Mathematics for Physics I

 1. Algebra

Common Formulas:

1. $ (a+b)^2 = a^2+2ab+b^2 $

2. $ (a+b)^3 = a^3+b^3+3a^2b+3ab^2 $

3. $ (a^2-b^2)=(a+b)(a-b) $

4. $ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca $

5. $ (a+b)^2 + (a-b)^2 = 2(a^2+b^2) $

6. $ (a+b)^2 - (a-b)^2 = 4ab $

7. $ (a-b)^3 = a^3-b^3-3a^2b+3ab^2 $

Solving Quadratic Equation: Let any quadratic equation be $ ax^2+bx+c = 0 $. Roots of the equation are given by,  $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $

If $ {b^2-4ac} = 0 $ then roots are real and equal.

If $ {b^2-4ac} > 0 $ then roots are distinct and real.

If $ {b^2-4ac} < 0 $ then roots are imaginary.

If $ \alpha $ and $ \beta $ are two roots of the equations, then  

Sum of roots: $ \alpha + \beta = \frac{-b}{a} $

Product of roots: $ \alpha \times \beta = \frac{c}{a} $

Difference of roots: $ \alpha - \beta = \frac{\sqrt{b^2 - 4ac}}{a} $

For example, Let $ x^2+x+1=0 $ is a quadratic equation and we need to find the roots of the equation.

For the given equation, $ b^2-4ac = 1 - (4 \times 1 \times 1) = 1-4 = -3  < 0 $

This means that the roots of the equation are imaginary. 

Roots will be, $ x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 - \sqrt{-3}}{2}, \frac{-1 + \sqrt{-3}}{2} $

For $ \sqrt{-1} = i $, then roots will be, \[x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 - \sqrt{3}i}{2}, \frac{-1 + \sqrt{3}i}{2}\]

Binomial Expansion:

If we need to expend $ (1+x)^n $ in powers of x where n is positive integer, we expand it binomially.

Expansion will be 

\[(1+x)^n = 1 + _{1}^{n}{x} + _{2}^{n}{x^2} + ....... + _{i}^{n}{x^i} + ..... + x^n \]

It can be written as \[(1+x)^n = \Sigma _{j=0}^{n} (_{j}^{n} C)(x^j)\]

where \[_{j}^{n} C = \frac{n!}{(n-j)! j!}, n! = n(n-1)(n-2).......3.2.1\] and $ 0! = 1 $.

The number of terms in the expansion of $ (1+x)^n $ are (n+1)

Binomial expansion for any index, i.e. if n is not a positive integer.

\[(1+x)^n = 1 + \frac{n}{1!}x + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + .......... \infty terms\]

If |x| << 1 then $ (1+x)^n = 1 + nx $ i.e. we can ignore highest power of the expansion.

For example: Expand $ (1 + x ) ^ {-2} $.

$ (1+x)^{-2} = 1 + \frac{-2}{1!}x + \frac{-2(-2-1)}{2!}x^2 + \frac{-2(-2-1)(-2-2)}{3!}x^3 + .......... \infty = 1 -2x -3x^2 - 4x^3 + ......  $. 

Try Yourself:

Q1. Find the roots of the equations:

(a) $ x^2 + 2x + 3 = 0 $

(b) $ x^2 - 2x - 3 = 0 $

(c) $ x^2 + 30x  + 1 = 0 $

(d) $ 2x^2 + x + 1 = 0 $

 

Q2. Expand following:

(a) $ (1+x)^7 $

(b) $ (1+x)^{-7} $

(c) $ (1+y)^{-1} $

(d) $ (1+z)^{-10} $

***Solutions of the above problems will be uploaded soon..... 

2. Trigonometry

Relation between arc length, l, radius of the circle, r, and angle $ \theta $ subtended by the arc at the center, \[l = r \theta\]

 

Usefull Trigonometric Formulas for Right Angle Triangle 

Let any right angle triangle with Right angle at B, as shown in the figure click here to see large image. 

1. $ sin A = \frac{a}{b} $

2. $ cosA = \frac{c}{b} $

3. $ tanA = \frac{sin A}{cosA}=\frac{a}{c} $

4. $ cosecA = \frac{1}{sinA}=\frac{b}{a} $

5. $ secA = \frac{1}{cosA}= \frac{b}{c} $

6. $ cot A = \frac{1}{tanA}= \frac{cosA}{sinA}= \frac{c}{a}  $

7. $ sin^2 A + cos^2 A = 1 $ 

8. $ 1 + tan^2 A = sec^2 A $

9. $ 1+ cot^2 A = cosec^2 A $

Above formulas are valid only for Right angle triangle.

In general, for any triangle ABC where A, B and C are the angles, and a, b, and c are the sides opposite to angle A, B and C respectively. 

1. $ \frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c} $

2. $ cos A = \frac{b^2+c^2-a^2}{2bc} $ 

Value of some trigonometric functions:

	0	30	45	60	90
sin	0	$ \frac{1}{2} $	$ \frac{1}{\sqrt{2}}$	$ \frac{\sqrt{3}}{2}$	1
cos	1	$ \frac{\sqrt{3}}{2}$	$ \frac{1}{\sqrt{2}}$	$ \frac{1}{2} $	0
tan	0	$\frac{1}{\sqrt{3}}$	1	$ \sqrt{3} $	$ \infty $

Compound Formula:

1. $ sin(A\pm B) = sinAcosB \pm sinBcosA $

2. $ cos(A\pm B) = cosAcosB \mp sinAsinB $

3. $ tan(A\pm B) = \frac{tanA\pm tanB}{1 \mp tanAtanB} $

4. $sin2A = 2sinAcosA $

5. $ cos2A = cos^2A - sin^2 A = 1 - 2sin^2 A = 2cos^2 A - 1 $

6. $tan2A = \frac{2tanA}{1-tan^2A} $

In the first quadrant, all trigonometric functions have positive values. In second quadrant, sine and cosec are positive and all others are negative. In third quadrant, tan and cot are positive and all others are negative. In fourth quadrant, cos and sec are positive and all other are negative.

1. $ sin(-\theta) = - sin \theta $

2. $ cos(-\theta) = cos \theta $

3. $ tan(-\theta) = - tan \theta $

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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 $.

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Kinematics

1. The branch of Physics which deal with the study of motion of material objects is called Mechanics. Mechanics can be broadly classified into following branches :-

(i) Statics: It is a branch of mechanics that deals with the study of material objects at rest. 

(ii) Kinematics: It is that branch  of mechanics which deals with the study of the motion of material objects without taking into account the factors (i.e., nature of forces, nature of bodies, etc.) 

(iii) Dynamics: It is that branch of mechanics which deals with the study of motion of objects taking into account the factors which cause motion.                  

2. Rest: An object is said to be at rest if it does not change its position with time, with respect to its surroundings/observer.

3. Motion: An object is said to be in motion if it changes its position with time, with respect to its surroundings/observer.

4. Types of motion: 

(i) Rectilinear or translatory motion: Rectilinear motion is that motion in which a body, which is not a point mass body is moving along a straight line. Translatory motion is that motion in which a body, which is not a point mass body is moving such that all its constituent particles move simultaneously along parallel straight lines and shift through equal distance in a given interval of time. 

(ii) Circular or Rotatory motion: A circular motion is that motion in which a particle or a point mass body is moving on a circle. A rotatory motion is that motion in which a body, which is not a point mass body, is moving such that all its constituent particles move simultaneously along concentric circles, whose centers lie on a line, called the axis of rotation and shift through equal angle in a given time.

(iii) Oscillatory or Vibratory motion: Oscillatory motion is that motion in which a body moves to and fro or back and forth repeatedly about a fixed point (called mean position) in a definite interval of time. If in the oscillatory motion, the amplitude is very small, i.e., microscopic, the motion of body is said to be a vibratory motion.

5. Point mass object: An object can be considered as a point object if, during motion in a given time, it covers distances much greater than its own size.

6. Frame of reference: The frame of reference is a system of coordinate axes attached to an observer having a clock with him, with respect to which, the observer can describe position, displacement, acceleration etc. of a moving object. Inertial frame of reference is one in which Newton’s first law* of motion holds good. The non-inertial frame of reference is one in which Newton’s first law of motion does not hold good.

7. One dimensional motion: The motion of an object is said to be one-dimensional motion if only one out of the three coordinates specifying the position of the object changes with respect to time. For example, the motion of a train along a straight railway track, an object dropped from a certain height above the ground, a man walking on a level and narrow road, oscillations of a mass suspended from a vertical spring etc. belong to one-dimensional motion.

8. Two-dimensional motion: The motion of an object is said to be dimensional motion if two out of the three coordinates specifying the position of the object change with respect to time. For example, an insect crawling over the floor.

9. Three-dimensional motion: The motion of an object is said to be three-dimensional motion if all the three coordinates specifying the position of the object change with respect to time. For example : a kite flying on a windy day, the random motion of a gas molecule, a flying airplane or bird etc. belong to this type of motion.

10. The point followed by a point object during its called trajectory.

11. Scalar quantities or scalars: The physical quantities which have only magnitude but no direction, are called scalar quantities or scalars. For example, distance, length, work, charge, current, speed etc

12. Vector quantities or vectors: The physical quantities which have magnitude as well as direction are called vector quantities or vectors. Geometrically or graphically, a vector is represented by a straight line with an arrowhead, i.e. arrowed line.

13. Path length and displacement: The path length of an object in motion in a given time is the length of actual path traversed by a object in the given time. The displacement of an object in a motion of a given time is defined as the change in position of the object, i.e., the difference between the final and initial positions of the object of a given time. It is the shortest distance between the two positions of the object and its direction is from initial to final position of the object, during the given interval of time. It is represented by the vector drawn from the initial position to its final position. Path length is scalar and displacement is a vector quantity. Both are measured in meter (m).

14.  Speed: Speed of an object in motion is defined as a ratio of total path length (i.e., actual distance covered ) and the corresponding time taken by the object, i.e., 

             Speed = (total path length) / (time taken)    

15. Uniform speed: An object is said to be moving with a uniform speed, if it covers equal distances in equal intervals of time, howsoever small these intervals may be.

16. Variable speed: An object is said to be moving with a variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time, howsoever small these intervals may be.

17. Average speed: When an object is moving with a variable speed, then the average speed of the object is that constant speed with which the object covers the same distance in a given time as it does while moving with variable speed during the given time. Average speed for the given motion is defined as the ratio of the total distance traveled by the object to the total time taken i.e.,

                                             Average speed  =  (total distance traveled) / (total time taken)

If a particle travels distances S₁,S₂,S₃ etc. with speed v₁,v₂, v_3, etc. respectively, in same direction then total distance travelled = S₁ + S₂ + S₃ +…….  

Total time taken, \[t = \frac{S_1}{v_1}+\frac{S_2}{v_2}+\frac{S_3}{v_3}+.......\]

Total distance, \[S = S_1 + S_2 + S_3 +.....\]

Average velocity, \[V_{av} =\frac{S_1 + S_2 + S_3 +.....}{\frac{S_1}{v_1}+\frac{S_2}{v_2}+\frac{S_3}{v_3}+....... }\]

18. Instantaneous speed of an object at an instant of time t is defined as the limit of the average speed as the time interval () at the given instant of time, becomes infinitesimally small.                                    Instantaneous speed, \[\lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}\]

19. Velocity of an object in motion is defined as the ratio of displacement and the corresponding time interval taken by the object, i.e., Velocity = (displacement)  / (time interval). 

Uniform velocity is that velocity of an object with which, it undergoes equal displacements in equal intervals of time howsoever small these intervals may be.

20. Graph: 

If an object is at rest, then the position-time graph is a straight line parallel to the time axis. see figure

If the object is in uniform motion along a straight line, starting from origin O, then the position-time graph is straight-line inclined to the time axis.   see figure

If an object is moving with constant negative velocity starting from a positive position then the position-time graph is a straight line. see figure

If the object is in non-uniform motion along a straight line then the position-time graph is a curve other than a straight line.    see figure

21. Displacement of the body is equal to the area of the velocity-time graph, during a given interval of time which is added with the proper sign. The area above the time axis is taken as positive while below is taken as negative in the velocity-time graph. 

22. If the body is moving along the straight line then the magnitude of the velocity and speed is equal and distance and displacement are also same. In uniform motion, the slope of the line gives the velocity of the object. Speed can never be negative.

23. The slope of velocity-time graph gives the acceleration.

24. It two bodies are moving with unequal velocities, their position-time graph must intersect each other.

25. Relative velocity of object B w.r.t. object A, \[\vec{v_{AB}} = \vec{v_B} - \vec{v_A}\]

26. Acceleration of an object in motion is defined as the ratio of change in velocity and the corresponding time taken by the object, i.e., Acceleration = (change in velocity) / (time taken)

27. Uniform acceleration. An object is said to be moving with a uniform acceleration if its velocity changes by equal amounts in equal intervals of time.

28. Variable acceleration. An object is said to be moving with a variable acceleration of its velocity changes by unequal amounts in equal intervals of time.

29. Average acceleration. When an object is moving with variable acceleration, then the average acceleration of the object for the given motion is defined as a ratio of the total change in velocity of the object during motion to the total time taken i.e., \[\vec{a_{av}} = \frac{\Delta \vec{v} }{\Delta t}\]

The slope of straight line joining two points on the velocity-time graph gives the average acceleration of the object between these two points. The average acceleration can be positive or negative depending upon the sign of the slope of the velocity-time graph. It is zero if the change in velocity of the object in the given interval of time is zero.

30. Instantaneous acceleration. When an object is moving with variable acceleration, then the object possess different acceleration at different instants.  \[\vec{a} = \lim_{\Delta t \to 0} (\frac{\Delta \vec{v} }{\Delta t}) = \frac{d\vec{v}}{dt}\]Instantaneous acceleration is also defined as the tangent to the velocity time graph at a position, corresponding to given instant of time.

31. The velocity-time graph for the motion for uniform velocity is parallel to the time axis. Velocity time graph of the accelerated motion is straight-line inclined with x-axis.

32. Equation of motion: \[v = u + at\] \[s = ut + \frac{1}{2}at^2\] \[v^2 = u^2 + 2as\]

where u is initial velocity, v is final velocity, a is acceleration, t is time, and s is displacement.

33. Distance travelled in nth second of uniformly accelerated motion is given by\[D_n = u + \frac{a}{2}(2n-1)\]

34. For motion under free fall, the equations of motion will be modified as \[v = gt\] \[h = \frac{1}{2}gt^2\] \[v^2 = 2gh\] assuming initial velocity is zero.

35. Relation time is the time which a person takes to observe, think, and act.

36. Acceleration of the body is given by the first derivative of velocity and second derivative of the position. Velocity is the first derivative of the position. Distance/Displacement is given by integration of the velocity with time. While velocity is given by integration of the acceleration.

 37. Polar vectors. These are those vectors that have a starting point or a point of application.

38. Axial vectors. These are those vectors that represent the rotational effect act along the axis of rotation in accordance with right-hand screw rule.

39. Modulus of a vector. The magnitude of a vector is called the modulus of that vector. The magnitude of vector $ \vec{A} $ is given by | A |. 

40. Unit vector a unit vector is the given vector is a vector of unit magnitude and has the same direction as that of the given vector. A unit vector in a given direction is also defined as a vector in that direction divided by the magnitude of the given vector. It is unitless and dimensionless vector and represents direction only.     \[\hat{A} = \frac{\vec{A}}{|A|}\]

41. Equal vectors. Two vector are said to be equal magnitude and same direction.

42. Negative vector. A negative vector of a given vector is a vector of same magnitude but acting in a direction opposite to that of the given vector.

43. Co-initial vectors. The vectors are said to be co-initial if their initial point is common.

44. Collinear vectors. These are those vectors which are having equal or unequal magnitudes and are acting along the parallel straight lines.

45. Coplanar vectors. These are those vectors which are acting in the same plane.

46. The multiplication of a vector A by a real number n becomes another vector n A. its magnitude becomes n times the magnitude of the given vector. Its direction is the same or opposite as that of A, according as n is a positive or negative real number. When a vector A is multiplied by a scalar S, it becomes a vector S A, whose magnitude is S times the magnitude of A and it acts along the direction of A . the unit of S A, is different from the unit of vector A. 

46. Resultant vector of two or more vectors is defined as that single vector which produced the same effect as is produced by individual vectors together.

47. Vector addition, It is state that the vectors to be added are arranged in such a way so that the head of first vector coincides with the tail of second vector, whose head coincides with the tail of third vector and so on, then the single vector drawn from the tail of the first vector to the head of first vector represent their resultants vector.

48. Triangle law of vectors states that if two vectors acting on a particle at the same time are represented in magnitude and direction by two sides of a triangle taken in one order, their resultant vector is represented in magnitude and directed by the third side of the triangle taken in the opposite order. Let two vectors $ \vec{A} $ and $ \vec{B} $ acting at an angle $ \theta $ is given by \[R = \sqrt{A^2 + B^2 + 2ABcos\theta }\] Direction of resultant vector with vector A is given by \[tan\alpha = \frac{Bsin\theta}{A + Bcos\theta}\]

49. Parallelogram law  of vector s state that if two vectors acting on a particle at the two adjacent side of a parallelogram drawn from a point, their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point. Resultant of two vector is same as given by triangle law of vector addition.

50. Polygon law of vectors states that if any number of vectors, acting on a particle at the same time are represented in magnitude and direction by various sides of an open polygon taken in the same order, their resultant is represented in magnitude and direction by the closing side of the polygon taken in opposite order.

51. Lami’s theorem. It states that if three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. i.e., \[\frac{A}{sin\alpha} = \frac{B}{sin\beta} = \frac{C}{sin\gamma}\]

52. The object is in equilibrium if there is no linear motion of the object i.e., the resultant force on the object is zero, There is no rotational motion of the object i.e., the torque due to forces on the object is zero and There is minimum potential energy of the object for stable equilibrium.

53. A single vector which balances two or more vectors acting on a body at the same time is called an equilibrant vector.

54. It is that vector which is zero magnitude and an arbitrary direction. A zero vector is represented by 0 (arrow over the number zero). It is also called null vector. When a vector is multiplied by zero, the result is a zero vector. i.e. 0 (A) = 0. The result of  addition of a vector to its own negative vector is a zero vector, i.e. A + (-A) =0.

55. Resolution of a vector, It is the process of splitting a single vector into two or more vectors in different directions which together produce the same effect as it produced by the single vector alone. The vectors into which the given single vector is split are called component vectors.

56. Uniqueness of component vectors. It is to be noted that the resolution of vector R into two component vectors along the direction of vector A and B is unique. 

57. A body is said to be projectile if it has motion in two dimensions i.e. a projectile should have two-component velocities in two mutually perpendicular directions.  

58. For horizontal projectile, Equation of trajectory is given by \[y = \frac{g}{2u^2}x^2\]Time of flight is given by, \[T = \sqrt{\frac{2h}{g}}\]Horizontal Range, i.e. distance covered along horizontal axis is given by,\[x = u\sqrt{\frac{2h}{g}}\]Velocity of object at any time is given by,\[v = \sqrt{u^2 + g^2t^2}, tan\beta = \frac{gt}{u}\]

59. For angular projectile, Equation of the trajectory is given  by, \[y = xtan\theta - (\frac{1}{2}\frac{g}{u^2cos^2\theta})x^2\] Time of flight is given by,\[T = \frac{2usin\theta}{g}\]Maximum height obtained by the projectile is, \[H = \frac{u^2sin^2\theta}{2g}\]Range of the projectile is given by,\[R = \frac{u^2sin2\theta}{g}\] where $ \theta $ is angle of projection.

60. Horizontal range of projectile is same when angle of projection is (i) $ \theta $ and 90⁰ – $ \theta $ or (ii) (45⁰ + $ \theta $) and (45 degree – $ \theta $).

61. At the higher point, the projectile possesses velocity only along horizontal direction. At the highest point of the projectile path, the velocity and acceleration are perpendicular to each other.

61. In projectile motion, the particle return to the ground at the same angle and with the same speed with which it was projected. In projectile motion, kinetic energy is maximum at the point of projection or point of reaching the ground and is minimum at the highest point. There are two times for which the projectile travels the same vertical distance and the sum of these timings is equal to the total time of flight of projectile. 

62. The time of flight, the horizontal range and max. height are independent of mass of projectile. The maximum height attained by projectile is equal to one fourth of its maximum range.

63. Angular displacement of the object moving around a circular path is defined as the angle traced out by the radius vector at the center of the circular path in a given time. It is denoted by $ \theta $.

64. Angular velocity, of an object in circular motion is define das the time rate of change of its angular displacement. i.e. \[\omega = \frac{d \theta }{d t}\]

65. Angular acceleration, of an object in circular motion is defined as the time rate of change of its angular velocity. i.e. \[\alpha = \frac{d \omega }{d t} = \frac{d^2 \theta }{d t^2}\]

66. Uniform circular motion,  When a point object is moving on a circular path with a constant speed (i.e. it covers equal distance of the circumference of the circle in equal intervals of time), then the motion of the object is said to be a uniform circular motion Time period in circular motion, is defined as the time taken by the object to complete one revolution on its circular path. Frequency in circular motion is defined as the number of revolutions completed by the object on its circular path in a unit time.

67. Relation between the time period and frequency, $ \nu T = 1 $

68. Relation between angular velocity, frequency and time period $ \omega = \frac{\theta}{t} = \frac{2\pi}{T} = 2\pi \nu $

69. Centripetal acceleration,  Acceleration acting on the object undergoing circular motion is called centripetal acceleration. When a body is moving with a constant angular velocity, its angular acceleration is zero. It is given  by, \[|\vec{a}| = \omega^2 r = \frac{v^2}{r}\]

70. Uniform circular motion is an accelerated motion. The work done by the centripetal force is zero. The centripetal force does not increase the kinetic energy and angular momentum of the particle moving in a circular path.

71. Without centripetal force, a body cannot move on a circular path. In different types of circular motion, the centripetal force is provided by different means.

72. Total acceleration of the body moving in circle is given by,  \[\vec{a} = \vec{a_c}+\vec{a_T}\]

\[|\vec{a}| = \sqrt{(a_c^2 + a_T^2)}\]

Video Lecture:

1 Kinematics, Frame of reference, scalar and vectors  watch video

2 Distance and displacement watch video

3 Speed and velocity watch video

4 Acceleration and equation of motion by graphical method watch video

5 Questions based on graphical method watch video

6 Equations by calculus method watch video 

7 Vector definitions and triangle law of vector addition watch video

8 Zero vector, lami's theorem and vector resolution watch video

9 Dot and cross product of two vectors watch video

10 Distance covered in n-th second and motion in vertical direction watch video

11 Relative velocity 1 watch video

12 Relative velocity 2 watch video

13 Horizontal projectile watch video

14 Angular projectile watch video

15 Circular motion watch video 

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Measurement and Error

1 Radian (rad): One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. \[d\theta =\frac{ds}{r}\]SI unit is radian.

2. Steradian (sr): One steradian is the solid angle subtended at the center of a sphere, by that surface of the sphere, which is equal in area, to the square of the radius of the sphere. \[d\Omega = \frac{dA}{r^2}\]SI unit is steradian

3. Astronomical Unit (AU): It is the average distance of the centre of the sun from the centre of the earth. 1 AU =1.496 x 10¹¹ m =1.5 x 10¹¹ m

4. Light year (ly): One light year is the distance travelled by light in vacuum in one year. 1 ly = 9.46 x10¹⁵m.

5. Parsec : One parsec is the radius of the circle at the centre of which an arc of the circle, 1 AU long subtends an angle of 1”. 1 parsec = 3.1 x 10¹⁶ m

6. Relation between AU, ly and par sec, 1 ly = 6.3 x 10⁴ AU, 1 par sec = 3.26 ly

7. In the micro-cosm measurement,

(i)                  1 micron = 1 μ or 1 μm = 10^-6 m

(ii)                1 nanometer = 1 nm =10^-9 m

(iii)               1 angstrom = 1A⁰ = 10^-10 m

(iv)               1 fermi = 1 femtometer = 1 fm = 10^-15 m

8. For measuring very small area,

1 acre = 4047 m²

1 are (a) = 10² m²

1 hactare = 10⁴ m²

9. For measuring heavy masses,

(i)                  1 tonne or  1 metric ton = 1000 kg

(ii)                1 quintal = 100  kg

(iii)               1 slug = 14.57 kg

(iv)               1 lb = 0.4536 kg

10. For measuring very small masses, 1 atomic mass unit = 1 a.m.u. or 1 u = 1.66 x 10^-27 kg.

11. Some practical units of standard of time are :

(i)Solar day: It is the time interval between two successive passage of the sun across the meridian.

(ii) Sedrial day: It is the time interval between two successive passages of a fixed star across the meridian. 

(iii) Solar year (or year) is the time taken by the earth to complete one revolution around the sun in its orbit. 

1 solar year = 365.25 average solar days = 366.25 sedrial days

The year in which there is total solar eclipse is called a tropical year. The year which is divisible by 4, and I which month of February has 29 days, is called a leap year. One hundred years make up 1 century. 

(iv) Lunar month. It is the time taken by moon to complete one revolution around the earth in its orbit. 1 Lunar month = 27.3 days. 

(v) Shake: It s the smallest practical unit of time. 1 shake = 10⁸s

12. Parallax method: Parallax is the name of the name given to change in the position of an object with respect to the background when the object is seen from two different positions. The distance between the two-position (i.e., points of observation) is called the basis. \[\Theta = \frac{b}{x}\] where b is the arc length, x is the radius and $ \Theta $ is angle subtended. The parallax method has been used for measuring the distance of stars of which are less than a hundred light-years away.

13. Error: The difference between the true value and the measured value of any physical quantity is called error. i.e. Error = [True Value] - [Measured Value].

14. There are three types of errors, namely, Systematic, Random and gross error.

15. Systematic Error: There error tend to be in any one direction either positive or negative. some of the systematic errors are: Instrumental error, Imperfection in experimental technique or procedure, Personal error, Least count error.

16. Random Error: These errors occur irregularly. It arises due to random and unpredictable variation in experimental conditions like Temperature, Pressure, voltage, etc. It can be minimized by repeating the experiments. 

17. Gross Error: These errors arise due to the carelessness of the observer. For example, Reading an instrument improperly, noting observations incorrectly, using wrong values in the calculation, etc.    

18. Absolute Error: It is the magnitude of the difference between the true value and the individual measured value of the quantity. Let physical quantity be measured n times and observed values be $ a_1, a_2,........,a_n $. Then, arithmetic mean of these value are, \[a_m = \frac{a_1+a_2+.....+a_n}{n} \implies a_m =\frac{1}{n}\Sigma_{i=1}^{i=n}a_i\]Then absolute error in any measured value is given by \[\Delta a_i = a_m - a_i\]

19. Means absolute error: It is arithmetic mean of the magnitude of absolute errors in all measurements of quantity. It is represented by $\Delta a_{mean} $. Thus, \[\Delta a_{mean}=\frac{|\Delta a_1|+|\Delta a_2|+.....+|\Delta a_n|}{n} \implies \Delta a_{mean} = \frac{1}{n}\Sigma_{i=1}^{i=n}|\Delta a_i |\]Hence final result of measurement may be written as  

  \[a=a_m \pm \Delta a_{mean}\]

20. Relative Error: It is defined as the ratio of mean absolute error to the mean value of the quantity measured. Thus, \[\delta a = \frac{\Delta a_{mean}}{a_m}\]

21. Error in Sum: Let x = a + b, then maximum absolute error in x is \[\Delta x = \pm (\Delta a +\Delta b)\]Hence maximum absolute error in sum of two quantities is equal to sum of the absolute errors in the individual quantities.

22. Error in difference: Let x = a - b,  then maximum absolute error in x is \[\Delta x = \pm (\Delta a +\Delta b)\] Hence maximum absolute error in difference of two quantities is equal to sum of the absolute errors in the individual quantities.

23. Error in Product: Let  $ x $ = a x b, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm (\frac{\Delta a}{a} +\frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities. 

24. Error in Product: Let  $ x = \frac{a}{b} $, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm (\frac{\Delta a}{a} +\frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities. 

25. Error in case of measured quantity raised to a power: Let  $ x = \frac{a_n}{b_m} $, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm ( n \frac{\Delta a}{a} + m \frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities. 

Video Lecture:

Measurement and errors: Watch video

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Unit and Dimensions

1. Measurement of any physical quantity involves comparison with a certain basic, arbitrary chosen, widely accepted reference standard called Unit. 

Mathematically, a measure of a quantity Q = nu, where u is the size of the unit, and n is the numerical value of the given measure.  

2. Fundamental quantities: Fundamental quantities are the base quantities. There are 7 fundamental quantities: 

(i) Length 

(ii) Mass

(iii) Time

(iv) Electric Current

(v) Thermodynamic Temperature

(vi) Amount of substance

(vii) Luminous Intensity.

3. Derived quantities: These quantities are formed using fundamental quantities like density, volume, force, etc.

4. Length: Unit is metre (m). Meter is defined as the length of the path traveled by light in vacuum during a time interval of $ \frac{1}{299792458} $ part of a second. 

 1 fermi  = $1f =   10^{-15} m $

 1 angstrom =  $ 1A = 10^{-10} m$

 1 nano-metre = $1nm = 10^{-10}m$

 1 micro-metre = $ 1\mu m = 10^{-6}m$

 1 mili-metre = $1mm = 10^{-3} m$

 1 Astronomical unit = $ 1AU  = 1.496 \times 10^{11}m$

 1 light-year = $ 1ly = 9.46 \times 10^{11} m$

 1 parsec = $ 3.08 \times 10^{16}m $

5. Mass: Unit is Kilogram(kg). The mass of a cylinder made of platinum-iridium alloy kept at the International Bureau of Weights and Measures is defined as 1 kg.

6. Time: Unit is second(s). One second is the time taken by 9 192 631 770 oscillations of the light (of a specified wavelength) emitted by a cesium-133 atom.

7. Electric Current: Unit is Ampere. If equal currents are maintained in the two wires so that the force between them is $ 2 x 10^{-7} $ newton per meter of the wires, the current in any of the wires is called 1 A

8. Thermodynamic Temperature: Unit is Kelvin(K). The fraction $ \frac{1}{273.16} $ of the thermodynamic temperature of the triple point of water is called 1 K.

9. Amount of the Substance: Unit is mole(mole). The amount of a substance that contains as many

elementary entities as there is the number of atoms in 0.012 kg of carbon-12 is called a mole. 

10. Luminous Intensity: Unit is Candela(cd). The SI unit of luminous intensity is 1 cd which is the luminous intensity of a blackbody of surface area $ \frac{1}{600 000} m^{2} $ placed at the temperature of freezing, platinum, and at a pressure of 101,325 $ {N/m^{2}} $, in the direction perpendicular to its surface. 

11. Dimensions: Dimensions are the powers to which fundamental quantities are raised to represent that quantity. It is represented by using a square bracket. 

Physical Quantities	Dimensions
Distance, Length, Displacement	$[M^0LT^0]$
Velocity, Speed	$[M^0LT^{-1}]$
Acceleration	$[M^0LT^{-2}]$
Force	$[MLT^{-2}]$
Linear momentum, Impulse	$[MLT^{-1}]$
Torque, Work, Kinetic Energy, Potential Energy, Energy,	$[ML^2T^2]$
Power	$[ML^2T^{-3}]$
Pressure, Stress, Modulus of Elasticity	$[ML^{-1}T^{-2}]$

12. Principle of homogeneity of Dimensions: A correct dimensional equation must be homogeneous i.e. dimensions on both sides are the same. 

13. Use of Dimension: To convert a unit from one system to another system, To find the relation between various physical parameters and to check whether the formula is dimensionally correct or not.

Example1:  Find the dimension of the constants a and b in Van Der Wall Equation

i.e. $ (P + \frac{a}{V^2})(V-b) = RT $         

Solution:  Using principle of homogeneity,  

Dimension of b  = Dimension of V (volume) = $ [{ L^3 }] $

Dimension of P (pressure) = Dimension of $ (\frac{a}{V^{2}}) $                               

Dimension of a = dimension of $ PV{^2} $  = $[ML^{-1}T^{-2}] [L^{3}]^{2} $= $ [ML^{5}T^{-2}] $          

 

Example 2: The value of the gravitational constant is G = $ 6.67 * 10^{-11} $ $ Nm{^2}kg^{-2} $. Convert it into a system based on kilometer, tonne and hour as base units.  

Solution: Dimnsional formula of  G is $ [M^{-1}L^{3}T^{-2}] $

$ n_2 = n_1 [\frac{M_1}{M_2}]^{-1}[\frac{L_2}{L_1}]^{3} [\frac{T_2}{T_1}]^{-2} $  

$ n_1 = 6.67 * 10^{-11}, M_1 = 1 kg, M_2 = 1 tonne = 1000kg, $

$T_1 = 1s, T_2 = 3600s, L_1 = 1m and L_2 = 1000m $

$ n_2 = 6.67*10^{-11}[\frac{1}{1000}]^{-1}[\frac{1}{1000}]^{3} [\frac{1}{3600}]^{-2} = 8.64 * 10^{10} $

Example 3: The frequency f of a stretched string depends upon the Tension (T), length (l) and the linear mass density $ /mu $. Find the relation for frequency. 

Solution: Let frequency depends on T, l, and $ \mu $ as follow:

                           $ f = kT^{a}l^{b}{\mu ^{c}} $                      where k is constant.

            

writing dimension formula of both sides,

$ [M^{0}L^{0}T^{-1}] $ = $ [MLT^{-2}]^{a}[L]^{b}[ML^{-1}]^{c} $  =  $ [M^{a+c}L^{a+b-c}T^{-2a}] $

Comparing dimensions on both sides, 

                                     a + c  =  0

                                a + b - c  =  0

                                        -2a  =  -1

solving these we get,  a = $\frac{1}{2} $, b = -1 and c = $ \frac{-1}{2} $

so relation will be,   $f = \frac{k}{l} \sqrt {\frac{T}{\mu}} $ 

Video Lecture:

Fundamental and Derived Quantities, Dimensions, How to find dimension of any physical Quantity, Formula validation by dimensions, Deriving relation between physical quantities Unit conversion Watch video

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