7. ˆˆˆˆˆˆˆˆˆˆˆˆ 1, 0 iijjkkijjkik ⋅=⋅=⋅=⋅=⋅=⋅= .
8. The scalar product of two non-zero orthogonal (i.e., perpendicular) vectors is zero.
9. The scalar product of two vectors A and B varies from AB to (–AB).
10. Scalar component of A along BA AB B == cos θ .
11. Vector component of A along ˆ AB BB B =
12. Scalar component of B along
13. Vector component of B along ˆ AB AA A
14. Vector component of A perpendicular to
15. Angle between two vectors θ =
Condition for two vectors to be parallel: ☞ If
ab and are parallel, then
Position Vector and Displacement Vector
1. If coordinates of point A are (x1, y1, z1) and coordinates of point B are (x2, y2, z2), then rA = position vector of A 111
, and 212121
ˆˆ ()()() BA rrrxxiyyjzzk =−=−+−+−=
displacement vector from A to B.
2. Position vector of the middle point of the line segment AB is given by
Cross product of two vectors: ☞ The cross product of two vectors is multiplication of two vectors, such that the yield is a vector quantity. Let CAB =× , then CCAB == || sin θθwhereis the angle between AB and . Direction of C is perpendicular to both AB and given by the RightHandLaw. We can also say that C is perpendicular to the plane containing AB and .
1. Vector product is not commutative. It is anticommutative, i.e.,
ABBA ×= −× .
2. Cross product of two vectors of given magnitudes has maximum value when they act at 90º.
3. Cross product of two parallel or antiparallel vectors is a null vector. A vector whose magnitude is zero, and has any arbitrary direction, is called as null vector or zero vector.
4. ˆˆ
,, ijkjki ×=×=
, jikiijjkk ×=−×=×=×= a null vector.
5. The magnitude of the vector product of two vectors A and B varies from 0 to AB.
6. If A and B are parallel, then
AB×= 0.
7. If
ABABAB ≠≠ ×= ⇒ 00 0 ,, || then
8. Angle θ between vectors A and B is given by sin ||
= ×
9. The geometrical meaning of vector product or cross product of two vectors is the area of the parallelogram formed by the two vectors as its adjacent sides.
10. If d1 and d 2 are the diagonals of the parallelogram, then it can be easily shown that the area of the parallelogram =× 1 2 12 dd .
11. The diagonals of a parallelogram make four triangles with sides dd 12 22 and and area of each triangle
12. Lagrange’s identity: || () ||||
Lami’s Theorem
If a body is in equilibrium under three coplanar concurrent forces, then each force is proportional to ‘sine’ of the angle between remaining two forces. That is:
Unit Vector
A vector whose magnitude is unity is called a unit vector. The unit vector in the direction of A , is denoted by  and is given by:
1. Unit vector has no unit, but magnitude of a vector has unit.
2. If ˆˆ and ij be the vector along x and y-axes respectively, then unit vector along a line which makes an angle θ with the positive direction of x-axis in anti-clockwise direction is cos ˆˆ sin ijqq + . If θ is made in clockwise direction then unit vector is cos θθ ij sin .
3. If αβ and be the unit vectors along any two lines, then αβ + and αβ are the vectors along the lines which bisect the angle between these lines.
4. A unit vector perpendicular to both ˆ and is ABC =± × ×
If vectors are given in terms of ˆˆˆ , and ijk
Let and , then xyzxyz aaiajakbbibjbk =++=++
1.
2.
3.
ˆˆ ()()() xxyyzz ababiabjabk +=+++++
ˆˆ –()()() xxyyzz ababiabjabk =−+−+−
4. Component of
Triple Product of Vectors
(A) Scalar triple product
1. If the three vectors be coplanar, their scalar triple product is zero, i.e.,
ABC ⋅× = ()
2. Value of a scalar triple product does not change when cyclic order of vectors is maintained. Thus,
()() ()
i.e., [] [] []
ABCBCACAB == Also, [] []
ABCBAC =−
3. If two of the vectors be equal, the scalar triple product is zero, i.e., [] [] .
0
4. If two vectors are parallel, the scalar triple product is zero. Let A and B are parallel, we can have BkA = , where k is a scalar. Then, [] ()
ABCkAAB =× = 0
5. The scalar triple product of the orthogonal vector triad is unity, i.e.,
[]()1 ijkijk=×⋅=
6. Scalar triple product
ABC ⋅×() represents the volume of parallelepiped, with the three vectors forming its three edges.
(B) Vector triple product: If ABC ,and are three vectors, then
ABCBCACAB ×× ×× ×× (),( )( ) and , are the examples of vector triple product.
ABCACBABC ×× =⋅ −⋅ () () ()
Polar vector: ☞ If the direction of a vector is independent of the co-ordinate system, it is called a polar vector, e.g., displacement, velocity, acceleration, etc.
Axial or pseudo vector: ☞ If the direction of a vector changes with the change of reference frame from right-handed to left-handed frame, it is called axial or pseudo vector, e.g., angular displacement, angular veclocity, etc.
Scalar and Vector Field: Gradient, Divergence, Curl
(A) Scalar Field: If a scalar changes from point to point in space, we say that there is a scalar field. For example, if we heat a rod at one end, the temperature of the rod in the steady state will vary from point-to-point and we say that there is a scalar field and that scalar is temperature. Vector Field: If a vector changes from point-to-point in space, we say that there is a vector field. For example, velocity of liquid flowing through a tube, magnetic field, electirc field, etc.
(B) The vector differential operator (del vector): The operator defined as
is called nabla or del vector. It is attributed to all the properties of a vector, and, at the same time, it is supposed to act as an operator. The most striking property of it is that it remains invariant under rotation of coordinate system.
(C) Gradient: If we operate with ∇ on a scalar ϕ, we obtain a vector which is called the gradient of the scalar. That is:
The gradient of a scalar is the rate of space variation along the normal to the surface on which it remains constant, or say, it is the directional derivative of the scalar along normal to the surface on which it remains constant. That is, grad ˆ d n
j j = , where d dn ϕ is the derivative of j along the normal and n ˆ is the unit vector along the normal.
(D) Divergence of a vector: If we make ‘del dot operation’ on a vector, we obtain a scalar which is called the divergence of the vector. That is:
(E) Curl or rotation of a vector: If we make ‘del cross operation’ on a vector, we get a vector which is called the curl of the vector. That is:
Tensor: ☞ A physical quantity which has different values in different directions at the same point is called a tensor. Pressure, stress, modulii of elasticity, moment of inertia, radius of gyration, refractive index, wave velocity, dielectric constant, conductivity, resistivity and density are a few examples of tensor. Magnitude of tensor is not unique.
exerCise (Multiple-ChoiCe Questions)
1. The (x, y, z) coordinates of two points A and B are given respectively as (0, 3, –1) and (–2, 6, 4). The displacement vector from A to B may be given by:
(a) ˆ ˆˆ 264ijk−++ (b)
233ijk−++ (c)
235ijk
2. Two vectors A1 and A2 each of magnitude A are inclined to each other, such that their resultant is equal to 3 A. Then the resultant of A1 and A2 , is:
(a) 2A (b) 3 A (c) 2 A (d) A
3. The maximum and minimum magnitude of the resultant of two given vectors are 17 units and 7 units, respectively. If these two vectors are at right angles to each other, the magnitude of their resultant is:
(a) 14 (b) 16 (c) 18 (d) 13
4. If vectors ˆˆˆˆˆˆ 35 and 3 ijkijak −+−− are equal vectors, then the value of a is: (a) 5 (b) 2 (c) –3 (d) –5
5. Given,
and . AijkBijk =++=−−−
() AB will make angle with A as: (a) 0º (b) 180º (c) 90º (d) 60º
6. If ABCAB ++ =× 0, then is:
(a) BC × (b) CB × (c) AC × (d) none of these
7. Two forces in the ratio 1: 2 act simultaneously on a particle. The resultant of these forces is three times the first force. The angle between them is:
(a) 0° (b) 60° (c) 90° (d) 45°
8. Resultant of two vectors AB and is of magnitude P. If B is reversed, then resultant is of magnitude Q. What is the value of P2 + Q2?
9. The two vectors have magnitudes 3 and 5. If angle between them is 60°, then the dot product of two vectors will be:
(a) 7.5 (b) 6.5 (c) 8.4 (d) 7.9
10. If ABC =+ and the magnitudes of AB , and C are 5, 4 and 3 units respectively, the angle between AC and is:
(a) cos
11. Two vectors AB and are such that ABC += and ABC 22 2 += .If θ is the angle between positive directions of AB and , then mark the correct alternative:
(a) θ =° 0 (b) θ π = 2 (c) θ π = 2 3 (d) θπ =
12. Which of the following operations make no sense in case of scalars and vectors?
(a) Multiplying any vector by a scalar.
(b) Adding a component of vector to the same vector.
(c) Multiplying any two scalars.
(d) Adding a scalar to a vector of the same dimensions.
13. Let ˆˆ cossin,AiAjAqq=+ be any vector. Another vector B which is normal to A is:
(a) ˆˆ cossiniBjBqq +
(b) ˆˆ sincosiBjBqq +
(c) ˆˆ sincosiBjBqq
(d) ˆˆ sincosiAjAqq
14. Which of the following is not essential for the three vectors to produce zero resultant?
(a) The resultant of any two vectors should be equal and opposite to the third vector.
(b) They should lie in the same plane.
(c) They should act along the sides of a parallelogram.
(d) It should be possible to represent them by the three sides of triangle taken in order.
15. Given that ABC ++ = 0. Which of the following options is correct?
(a) |||| ||
21. The resultant of ABR + is 1 On reversing the vector B , the resultant becomes R2 . What is the value of RR 1 2 2 2 + ?
(a) AB22 + (b) AB22
(c) 2 22 () AB + (d) 2 22 () AB
22. A vector of length l is turned through the angle q about its tail. What is the change in the position vector of its head?
ABC += (b) || ||
ABC −= (d) || || ABC −=
ABC += (c) |||| ||
16. Given that CABC =+ and makes an angle α with AB andwith β Which of the following options is correct?
(a) α cannot be less than b
(b) αβ<< ,if AB
(c) αβ<> ,if AB
(d) αβ<= ,if AB
17. Which of the following operations will not change a vector?
(a) Rotation in its own plane
(b) Rotation perpendicular to its plane
(c) Rotation about the tail
(d) None of the above
18. A is directed along north and B is directed along south-west. If CAB =+ , then which of the following relations are correct?
(a) l cos (q/2) (b) 2l sin (q/2) (c) 2l cos (q/2) (d) l sin (q/2)
23. The diagonals of a parallelogram are 2i ˆ and 2j ˆ . What is the area of the parallelogram?
(a) 0.5 unit (b) 1 unit
(c) 2 unit (d) 4 unit
24. A parallelogram is formed with a and b as the sides. Let d1 and d 2 be the diagonals of the parallelogram. Then ab22+= … :
(a) dd 1 2 2 2 + (b) dd 1 2 2 2
(c)
25. Resultant of three non-coplanar non-zero vectors ab , and c : (a) always lies in the plane containing ab + . (b) always lies in the plane containing ab . (c) can be zero (d) cannot be zero
AB +
(a) C must be equal to ||
(b) C must be greater than ||
AB +
(c) C must be greater than ||
AB
(d) C must be equal to || AB
19. What is the component of ˆˆˆˆ 34along ? ijij ++
26. [] abc is a scalar triple product of three vectros ab , and c then [] abc is equal to (a) [] abc (b) [] cab (c) [] acb (d) [] acb
27. The number of vectors of unit length perpendicular to vectors a = (1, 1, 0) and b = (0, 1, 1) is (a) one (b) two (c) three (d) infinite
28. If the vector, ˆˆˆˆˆˆ , aijkibjk ++++ and ˆ ˆˆ ijck ++
20. Component of ˆˆ 34jj + perpendicular to ˆˆ ij + and in the same plane as that of ˆˆ 34ij + is: (a) 1 ˆˆ () 2 ji (b) 3 ˆˆ () 2 ji
() 2 ji (d) 7
() 2 ji
(a ≠ b,c ≠ 1) are coplanar, then the value of 1 1 1 1 1 1 + + abc is:
(a) –1 (b) 0 (c) 1 (d) 3
29. If ˆˆ ˆˆˆˆ ()()() uiaijajkak =××+××+×× then: (a) u is a unit vector (b) ˆ ˆˆ uaijk =+++
(c) ua = 2 (d) ˆ ˆˆ 8() uijk =++
30. The vector sum of two forces is perpendicular to their vector differences. In that case, the forces:
(a) can not be predicted.
(b) are perpendicular to each other.
(c) are equal to each other in magnitude.
(d) are not equal to each other in magnitude.
31. A and B are two vectors given by ˆˆ 23 Aij =+ and ˆˆ Bij =+ The magnitude of the component of A along B is:
(a) 5 2 (b) 3 2 (c) 7 2 (d) 1 2
32. If a and b are two vectors, then the value of () () abab +× is:
(a) ab × (b) ba × (c) −×2() ba (d) 2() ba ×
33. What should be the angle between ∆ A and A , so that || ||∆∆ AA = ?
(a) 0° (b) 30° (c) 60° (d) 90°
34. The direction of a vector A is reversed. What are the values of ∆ A and ∆ || A ?
(a) +20 A, (b) + A,0 (c) 20 A, (d) A,0
hints and explanations
1. (c) ˆˆˆˆˆˆ 03,264AB rijkrijk =+−=−++
Displacement vector from A to B is given by d
(264)(03)
2. (d) Let q be the angle between A1 and A2 . Resultant of A1 and A2 is RAAAA 2 1 2 2 2 12 2 =+ + cos θ or 32 22 2 AAAAA =+ + cos, θ or coscos θ == 1 2 60º or q = 60º.
The angle between A1 and A2 is (180º – 60º) = 120º
⇒+ += = 92 1625 20ABABcoscos] θθ or or cos θ = 0 ∴ θ = 90º.
2. The resultant of A × 0 will be equal to: (a) zero (b) A (c) zero vector (d) unit vector (1991)
Solution: (c) The cross product AB × is a vector, with its direction perpendicular to both A and B AB × is area. If side B is zero, area is zero.
A × 0 is a zero vector.
and is
If in case 0 is a scalar, then also the product is zero. But, a scalar × a vector is also a vector Hence, one gets a zero vector in any case.
3. The angle between the two vectors
345and345 AijkBijk =++=+−
(a) 90° (b) 180° (c) zero (d) 45°
Solution: (a)
ˆˆ 345and345. AijkBijk =++=+−
will be
(345)(345) (3)(4)(5)(3)(4)(5) ijkijk +++−
++++ = +− == ° 91625 50 0or09 θ .
4. Identify the vector quantity among the following.
(a) distance (b) angular momentum (c) heat (d) energy (1997)
Solution: (b) Since the angular momentum has both magnitude and direction, it is a vector quantity.
5. If a unit vector is represented by ˆ ˆˆ 0.50.8ijck −+ then the value of c is:
(a) 00 1. (b) 01 1. (c) 1 (d) 03 9. (1999)
Solution: (b) For a unit vector ˆˆ,||1 nn = 22 2 ˆ ˆˆ |0.50.8|10.250.641 ijckc −+=⇒++= or c = 01 1.
6. If || || || ABAB += + , then angle between A and B will be:
ˆˆˆˆˆˆ (238)(44)0 ijkijk a ++⋅−++= or, /. −+ += ⇒+ = ⇒= 8128 04 80 12 aa a
9. If the angle between the vectors ABandis, θ the value of the product () BAA ×⋅ is equal to: (a) BA2 sinq (b) BA2 cosθ (c) BA2 sinqcosq (d) zero (1989, 2005)
Solution: (d) Let ABC ×=
The cross product of ABand, perpendicular to the plane containing ABand, i.e., perpendicular to B . If a dot product of this cross product and A is taken, as the cross product is perpendicular to ACA,. ×= 0
Therefore, product of () . BAA ×⋅ = 0
10. The vectors AB and are such that || || ABAB +=
ABAB += The angle between the two vectors is: (a) 45º (b) 90º (c) 60º (d) 75º (1991, 96, 2006)
Solution: (b) Let q be angle between AB and || ||,| || |
ABABABAB += =− then 22 or ()() ()()
ABABABAB +⋅ += −⋅ or
AAABBABB AAABBABB ⋅+⋅+⋅+ ⋅ =⋅ −⋅−⋅ +⋅ or 40 0 90 AB coscosº º θθ θ == = or or
11. AB and are two vectors and q is the angle between them, if || (), ABAB ×= ⋅ 3 the value of q is:
(a) 45º (b) 30º (c) 90º (d) 60º (2007)
Solution: (d) || (. ) ABAB ×= 3
∴ ABABsincosθθ = 3 or, tan θ = 3 or θ == ° tan( ). 1 360
1. Assertion: A physical quantity that has both magnitude and direction is not necessarily a vector quantity.
Reason: For a physical quantity to be vector, the commutative law must hold for the addition of such two physical quantities.
2. Assertion: Any two vectors can be added.
Reason: The vectors are added by applying the laws of algebra.
3. Assertion: The resultant of any three vectors lying in the same plane is zero.
Reason: Any three vectors lying in the same plane can be represented by the three sides of a triangle taken in order.
4. Assertion: The magnitude of the resultant of two vectors is always greater than magnitude of the individual vectors.
Reason: It is in accordance with the laws of algebra.
5. Assertion: The magnitude of the resultant of two vectors PQ and is maximum (P + Q), when the two vectors act in the same direction
and minimum (P – Q), when they act in opposite directions.
Reason: The resultant of two vectors can be found by using the relation RPQPQ =+ + 22 2cos θ
6. Assertion: If || ||, ABAB += then vectors A and B must be at right angles to each other.
Reason: The vectors AB + and AB are always at right angles to each other.
7. Assertion: If ABAB += , then vector B must be a zero vector.
Reason: It is because, by definition of null vector AA ±= 0
Hints:
1. (a) Both are true.
2. (d) A vector can be added to another vector of same nature only.
3. (d) Both are false.
4. (d) Both are false.
5. (b) When the two vectors act in the same direction, q = 0º and when they act in opposite directions, q = 180º. By setting q = 0º and q = 180º, from the relation
RPQPQ =+ + 22 2cos, θ
it can be obtained that the resultant is (P + Q) and (P – Q) in the respective cases.
6. (c) The assertion is true but the reasoning is false.
7. (a) Both are true.
Board text drills
1. State for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
Solution: Scalars are: volume, mass, speed, density, number of moles and angular frequency.
Vectors are: acceleration, velocity, displacement and angular velocity.
2. Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton’s third law, relative velocity.
Solution: Scalar quantities are: work and current
3. Pick out the only vector quantity in the following list: temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
Solution: Vector quantity is: impulse
4. Read each statement, below, carefully and state with reasons if it is true or false:
(a) The magnitude of a vector is always a scalar.
(b) Each component of a vector is always scalar.
(c) The total path length is always equal to the magnitude of the displacement vector of a particle.
(d) The average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time.
(e) Three vectors not lying in a plane can never add up to give a null vector.
Solution:
(a) True. Magnitude of a vector is a scalar. For example, velocity of a car is 20 m/s. Here, magnitude of the velocity is 20 which is scalar.
(b) False. As each component of a vector is not always scalar.
(c) False. A body moves on a circular path of radius r returns to its original position after
some time. Here, magnitude of the total path is 2πr while magnitude of displacement is zero.
(d) True. The total path length is either greater of equal to the magnitude of the displacement.
(e) True. As to get a null vector, the third vector should have the same magnitude and opposite direction to the resultant of the two vectors.
5. Establish the following vector inequalities geometrically or, otherwise:
(a) |a + b| ≤ |a| + |b|
(b) |a + b| ≥ ||a|
(c) |a – b|
(d) |a – b| ≥ ||a| – |b||
When does the equality sign, above, apply?
Solution:
(a) || || abR += is given by || cos Rabab =+ + 22 2 θ when cos, º, || || θθ== =+ =+ + 10 2 22 Rab abab
=− ° () || ||abab if cos =1or = 0 θθ >− =− ° () || ||abab if cos < 1or > 0 θθ
Combining the above results, we get: || || || || abab −≥
The equality sign will apply, if q = 0º or the two vectors ab and are parallel or ab and are anti-parallel.
6. Given abcd ++ += 0, which of the following statements are correct:
(a) abcd ,, and must each be null vectors.
(b) The magnitude of () ac + equals the magnitude of () bd + .
(c) The magnitude of a can never be greater than the sum of the magnitudes of acd ,. and (d) bc + must lie in the plane of ad and , if ad and are not collinear and in the line of ad and , if they are collinear?
Solution:
(a) Wrong (b) Correct (c) Correct (d) Correct
7. ˆˆ and ij are units vectors along x- and y-axis respectively. What is the magnitude and direction of the vectors ˆˆˆˆ and ijij +− ? What are the components of vector ˆˆ 23 Aij =+ = along the direction of ˆˆˆˆ and ijij +− ?
Solution: (i) ˆˆ and ij a are unit vectors at right angles to each other represented by OAOB →→ and . Their resultant is given by OC → is R =+ +× ×× °= 11 21 1902 22 cos and direction will be, tan ˆ || || Bj Ai a ==
== °1or45 α .
Thus, magnitude and direction of the vector ˆˆ is2 ij + is along 45º with x-axis.
Now, ˆ j can be represented by OB ′ → . Resultant of ˆˆ and ij
=+ −+ ××× °= ′ → 11 21 1902 22 () cos is OC and direction of the resultant is at –45° with x-axis.
(ii) The vector ˆˆ 23ij + can be represented by OP → which has x-component = 2 and y component = 3. It makes an angle q with the x-axis such that
tantan . θθ==
3 2 3 2 1 or The angle between
ˆˆ and Aij + .
[between OP and OC] = ∠POC = q – 45°
The angle between ˆˆ and is 45 AijPOC q ′ −∠=+°
Now, the magnitude of OPA → == += 23 13 22
The component of A along the direction of ˆˆ ij+= component of ˆˆ ij + along AC
=∠ =− 13 1345coscos (º ) POC θ
=+134545 [cos cossin sin] θθ
Note: In a right-angled ∆= OPQOPOQ ,, 13 ==23 ,. QP
Hence, sin. θθ== 3 13 2 13 andcos
The component of A along the direction of ˆˆ ij−= component of AAC along ′
=∠ =+ 13 1345 cos, cos( º) POC θ
=−134545 [cos cossin sin] θθ
8. Which of the following quantites are independent of the choice of orientation of the coordinate axes?
abababc xy ++ +− ,, [], 32 angle between a and ca , λ where l is a scalar.
Solution: All the quantities except (3a x + 2b y) are independent of the choice of orientaion of the coordinate axis.
9. A vector has magnitude and direction.
(a) Does it have a location in space?
(b) Can it vary with time?
(c) Will two equal vectors ab and at different locations in space necessarily have identical physical effects? Give examples in support of your answer.
Solution:
(a) The answer to this query is ‘No’. Vectors do not have a location in space as everything is moving and, thus has no fixed frame of reference in space. For example, the sun with its solar system is moving in the space.
(b) Yes, it can vary with time. For example, velocity and acceleration vectors vary with time.
(c) No. two identical vectors ab and at different locations in space will not have identical physical effects. A good example is a ball thrown at the moon and at the earth with same force will cover different maximum heights due to the difference in gravitational force at moon and earth.
10. A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?
Solution: A physical quantity that has both magnitude and direction does not mean that it is a vector quantity. Finite rotation of a body about an axis is not a vector quantity because finite rotation does not obey the commutative law of addition.
11. Can you associate vectors with: (a) the length of a wire bent into a loop; (b) a plane area; (c) a sphere. Explain.
Solution:
(a) We can associate a vector with the length of a wire bent into a loop.
(b) We can associate a vector with a plane area
Alblb =× ,where and are the length and breadth vectors.
(c) We can associate a null vector with a sphere.
C hapter 2
Kinematics of 1-D, 2-D, 3-D and Circular Motion
Motion in a Straight Line: Speed and Velocity, Position-time Grapgh, Velocity-time Graph, Accelerationtime Graph ■ Uniform and Non-uniform Motion ■ Average Speed and Instantaneous Velocity ■ Uniformly and Non-uniformly Accelerated Motion ■ Relations for Uniformly Accelerated Motion ■ Relative Velocity as Rate of Change of Separation ■ Closest Distance of Approach Between Two Moving Bodies ■ Motion in a Plane or Two-dimensional Motion: Projectile Motion, Kinematics of Circular Motion
