Chapter no. 6 linear mo

Chapter No. 6 Linear Motion Page 1

CHAPTER No. 6

KINEMATICS: LINEAR MOTION

6. Introduction:

6.1 Dynamics:

It is the branch of Applied Mechanics which deals with the analysis of the bodies in

motion. It is divided into two branches or parts:

a) Kinematics and

b) Kinetics

6.1.1 Kinematics:

It is the study of the motion of bodies without consideration of the causes of motion

such as mass of the body and forces acting on the body.

6.1.2 Kinetics:

It is the study of the motion of bodies with consideration of the causes of motion such

as mass of the body and forces acting on the body.

1. Introduction to various types of motions:

It covers the study of

1.1 Rectilinear Motion

1.2 Motion under Gravity

1.3 Relative Motion (without consideration of the forces producing the change in motion)

Before going to study the types of motion, it is very important to study some terminologies,

which are mentioned as below:


a) Force:

An external agency which causes change in the motion or in the state of the particle or body is

known as force.

Fig. a

The above figure shows the position of a ball due to application of a force F before and after

applies to it.

b) Motion:

The action of changing the position of a body is known as a motion.

c) Path:

The curve followed by a particle during its motion in space is known as the path. It can be

rectilinear (straight line) or curved.

d) Displacement (s):

If the body is moved from initial position A to final position B as shown in figure below then a

straight line distance AB is known as the displacement. So it is a movement of the particle from

initial point to final point measured along a straight line. Following figure shows that the body

may follow various paths to move from A to B but the shortest path or distance between A and B

is the displacement. Displacement is a vector quantity having magnitude and direction. Its S.I.

unit is metre (m).


Fig. 6.1

e) Velocity:

A velocity is defined as the rate of change of displacement with time.

V = 𝑑𝑠

𝑑𝑡

Also it is defined as the distance covered per unit time in the given direction.

V = 𝑠

𝑡

Its S.I. unit is metre/second (m/s); other units are km/hour or kmph. It is a vector quantity

having magnitude and direction.

f) Average Velocity:

It is defined as the ratio of the resultant displacement to the total time required to cover it.

Here resultant displacement is sum of velocities at different time.

Average Velocity =𝑅𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡

𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒 𝑇𝑎𝑘𝑒𝑛

g) Uniform Velocity:

If the velocity of a particle is constant in magnitude and direction with respect to time, then it is

known as uniform velocity i.e. equal distances are covered in equal time intervals.


h) Speed:

Speed is defined as the rate of covering the distance with respect to time irrespective of its

direction.

Speed = Distance or lenght of path

Time Required

Speed is a scalar quantity. S.I. unit is m/s; other units are km/hour or kmph.

i) Acceleration:

Acceleration is defined as the rate of change of velocity with respective to time.

Acceleration = Change in Velocity

Time =

dv

dt

It is also a vector quantity having magnitude and direction. S.I. unit is m/sec2. Other unit is

cm/sec2.

j) Retardation:

If velocity decreases with time, the acceleration becomes negative which is known as retardation

or deceleration.

k) Uniform Acceleration:

If the velocity of a body changes by equal magnitude in equal intervals continuously, the

acceleration is known as uniform acceleration or constant acceleration.

l) Variable Acceleration:

If the change in velocity per unite time is not constant in a continuous motion, then the

corresponding acceleration is known as variable acceleration.


2.1 Rectilinear Motion:

Motion of the particle along the straight line path is known as a rectilinear motion.

a) Uniformly Accelerated Rectilinear Motion

b) Motion Under Variable Acceleration ( as a function of time, velocity or displacement)

a) Motion with uniform acceleration:

Fig. 6.2

Following Figure 6.2 shows a particle in a straight line (rectilinear) motion, travelling a distance

‘s’ from A to B in time ‘t’.

S= displacement in m

u= initially velocity in m/s

v= final velocity in m/s

t= time required in sec.

The velocity is changes uniformly from ‘u’ to ‘v’ during time ‘t’. So the acceleration ‘a’ is

uniform or constant.

Equations of motion with uniform Acceleration:

1) Change in Velocity = (v-u) in time ‘t’

∴ Acceleration = a = (v − u)

t

∴ at = v − u


∴ 𝐯 = 𝐮 + 𝐚𝐭 ------------- (1)

i.e. Final Velocity = Initial velocity + Change in Velocity

‘a’ is positive if velocity increases

‘a’ is negative if velocity decreases i.e. retardation acceleration

2) Average velocity = (u+v)

2

∴ Distance Travelled = s = Average Velocity x time

∴ S = (u+v)

2 x t

∴ S = (u+(u+at))

2 x t

∴ S = 𝐮𝐭 +𝟏

𝟐 𝐚𝐭𝟐 ---------- (2)

3) From equation v = u + at

Squaring both sides,

v2 = (u + at) 2

= u2 + 2uat + v2

= u2 + 2a (ut + 1

2 at2)

v2 = u2 + 2 as ------- (3)

Therefore the basic equations of rectilinear motion with uniform acceleration are

1) 𝐯 = 𝐮 + 𝐚𝐭

2) S = 𝐮𝐭 +𝟏

𝟐 𝐚𝐭𝟐

3) V2 = u2 + 2 as

2. MOTION UNDER GRAVITY

2.1 Gravitational Motion

1) Every body or a particle experiences a force of attraction of the earth. This force of

attraction is called as Gravitational force.


2) According to Newton’s second law of motion this force produces acceleration in the

body which is directed towards the center of the earth. This acceleration is known as

gravitational acceleration.

3) The distance travelled by the freely falling body is relatively very small on the surface of

the earth. So the gravitational acceleration .It is treated as constant which is denoted as

‘g’ and its value are assumed constant as 9.81 m/s2.

2.2 Freely Falling Body:

Following figure 6.3 shows a body or a particle P which falls freely down in the vertical

direction. It is under the action of a force of gravity only. Such a body is known as freely

falling body.

Fig. 6.3

The velocity of such a body increases uniformly from zero as it moves vertically downwards.

The change in the velocity is constant with respect to time. So the body is under constant

gravitational acceleration. Such rectilinear motion is known as motion under gravity with

constant gravitational acceleration.


2.3 Equations of Motion under Gravity:

The motion under gravity is a rectilinear motion with uniform acceleration. Therefore its

equation of motion can be obtained by substituting ‘g’ in place of ‘a’ in the three basic

equations of motion.

For freely falling body for vertically downward motion:

1) v = u + gt

2) S = ut +1

2 gt2

3) V2 = u2 + 2 g s

For vertically upward motion:

1) v = u − gt

2) S = ut −1

2 gt2

3) V2 = u2 - 2 g s

Sign Convention:

Acceleration due to gravity is considered as positive when body is moving downward and the

same is considered as negative when the body is moving upward.

3. MOTION UNDER VARIABLE ACCELERATION:

1) If the change in the velocity of the body is not constant w.r.t. time the motion has variable

acceleration.

2) For the study of such motion, the equation of motion should be given in terms of

displacement or velocity and acceleration and time.

3) Then the displacement, velocity and acceleration can be calculated by using two methods.

i) Differentiation Method

ii) Integration Method

i) Differentiation Method

The method is useful in finding velocity and acceleration if the equation of motion is

given in terms of displacement and time. First differentiation of this equation w.r.t

time gives the acceleration and second differentiation gives the acceleration.


e.g. if s= 4t3+ 3t2+ 2t+1 -------(1)

Then diff. eqn (1) w.r.t. time ‘t’

𝑑𝑠

𝑑𝑡 = v = 12 t2+ 6t+2 ------ (2)

Diff. eqn (2) w.r.t. time ‘t’

𝑑𝑣

𝑑𝑡 =

d

dt (

ds

dt) = a = 24 t + 6 ------ (3)

ii) Integration Method:

This method is useful when the equation of motion is given in terms of acceleration and time.

Successive integration of this equation w.r.t time gives the velocity and displacement in

terms of time.

e.g. if a = 24 t + 6 --------(1)

Integrating equation (1) w.r.t. time ‘t’

v = 12 t2+ 6t+2 +C1 -------- (2)

Again integrating equation (1) w.r.t. time ‘t’

s= 4t3+ 3t2+ 2t+1+C1t+ C2 ------- (6)

The constants of integration C1 and C2 can be calculated by applying initial conditions which

are given in the problem.

4. GRAPHICAL REPRESENTATION OF MOTION

A rectilinear motion can be studied graphically by plotting various curves of motion or

motion diagrams as listed below:

1) Displacement – Time Curve ( s-t curve)

2) Velocity – Time Curve ( v-t curve)

3) Acceleration– Time Curve ( a-t curve)


1) Displacement – Time Curve ( s-t curve):

i) Uniform Velocity:

Fig. 6.4.a

s-t curve for uniform velocity is a straight line giving constant slope as shown in Figure 6.4.a

Velocity = V = slope = tan θ = constant

ii) Variable Velocity:

Fig. 6.4.b

s-t curve for variable velocity is a curved line. The instantaneous velocity is given by slope of

the tangent to the curve as that instant as shown in Figure 6.4.b

𝑽 = 𝒅𝒔

𝒅𝒕 ---------Where v = which varies from point to point


2) Velocity – Time Curve ( v-t curve)

i) Uniform Velocity:

Fig. 6.4.c

Area under v-t curve gives the distance travelled in the given time.

Distance travelled = s = v x t = Area under the curve (see hatched portion)

ii) Variable velocity (Uniform acceleration):

Fig. 6.4.d


For variable velocity, v-t curve is a straight line with constant slope if the acceleration is uniform

as shown in Figure 6.4.d above.

Acceleration = a = tan θ = slope

The area under the curve gives the distance covered. (See hatched portion)

Iii) Variable velocity (Variable acceleration):

Fig. 6.4.e

For variable acceleration, v-t curve is a curved line. So instantaneous acceleration is given by the

slope of curve at that instant as shown in Figure 6.4.e.

A = 𝒅𝒗

𝒅𝒕 = at instant t.

3) Acceleration– Time Curve ( a-t curve):

i) Uniform Acceleration:

A straight horizontal line on a-t curve shows uniform acceleration as shown in Figure

6.4.f. Area under this curve gives the change in velocity with respect to time.


Fig. 6.4.f

ii) Variable Acceleration:

Fig. 6.4.g

A curved line of a-t curve indicates the variable acceleration as shown in figure 6.4.g. Area under

this curve in a particular time interval gives the change in velocity in that interval of time.


5. CONCEPT OF RELATIVE MOTION:

6.1 Relative Velocity (Vr):

Definition:

When the distance between any two particles is changing either in magnitude or in direction

or in both, each particle is said to have motion or velocity relative to each other. Such velocity is

known as relative velocity (Vr) and the corresponding motion is known as relative motion.

i) If two particles A and B are in motion with the velocities Va and Vb respectively, then

the velocity of A as seen by observer placed on B is known as relative velocity of A

with respect to B.

ii) Similarly, the velocity of B as seen by observer on A is known as relative velocity of

B w.r.t. to A.

a) MOTION IN SAME DIRECTION:

If the two velocities are in the same direction such that Va > Vb as shown in figure 6.6.a then

the relative velocity Vr of A w.r.t. B is given by Vr= Va – Vb in the same direction.

Fig 6.6.a Fig 6.6.b

b) MOTION IN SAME DIRECTION:

If the two velocities are in the opposite direction as shown in figure 6.6.b then the relative

velocity is given by Vr= Va + Vb along the same line of motion.


DETERMINTION OF LEAST DISTANCE BETWEEN MOVING BODIES:

Velocities 𝑉𝑎 and 𝑉𝑏 of the two moving bodies A and B respectively are given at any instant.

The positions A and B are also known at the given instant so that distance 𝐴𝐵 is known.

The relative velocity of B w.r.t. A i.e.

Draw a perpendicular AC from A on

Knowing and angles and in ABC can be calculated.

Form right angled the least distance or the shortest distance between the two moving bodies

A and B is given by length AC = AB cos

= AB sin


Date post:	21-Jul-2015
Category:	Engineering
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Chapter no. 6 linear mo

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