Engineering Mechanics : Kinematics and Dynamics of Particles and Rigid Bodies in Plane Motion

By Akhil Gupta|Updated : November 21st, 2021

                                                                                                                                                               

Plane Motion: A rigid body is said to perform plane motion when all parts of the body move in a 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 curve path.

                                                                    

Plane Motion: A rigid body is said to perform plane motion when all parts of the body move in a 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 curve path.

Straight Line Motion: In a straight line motion, acceleration is constant both in magnitude and direction. Three equations which we usually apply in a straight line motion are:

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

Distance travelled in nth second:

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

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  • 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.

Trajectory of a projectile motion can be given by the following equation: 

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where y is the vertical distance and x is the horizontal distance. Other symbols have their usual meanings.

Angular acceleration and Angular Velocity: Consider a rod pivoted at a point and rotating about it.

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Angular  velocity (change in angular displacement per unit time) =

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Where θ = angle between displacement.

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

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

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Relation between Linear and Angular Quantities

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

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et and eare tangential and radial unit vector.

 

Where ar = centripetal acceleration

at = tangential acceleration

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Centre of Mass of Continuous Body: For a continuous body, centre of mass of can be defined as

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