Short Notes: Kinematics and Dynamics of Particles and Rigid Bodies in Plane Motion | Short Notes for Mechanical Engineering PDF Download

 Page 1


Kinematics and Dynamics of Particles and Rigid Bodies in 
Plane Motion
Plane Motion: When all parts of the body move in a parallel planes then a rigid body 
said to perform plane motion.
• The motion of rigid body is said to be rigid body is said to be translation, if 
every line in the body remains parallel to its original position at all times.
• In translation motion, all the particles forming a rigid body move along parallel 
paths.
• If all particles forming a rigid body move along parallel straight line, it is 
known as rectilinear translation.
• If all particles forming a rigid body does not move along a parallel straight line 
but they move along a curve path, then it is known as curvilinear translation.
Straight Line Motion: It defines the three equations with the relationship between 
velocity, acceleration, time and distance travelled by the body. In straight line 
motion, acceleration is constant.
v = u + at 
1
s = ut + — at'
2
v*= u2+ las
Where, u = initial velocity, v = final velocity, a = acceleration of body, t = time, and s 
= distance travelled by body.
Distance travelled in nth second:
s = u+ - a (2 n - 1)
Page 2


Kinematics and Dynamics of Particles and Rigid Bodies in 
Plane Motion
Plane Motion: When all parts of the body move in a parallel planes then a rigid body 
said to perform plane motion.
• The motion of rigid body is said to be rigid body is said to be translation, if 
every line in the body remains parallel to its original position at all times.
• In translation motion, all the particles forming a rigid body move along parallel 
paths.
• If all particles forming a rigid body move along parallel straight line, it is 
known as rectilinear translation.
• If all particles forming a rigid body does not move along a parallel straight line 
but they move along a curve path, then it is known as curvilinear translation.
Straight Line Motion: It defines the three equations with the relationship between 
velocity, acceleration, time and distance travelled by the body. In straight line 
motion, acceleration is constant.
v = u + at 
1
s = ut + — at'
2
v*= u2+ las
Where, u = initial velocity, v = final velocity, a = acceleration of body, t = time, and s 
= distance travelled by body.
Distance travelled in nth second:
s = u+ - a (2 n - 1)
Projectile Motion: Projectile motion defines that motion in which velocity has two 
components, one in horizontal direction and other one in vertical direction. 
Horizontal component of velocity is constant during the flight of the body as no 
acceleration in horizontal direction.
Let the block of mass is projected at angle 0 from horizontal direction.
R range
Projectile motion 
Maximum height 
, u ‘ s in '0
Time of flight
_ 2ttsinfl
a
5
Range
„ u‘ sin 26 
R = -----------
g
Where, u = initial velocity.
• At maximum height vertical component of velocity becomes zero.
• When a rigid body move in circular path centered on the same fixed axis, then 
the particle located on axis of rotation have zero velocity and zero 
acceleration.
• Projectile motion describes the motion of a body, when the air resistance is 
negligible.
Rotational Motion with Uniform Acceleration: Uniform acceleration occurs when 
the speed of an object changes at a constant rate. The acceleration is the same 
over time. So, the rotation motion with uniform acceleration can be defined as the 
motion of a body with the same acceleration over time.
Let the rod of block rotation about a point in horizontal plane with angular velocity.
Rotational rru on
Angular
de
velocity (change in angular displacement per unit time) 
Angular acceleration
Page 3


Kinematics and Dynamics of Particles and Rigid Bodies in 
Plane Motion
Plane Motion: When all parts of the body move in a parallel planes then a rigid body 
said to perform plane motion.
• The motion of rigid body is said to be rigid body is said to be translation, if 
every line in the body remains parallel to its original position at all times.
• In translation motion, all the particles forming a rigid body move along parallel 
paths.
• If all particles forming a rigid body move along parallel straight line, it is 
known as rectilinear translation.
• If all particles forming a rigid body does not move along a parallel straight line 
but they move along a curve path, then it is known as curvilinear translation.
Straight Line Motion: It defines the three equations with the relationship between 
velocity, acceleration, time and distance travelled by the body. In straight line 
motion, acceleration is constant.
v = u + at 
1
s = ut + — at'
2
v*= u2+ las
Where, u = initial velocity, v = final velocity, a = acceleration of body, t = time, and s 
= distance travelled by body.
Distance travelled in nth second:
s = u+ - a (2 n - 1)
Projectile Motion: Projectile motion defines that motion in which velocity has two 
components, one in horizontal direction and other one in vertical direction. 
Horizontal component of velocity is constant during the flight of the body as no 
acceleration in horizontal direction.
Let the block of mass is projected at angle 0 from horizontal direction.
R range
Projectile motion 
Maximum height 
, u ‘ s in '0
Time of flight
_ 2ttsinfl
a
5
Range
„ u‘ sin 26 
R = -----------
g
Where, u = initial velocity.
• At maximum height vertical component of velocity becomes zero.
• When a rigid body move in circular path centered on the same fixed axis, then 
the particle located on axis of rotation have zero velocity and zero 
acceleration.
• Projectile motion describes the motion of a body, when the air resistance is 
negligible.
Rotational Motion with Uniform Acceleration: Uniform acceleration occurs when 
the speed of an object changes at a constant rate. The acceleration is the same 
over time. So, the rotation motion with uniform acceleration can be defined as the 
motion of a body with the same acceleration over time.
Let the rod of block rotation about a point in horizontal plane with angular velocity.
Rotational rru on
Angular
de
velocity (change in angular displacement per unit time) 
Angular acceleration
d + d 2 0
Where 0 = angle between displacement.
In case of angular velocity, the various equations with the relationships between 
velocity, displacement and acceleration are as follows.
9 = at 
a = 0
a = < y 0+ at 
9 = a n t + - a t 2
v
co2 = col+ 2a9
Where u)0 = initial angular velocity, c j = final angular velocity, a = angular 
acceleration, and 0 = angular displacement.
Angular displacement in nth second:
0,= o)o+ ^ a (2 n -l)
Relation between Linear and Angular Quantities
There are following relations between linear and angular quantities in rotational 
motion.
|e ,l= \et\= 1
er and et are radial and tangential unit vector.
Linear velocity
v = rio er
Linear acceleration (Net)
dr
a = co're.------e,
r dt '
Tangential acceleration
dr
(rate of change of speed) 
Centripetal acceleration
V
a = o ' r - —
r
(v v = ro')
Net acceleration,
a= yja‘ + a;
- fzrw
~ t r . + U 1
Where ar = centripetal acceleration
Page 4


Kinematics and Dynamics of Particles and Rigid Bodies in 
Plane Motion
Plane Motion: When all parts of the body move in a parallel planes then a rigid body 
said to perform plane motion.
• The motion of rigid body is said to be rigid body is said to be translation, if 
every line in the body remains parallel to its original position at all times.
• In translation motion, all the particles forming a rigid body move along parallel 
paths.
• If all particles forming a rigid body move along parallel straight line, it is 
known as rectilinear translation.
• If all particles forming a rigid body does not move along a parallel straight line 
but they move along a curve path, then it is known as curvilinear translation.
Straight Line Motion: It defines the three equations with the relationship between 
velocity, acceleration, time and distance travelled by the body. In straight line 
motion, acceleration is constant.
v = u + at 
1
s = ut + — at'
2
v*= u2+ las
Where, u = initial velocity, v = final velocity, a = acceleration of body, t = time, and s 
= distance travelled by body.
Distance travelled in nth second:
s = u+ - a (2 n - 1)
Projectile Motion: Projectile motion defines that motion in which velocity has two 
components, one in horizontal direction and other one in vertical direction. 
Horizontal component of velocity is constant during the flight of the body as no 
acceleration in horizontal direction.
Let the block of mass is projected at angle 0 from horizontal direction.
R range
Projectile motion 
Maximum height 
, u ‘ s in '0
Time of flight
_ 2ttsinfl
a
5
Range
„ u‘ sin 26 
R = -----------
g
Where, u = initial velocity.
• At maximum height vertical component of velocity becomes zero.
• When a rigid body move in circular path centered on the same fixed axis, then 
the particle located on axis of rotation have zero velocity and zero 
acceleration.
• Projectile motion describes the motion of a body, when the air resistance is 
negligible.
Rotational Motion with Uniform Acceleration: Uniform acceleration occurs when 
the speed of an object changes at a constant rate. The acceleration is the same 
over time. So, the rotation motion with uniform acceleration can be defined as the 
motion of a body with the same acceleration over time.
Let the rod of block rotation about a point in horizontal plane with angular velocity.
Rotational rru on
Angular
de
velocity (change in angular displacement per unit time) 
Angular acceleration
d + d 2 0
Where 0 = angle between displacement.
In case of angular velocity, the various equations with the relationships between 
velocity, displacement and acceleration are as follows.
9 = at 
a = 0
a = < y 0+ at 
9 = a n t + - a t 2
v
co2 = col+ 2a9
Where u)0 = initial angular velocity, c j = final angular velocity, a = angular 
acceleration, and 0 = angular displacement.
Angular displacement in nth second:
0,= o)o+ ^ a (2 n -l)
Relation between Linear and Angular Quantities
There are following relations between linear and angular quantities in rotational 
motion.
|e ,l= \et\= 1
er and et are radial and tangential unit vector.
Linear velocity
v = rio er
Linear acceleration (Net)
dr
a = co're.------e,
r dt '
Tangential acceleration
dr
(rate of change of speed) 
Centripetal acceleration
V
a = o ' r - —
r
(v v = ro')
Net acceleration,
a= yja‘ + a;
- fzrw
~ t r . + U 1
Where ar = centripetal acceleration
at = tangential acceleration
Position of radial and 
tangential vectors
Centre of Mass of Continuous Body: Centre of mass of continuous body can be 
defined as
• Centre of mass about
J.vcfat j x dm
x . w =
f dm M 
Centre of mass about
J'jJVv —
J rdm f r dm
dm M 
Centre of mass about
Z.Zr
M
• CM of uniform rectangular, square or circular plate lies at its centre.
• CM of semicircular ring
• CM of semicircular disc
• CM of hemispherical shel
• CM of solid hemisphere
Law of Conservation of Linear Momentum
The product of mass and velocity of a particle is defined as its linear momentum
ip)-
Page 5


Kinematics and Dynamics of Particles and Rigid Bodies in 
Plane Motion
Plane Motion: When all parts of the body move in a parallel planes then a rigid body 
said to perform plane motion.
• The motion of rigid body is said to be rigid body is said to be translation, if 
every line in the body remains parallel to its original position at all times.
• In translation motion, all the particles forming a rigid body move along parallel 
paths.
• If all particles forming a rigid body move along parallel straight line, it is 
known as rectilinear translation.
• If all particles forming a rigid body does not move along a parallel straight line 
but they move along a curve path, then it is known as curvilinear translation.
Straight Line Motion: It defines the three equations with the relationship between 
velocity, acceleration, time and distance travelled by the body. In straight line 
motion, acceleration is constant.
v = u + at 
1
s = ut + — at'
2
v*= u2+ las
Where, u = initial velocity, v = final velocity, a = acceleration of body, t = time, and s 
= distance travelled by body.
Distance travelled in nth second:
s = u+ - a (2 n - 1)
Projectile Motion: Projectile motion defines that motion in which velocity has two 
components, one in horizontal direction and other one in vertical direction. 
Horizontal component of velocity is constant during the flight of the body as no 
acceleration in horizontal direction.
Let the block of mass is projected at angle 0 from horizontal direction.
R range
Projectile motion 
Maximum height 
, u ‘ s in '0
Time of flight
_ 2ttsinfl
a
5
Range
„ u‘ sin 26 
R = -----------
g
Where, u = initial velocity.
• At maximum height vertical component of velocity becomes zero.
• When a rigid body move in circular path centered on the same fixed axis, then 
the particle located on axis of rotation have zero velocity and zero 
acceleration.
• Projectile motion describes the motion of a body, when the air resistance is 
negligible.
Rotational Motion with Uniform Acceleration: Uniform acceleration occurs when 
the speed of an object changes at a constant rate. The acceleration is the same 
over time. So, the rotation motion with uniform acceleration can be defined as the 
motion of a body with the same acceleration over time.
Let the rod of block rotation about a point in horizontal plane with angular velocity.
Rotational rru on
Angular
de
velocity (change in angular displacement per unit time) 
Angular acceleration
d + d 2 0
Where 0 = angle between displacement.
In case of angular velocity, the various equations with the relationships between 
velocity, displacement and acceleration are as follows.
9 = at 
a = 0
a = < y 0+ at 
9 = a n t + - a t 2
v
co2 = col+ 2a9
Where u)0 = initial angular velocity, c j = final angular velocity, a = angular 
acceleration, and 0 = angular displacement.
Angular displacement in nth second:
0,= o)o+ ^ a (2 n -l)
Relation between Linear and Angular Quantities
There are following relations between linear and angular quantities in rotational 
motion.
|e ,l= \et\= 1
er and et are radial and tangential unit vector.
Linear velocity
v = rio er
Linear acceleration (Net)
dr
a = co're.------e,
r dt '
Tangential acceleration
dr
(rate of change of speed) 
Centripetal acceleration
V
a = o ' r - —
r
(v v = ro')
Net acceleration,
a= yja‘ + a;
- fzrw
~ t r . + U 1
Where ar = centripetal acceleration
at = tangential acceleration
Position of radial and 
tangential vectors
Centre of Mass of Continuous Body: Centre of mass of continuous body can be 
defined as
• Centre of mass about
J.vcfat j x dm
x . w =
f dm M 
Centre of mass about
J'jJVv —
J rdm f r dm
dm M 
Centre of mass about
Z.Zr
M
• CM of uniform rectangular, square or circular plate lies at its centre.
• CM of semicircular ring
• CM of semicircular disc
• CM of hemispherical shel
• CM of solid hemisphere
Law of Conservation of Linear Momentum
The product of mass and velocity of a particle is defined as its linear momentum
ip)-
p = mv
P = ^
F * —
Where, K = kinetic energy of the particle 
F = net external force applied to body 
P = momentum 
Rocket Propulsion
Let m0 be the mass of the rocket at time t = 0, m its mass at any time t and v its 
velocity at that moment. Initially, let us suppose that the velocity of the rocket is u.
• Thrust force on the rocket
F =
dm
~ d t
Where,
dm _ 
dt
rate at which mass is ejecting
vr = relative velocity of ejecting mass (exhaust velocity)
• Weight of the rocket w = mg
• Net force on the rocket
K : =F ,-W = V r
— dm
dt
-m g
Net acceleration of the rocket
F
a = — 
m
dv _ \ r — dm 
dt m dt , ^
v = u — gt + vr In — 
m
Where, m0 = mass of rocket at time f = 0
m = mass of rocket at time t