Kinematics and Dynamics of Particles and Rigid Bodies in Plane Motion Study Notes for GATE ME Exam

By Akhil Gupta|Updated : November 29th, 2021

Kinematics and Dynamics of Particles and Rigid Bodies in Plane Motion Study Notes for GATE ME Exam: In this article, know the important pointers for Kinematics and Dynamics of Particles and Rigid Bodies in Plane Motion like types and other terminologies, which can help you to score well in GATE ME, ISRO and other competitive exams.

Table of Content

Plane Motion

A rigid body is said to perform plane motion when all parts of the body move in parallel planes.

  • If every line in the body remains parallel to its original position at all times, the body is said to be in translation motion.
  • All the particles forming a rigid body move along parallel paths in translation motion.
  • A curvilinear translation motion takes place when all particles which form a rigid body do not move along parallel straight lines but move along a curved path.

Straight Line Motion

In a straight-line motion, acceleration is constant both in magnitude and direction. Three equations that we usually apply in a straight line motion are:

Kinematics-and-Dynamics_files (1)

u is initial velocity, v being final velocity, a being acceleration of the body, t being time, and s being distance travelled by the body.

Distance travelled in the nth second:

Kinematics-and-Dynamics_files (2)

Projectile Motion

Type of motion where velocity has two components, one in the vertical direction and another one in the horizontal direction. Component of velocity in the horizontal direction is constant during the flight of the body as no acceleration in the horizontal direction is present. Consider the following projectile motion:

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)

u is the initial velocity.

  • The vertical component of velocity becomes zero at the maximum height.
  • A particle located on the axis of rotation has zero velocity and zero acceleration (in case a rigid body moves in a circular path).
  • The air resistance is considered negligible during projectile motion.

Angular acceleration and Angular Velocity

Consider a rod pivoted at a point and rotating about it.

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 rotatory motion, the equations that were used in the straight-line motion, changes slightly to the following:

Kinematics-and-Dynamics_files (10)

ω0 being initial angular velocity, ω being final angular velocity, α being the angular acceleration, and θ being angular displacement.

Angular displacement in the nth second: Kinematics-and-Dynamics_files (11)

Relation between Linear and Angular Quantities

The  relationships between linear and angular quantities in rotational motion are listed below:

 Kinematics-and-Dynamics_files (12)

et and eare tangential and radial unit vectors.

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: For a continuous body, centre of mass 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)
  • For a uniform rectangular, square or circular plate, the CM lies at its centre.
  • CM of semicircular ring

Kinematics-and-Dynamics_files (24)

  • CM of semicircular disc

Kinematics-and-Dynamics_files (25)

  • CM of a hemispherical shell

Kinematics-and-Dynamics_files (26)

  • CM of a solid hemisphere

Kinematics-and-Dynamics_files (27)

Law of Conservation of Linear Momentum

 Linear momentum (p) can be defined as the product of mass and velocity.

Kinematics-and-Dynamics_files (28)

Where K = kinetic energy of the particle

F = net external force applied to the body

P = momentum

Rocket Propulsion- Consider a rocket of mass m0 at time t = 0, at any time t, v is its velocity and m is the mass at that moment. Initially, let the velocity of the rocket be 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 is the relative velocity of ejecting mass (exhaust velocity)

  • Weight of the rocket w = mg
  • The net force on the rocket Kinematics-and-Dynamics_files (33)
  • Net acceleration of the rocket

Kinematics-and-Dynamics_files (34)

m0 is the mass of the rocket at time t = 0

m is the mass of rocket at time t

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