Engineering Mechanics & Design of Steel Structures: Kinematics &Dynamics of Particles &Rigid Bodies

By Ashutosh Yadav|Updated : October 24th, 2021

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 Plane Motion: When all parts of the body move in a parallel plane then a rigid body said to perform plane motion.

  • The motion of 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.

Kinematics-and-Dynamics_files (1)

Where, u = initial velocity, v = final velocity, a = acceleration of body, t = time, and s = distance travelled by body.

Distance travelled in nth second:

Kinematics-and-Dynamics_files (2)

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 θ from horizontal direction.

Kinematics-and-Dynamics_files (3)

Maximum height Kinematics-and-Dynamics_files (4)

Time of flight Kinematics-and-Dynamics_files (5)

Range Kinematics-and-Dynamics_files (6)

Where, u = initial velocity.

  • At maximum height vertical component of velocity becomes zero.
  • When a rigid body move in circular path centred 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.

Kinematics-and-Dynamics_files (7)

Angular Kinematics-and-Dynamics_files (8) velocity  (change in angular displacement per unit time)

Angular acceleration Kinematics-and-Dynamics_files (9)

Where θ = angle between displacement.

In case of angular velocity, the various equations with the relationships between velocity, displacement and acceleration are as follows.

Kinematics-and-Dynamics_files (10)

Where ω0 = initial angular velocity, ω = final angular velocity, α = angular acceleration, and θ = angular displacement.

Angular displacement in nth second:

Kinematics-and-Dynamics_files (11)

Relation between Linear and Angular Quantities

There are following relations between linear and angular quantities in rotational motion.

Kinematics-and-Dynamics_files (12)

er and et are radial and tangential unit vector.

Linear velocity Kinematics-and-Dynamics_files (13)

Linear acceleration (Net)

Kinematics-and-Dynamics_files (14)

Tangential acceleration image001 (rate of change of speed)

Centripetal acceleration Kinematics-and-Dynamics_files (16) Kinematics-and-Dynamics_files (17)

Net acceleration, Kinematics-and-Dynamics_files (18)

Kinematics-and-Dynamics_files (19)

Where ar = centripetal acceleration

at = tangential acceleration

Kinematics-and-Dynamics_files (20)

Centre of Mass of Continuous Body: Centre of mass of continuous body can be defined as

  • Centre of mass about Kinematics-and-Dynamics_files (21)
  • Centre of mass about Kinematics-and-Dynamics_files (22)
  • Centre of mass about Kinematics-and-Dynamics_files (23)
  • CM of uniform rectangular, square or circular plate lies at its centre.
  • CM of semicircular ring

Kinematics-and-Dynamics_files (24)

  • CM of semicircular disc

Kinematics-and-Dynamics_files (25)

  • CM of hemispherical shel

Kinematics-and-Dynamics_files (26)

  • CM of solid hemisphere

Kinematics-and-Dynamics_files (27)

Law of Conservation of Linear Momentum

The product of mass and velocity of a particle is defined as its linear momentum (p).

Kinematics-and-Dynamics_files (28)

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.

Kinematics-and-Dynamics_files (30)

  • Thrust force on the rocket Kinematics-and-Dynamics_files (31)

Where, Kinematics-and-Dynamics_files (32) 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 Kinematics-and-Dynamics_files (33)
  • Net acceleration of the rocket

Kinematics-and-Dynamics_files (34)

Where, m0 = mass of rocket at time t = 0

m = mass of rocket at time 

 

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Ashutosh YadavAshutosh YadavMember since Jan 2020
ESE, GATE, SSC-JE and other Civil Engineering exams.
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