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

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:

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:

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:

Maximum height  

Time of flight  

Range 

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.

Angular  velocity  (change in angular displacement per unit time)

Angular acceleration 

Where θ = angle between displacement.

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

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: 

Relation between Linear and Angular Quantities

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

e_t and e_r are tangential and radial unit vectors.

Linear velocity 

Linear acceleration (Net) 

Tangential acceleration  (rate of change of speed)

Centripetal acceleration  

Net acceleration,  

Where a_r = centripetal acceleration

   a_t = tangential acceleration

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

• Centre of mass about 
• Centre of mass about 
• Centre of mass about 
• For a uniform rectangular, square or circular plate, the CM lies at its centre.
• CM of semicircular ring

• CM of semicircular disc

• CM of a hemispherical shell

• CM of a solid hemisphere

Law of Conservation of Linear Momentum

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

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.

• Thrust force on the rocket 

Where,  rate at which mass is ejecting,

v_r is the relative velocity of ejecting mass (exhaust velocity)

• Weight of the rocket w = mg
• The net force on the rocket 
• Net acceleration of the rocket

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

m is the mass of rocket at time t

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