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, v3, 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 video4 Acceleration and equation of motion by graphical method
watch video7 Vector definitions and triangle law of vector addition
watch video8 Zero vector, lami's theorem and vector resolution
watch video10 Distance covered in n-th second and motion in vertical direction
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