Introduction To Rigid Body And Rotational Motion

Introduction

Rigid Body

Rigid Body is one in which there is no change in the shape and size of the body when it is under the application of external force.

Types:

  • Translational motion (Linear Motion)
  • Rotational motion (Circular motion)

Translational Motion: Type of motion of the body, every particles of the body has the same velocity at a particular instant of time.
Example:
Rectangular block is slides down an inclined plane.

Recommended to read article regarding Physical World and Introduction To Physics.

Rotational Motion

Type of motion in which body rotates about a fixed axis. All the particles in a body remains fixed and describe concentric circles around the fixed axis.
This fixed axis is called Axis of Rotation.

Example:
Motion of the ceiling fan,  Potter’s wheel.

Centre of Mass

Centre of mass of a rigid body or a system of particles of a body is a point at which the entire mass of the body is supposed to concentrated.
The position of centre of mass of the system at any time is calculated using Newton’s equations of motion.

Centre of mass of two particle system:

Consider a rigid body having large number of particles, let us consider any two particles of mass
m1and m2.

Let x1 and x2 be the position vector of the system of two particles from origin O.
The centre of mass is at point C which is at a distance X from O.
The position vector of centre of mass of the two particle system is,

For a system of n particles,
Let xi be the position vector of the ith particle and mi be the corresponding mass of the ith particle, then the position vector X of the centre of mass is given by

Note:

1) if two particles of same mass then m1 = m2 the position vector of centre of mass is,
Equation (1) becomes

Therefore, centre of mass of two equal masses lies exactly middle between them.
2) Suppose that particles lies in plane, (xi,yi) be the coordinates of the point, where ith particle of the system is located. If X, Y be the co-ordinates of the centre of mass of the system,

3) suppose that particles lies in space, (xi, yi, zi ) be the co-ordinates of the point where ith particle is located.

Motion of centre of mass

The centre of mass of a system of particles move as if all the mass of the system is supposed to be concentrated at the centre of mass and all the external forces are apply at that point.
Let us consider system of n particle, position vector of a centre of mass R and total mass of the system is M.

Where V is the velocity of centre of mass and v1,v2,v3….vn be the velocity of the individual particle.
By differentiating above equation with respect to time will gives acceleration of the centre of mass.

Where A is the acceleration of centre of mass and a1,a2,a3…… an be the acceleration of the individual particle.
According to Newton’s second law,

Where Fext  is  sum of all external forces acting on the particles.
The Eq (9) states that the centre of mass of a system of particles moves as if all the mass of the system was concentrated at the centre of mass and all the external forces were applied at that point.

Linear momentum of a system of particles

We know that linear momentum of particle is defined as
P=mv          …………………………(10)
Newton’s second law for a single particle is,

Where F is force on particle.

Consider system of n particles with masses  m1,m2,m3,……mn  respectively and velocities v1,v2,v3….vn respectively. The particles interacting and have external forces acting on them. The linear momentum of first particle is m1v1 second particle  m2v2 and so on.

For the system of n particles, the linear momentum of the system is vector sum of all individual particles of the system.

The total momentum of a system of particles is equal to the product of the mass of the system and velocity of centre of mass.
Diff  Equation (13) with respect to time

2)  centre of mass moves with a constant velocity, that is moves uniformly in a straight line like a free particle.

Linear velocity in relation with angular velocity

In rotational motion of a rigid body about a fixed axis, every particle of the body moves in a circle, which lies in a plan perpendicular to the axis and has its centre on the axis.

Consider rigid body  rotating about a fixed axis(y axis). A particle describes a circle with centre C lying on axis. Let r be the radius of the circle which is perpendicular distance of the point P from the axis. The linear velocity v of the particle at P is tangent to the circle.

 Let P’ be the position of the particle after small interval of time 𝝙t and angular displacement during this time is the angle PCP’=𝝙𝝷.
Angular velocity(ω) of the particle is the rate of change of its angular displacement.
Angular velocity ω= 𝝙𝝷/𝝙t   as 𝝙t tends to zero , the ratio 𝝙𝝷/𝝙t  approaches to limit which is instantaneous angular velocity of the particles. That is

Angular velocity is a vector quantity its direction is determined by Right Hand Screw Rule.
In the above figure arc length  PP’=radius *  d𝝷
Diving by dt on both the side

Angular Acceleration

Angular acceleration in rotational motion can be defined as it is the time rate of change of angular velocity. It is denoted by α

SI unit is rad s-2 and Dimensional formula is [M0L0T-2]
Relation between angular acceleration and linear acceleration:
We know that relation between linear velocity and angular velocity is v=rω.
Differentiating this with respect to time,

Torque and Angular momentum

We know that motion of a rigid body is a combination of linear motion and rotational motion. If a body is fixed at a point it has only a rotational motion. To change the linear state we need force  then what is the analogue of force in rotational motion.

Ex :if a fan is switch on, the centre of the fan remain unmoved here fan rotates with an angular acceleration. Since centre of mass of the fan at rest no external external force acting on the fan. Here there is a production of angular acceleration even when external force is zero. At the same time we know that angular acceleration produces with  external force, then  what is the responsible for this angular acceleration? The answer is rotational analogue of force is moment of force Or Torque.

When force acting on a body gives us the turning effect of the force about the fixed point or about a fixed axis. It can be measured by the product of magnitude of force and perpendicular distance between action of force and axis of rotation.

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