Motion
Kinematics
The branch of mechanics dealing with the study of motion of a particle without
taking into account the force and energy.
Motion
A particle is said to be in motion if its position, relative to the
surrounding chains with respect to time.
Rest
A particle is said to be at rest if its position, relative to the surrounding
does not change with respect to time.
🠆 Rest and motion are relative terms not absolute.
Types of motion
1. Random motion :- In this type of motion the particle moves randomly, does
not move along a definite path.
Example :- The motion of dust particle in the wind or in air.
2. Translational motion :- In this type of motion, every particle of the body
has the same displacement.
🠆 Translational motion can be along a straight or along a curved path.
🠆 The motion along a straight line is called rectilinear motion.
🠆 The motion along a curved path is called curvilinear motion.
3. Rotational motion :- If the particle of the body revolve in a circle about
the same axis, then the motion is said to be rotational motion.
Example :- Rotation of earth on its axis.
4. Oscillatory or vibratory motion :- To and fro motion about a fixed point is
called oscillatory or vibratory motion.
Example :- Motion of pendulum of clock
Scalar quantity :- Physical quantity that can be defined "magnitude"
only are known as a scalar quantity.
Vector quantity :- Physical quantities that can be defined only if
both, "its magnitude" and "direction" are specified are called vector
quantities.
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Distance
The total path length travelled by a body in a given time interval is called
distance.
Distance = AB + BC + CD + DE + EF + FA
🠆 SI unit = metre (m)
🠆 It is a scalar quantity
Displacement
The change in position of the object along a particular direction during the
time from the initial position to the final position.
Displacement = (AF)
⃗
🠆 SI unit = metre (m)
🠆 It is a vector quantity.
Characteristics of displacement
🠆 The displacement of an object has the unit of length.
🠆 The displacement of an object in a given interval of time can be positive,
negative or zero.
🠆 The magnitude of the displacement of an object between two points gives the
shortest distance between those two points.
🠆 The displacement of the object between two point has a unique value.
🠆 The actual distance travel by the object in the given interval of time can
be equal to or greater than the magnitude of the displacement.
distance = / displacement /
(1). If direction is not charged
distance = / displacement /
(2). If direction is changed
distance > / displacement /
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Speed
The time rate of change of position of the object in any direction is called a
speed.
Or
The distance travelled by a body in per unit time in any direction.
Speed = distance / Time
🠆 SI unit = m/s = ms-1
🠆 CGS unit = cm/s = cms-1
It is a scalar quantity
Uniform speed
An object is said to be moving with a uniform speed if it cover equal distance
in equal interval of time however a small this interval may be.
Equal distance 8m
8m
8m 8m
Unequal time 4sec
4sec
4sec 4sec
Speed 2
m/s 2 m/s
2 m/s 2 m/s
Non uniform speed or variable speed
An object is said to be moving with variable speed if it covers equal distance
unequal interval of time or an unequal distance in equal intervals of time
however a small this interval may be.
Equal distance 5m
5m
5m
5m
Unequal time 1sec
3sec
2sec
4sec
Speed
5 m/s 5/3
m/s 2.5 m/s
1.25 m/s
Average speed
Average speed for the given motion is defined as the ratio of total
distance travelled by the body to the total time taken.
Case 1
If a particle travels distance S1, S2, S3
etc with the speed V1, V2, V3
respectively in same direction then,
Note :- The body travels equal distance with different speeds.
Case 2
If a particle travels with a speed V1, V2, V3 etc during time interval t1, t2, t3
respectively then,
Instantaneous speed
The speed of an object at a given instant of time is called its
instantaneous speed.
Let at on instant t,
moving covers a distance Δs in a small interval of time Δt. so that Δt → 0
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Velocity
The time rate change of displacement of the object.
Or
The speed of an object in a given direction is called velocity.
Velocity = displacement / time interval
🠆 SI unit = m/s
🠆 CGS unit = cm/s
🠆 It is a vector quantity.
🠆 The velocity of an object can be positive, negative and zero according
as its displacement is positive, negative and zero.
Uniform velocity
An object is said to be moving with the uniform velocity if it undergoes
equal displacement in equal intervals of time however a small this
interval may be.
| Displacement | 20 m | 20 m | 20 m | 20 m |
| Time interval | 4 sec | 4 sec | 4 sec | 4 sec |
| Velocity | 5 m/sec | 5 m/sec | 5 m/sec | 5 m/sec |
🠆 Uniform velocity is also called constant velocity.
Non uniform velocity or variable velocity
An object is said to be moving with the variable velocity if it undergoes
equal displacement in unequal intervals of time.
Or
An equal displacement in equal interval of time.
Or
Change the direction of motion while moving with constant speed.
Uniform motion in a straight line
An object is said to be in uniform motion if it undergoes equal
displacement in equal interval of time however a small these intervals may
be.
Consider an object in uniform motion along a straight line OX with a
velocity V
Let point O be the origin for position measurement.
Let A and B be the positions of the object at instant of time t1
and t2
respectively.
Important feature of uniform motion
• For a uniform motion along a straight line in a given direction
the magnitude of the displacement is equal to the actual distance covered
by the object.
• The velocity in uniform motion does not depend on the time
interval.
• The velocity in uniform motion is independent of choice of origin
• Velocity of an object is taken positive if object is moving
towards the right of origin and is taken to be negative if the object is
moving toward the left of origin.
• To force is required for an object to be in uniform motion.
• The average and instantaneous velocity have same value in a
uniform motion.
Non-uniform motion
An object is said to be in a non uniform motion if it undergoes equal
displacement in unequal intervals of time however a small these intervals
may be.
Acceleration
The time rate of change in velocity of the object is called
acceleration.
Or
The change in velocity in unit time is called acceleration.
• SI unit is m/s2 = ms2
• It is a vector quantity.
• Acceleration may be positive, negative and zero.
• Acceleration is negative if the velocity is decreasing.
• Acceleration is positive if the velocity is increasing.
• Acceleration is zero if the velocity is constant.
• The negative acceleration is also called retardation or
deacceleration.
If initial velocity is 'U' at time t1 and time t2
its final velocity become V
Equation of motion :-
equation of motion are valid only for uniform accelerated body.
u = initial velocity
v = final velocity
a = uniform acceleration
t = time taken by body
s = distance covered
1. v = u + at
2. s = ut + 1/2 at2
3. v2 = u2 +2as
(i). Analytical method
(ii). Graphical method
Analytical methods
1. v = u + at
according to definition of acceleration
a = v - u / t
at = v - u
u + at = v
v = u + at
2. s = ut + 1/2 at2
Distance = average velocity ✖ time
3. v2 = u2 + 2as
v = u + at
Both side squaring
v2 = (u + at)2
v2 = u2 + 2uat + a2t2
v2 = u2 + 2a (ut + 1/2 at2)
v2 = u2 + 2as
Distance traveled in nth sec of uniformly accelerated motion.
let u = initial velocity of the object
a = Uniform acceleration of the object
sn = distance travell by object in n
sec.
sn-1 = distance travell by object in n-1 sec.
Dn = distance travell
in nth sec.
Graphical methods
velocity time graph of a uniformly accelerated motion
AB shows acceleration
In △ABC
slop AB = tan Ө = BC/AC
a =
(v - u)/ t
at = v - u
v = u - at
Area under the curve of velocity - time graph gives
distance
Distance = area of OABD
S = Area of △ABC + Area of rectangle
Circular motion
Uniform circular motion :- When a point object is moving on a
circular path with a constant speed,it covers the equal distance on the
circumference of the circle in equal intervals of time then the motion
of the object is said to be a uniform circular motion.
• If uniform circular motion, the velocity of the object is changing its
direction continuously. hence, it is a case of uniformly accelerated
motion.
Angular displacement :-
the angle traced out by the radius vector at the axis of circular path
in a given time.
• SI unit = radian
• it is a vector quantity.
Angular velocity :- the time rate of change of its angular displacement
is called angular velocity.
• SI unit = radian/sec
• it is a vector quantity.
Time period :- the time taken by the object to complete one revolution
on its circular path is called time period.
• It is represented by 'T'
Frequency :- the number of revolution completed by the object on its
circular path in a unit time is called frequency.
• It is denoted by 𝝂 or f
Let a body take 'n' revolution in 't' second then
𝝂 = n/t
• SI unit = s-1 (per second)
Relation between time period and frequency :-
If n = 1 Then t = T
𝝂 = 1/T
Relation between angular velocity, frequency and time period.
when the object complete one revolution the angle trust at its axis of
circular motion is is 2π radius
ω = 0/t
t = T, ፀ = 2π rdian
ω = 2π/t
∵ 𝝂 = 1/T
ω = 2π𝝂
Relation between linear velocity and angular velocity :-
Angular acceleration
The time rate of change of its angular velocity is called angular
acceleration .
Relation between linear acceleration and angular acceleration :-
Centripetal acceleration :-
acceleration acting on the object undergoing uniform circular motion is
called centripetal acceleration.
• It always act on the object along the radius towards the centre of the
circular path.
Centripetal acceleration = ω2r
= V2/r
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