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Chapter 7 – System of Particles and Rotational Motion

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Earlier we have only considered the motion of a single particle, considering the object as a point mass. But in daily life, the real body has a finite size. We can consider them as rigid bodies having a definite shape. If you roll a cube-shaped rigid body on an inclined surface, it will show translational motion i.e. at any instant of time, every particle of the body has the same velocity. But what will happen if you have a cylindrical object? It will show translational motion and rotational motion. This helps us to study the system of particles which stands for groups of particles which are interrelated. Along with these, you will study about new terms like the center of mass, the radius of gyration, moment of inertia, parallel axes theorem, and perpendicular axis theorem.

Look around you, you will observe many examples of rotational motion, i.e. ceiling fan, washing machine, grinder, cycle, swings and many more. Such motion in which a rigid body rotates about a fixed axis and every particle of the body moves in a circle lying in a plane perpendicular to the axis and has its centre on the axis is termed as rotational motion. Axis of rotation is the line along which the body is fixed.

If the motion of a rigid body is not pivoted (fixed), such motion is either a pure translational motion or a combination of translational motion and rotational motion. The motion of a rigid body which is pivoted or fixed in some way is rotation.

Centre of mass of a body can be defined as the region or point where the entire mass of the body appears to be concentrated.

R = \frac{\sum m_{i}r_{i}}{M}

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.

 

 

 

 

 

 

Let two particles m1 and m2 be located on the x-axis.

The distance of m1 and m2 from the origin is x1 and x2 respectively

Centre of mass (C) is at distance X from the origin (O), then X is

X = \frac{m_{1}x_{1} + m_{2}x_{2}}{m1 + m2}

X can be regarded as the mass-weighted mean of x1 and x2. In this way, we can find the centre of mass of the two particle system.

If the two particles have the same mass m1 = m2 = m, then

X = \frac{mx_{1} + mx_{2}}{2m} = \frac{x_{1} + x_{2}}{2}

From the above equation we can conclude that for two particles of equal mass, the centre of mass lies exactly midway between them.

Total Linear momentum of particles (v) is the product of mass and velocity.

p = mv

The total momentum of a system of particles is equal to the product of the total mass of the system and the velocity of its centre of mass.

 \frac{d\textbf{P}}{dt} = M \frac{d\textbf{V}}{dt} = M\textbf{A}

From Newton’s Second law of motion; F=MA. Thus,

 \frac{d\textbf{P}}{dt} = \textbf{F}_{ext}

Consider an object undergoing rotational motion around a fixed axis. As a result of its motion, we can consider its kinetic energy. The velocity of the object will be replaced by angular velocity (ω). Now the kinetic energy of a rotating object could be derived using Moment of Inertia.

The amount of torque required for a specific angular acceleration in a rotational axis is called Moment of Inertia.

 K = \frac{1}{2} \omega^2 (\displaystyle\sum_{i=1}^{n} m_ir_i^2)

 I = \displaystyle\sum_{i=1}^{n} m_ir_i^2

With this definition,

 K = \frac{1}{2} I \omega^2

K= Kinetic energy

I = Moment of Inertia

ω = Angular velocity

You can see in the above diagram that the moment of inertia changes with respect to the axis of rotation. A new term comes which relates the moment of inertia with a total mass of the body – Radius of gyration.

The radius of gyration of a body about an axis is the distance from the axis of a mass point (its mass is equal to the mass of the whole body) and whose moment of inertia is equal to the moment of inertia of the body about the axis.

Moment of inertia of a rigid body depends on:

  1. Mass of the body
  2. Shape and size of the body
  3. Distribution of mass about the axis of rotation
  4. Position and orientation of the axis of rotation

Theorem of perpendicular axes is related to the moment of inertia. It is applicable to planar objects, i.e. flat bodies with little thickness as compared to other dimensions. Theorem of perpendicular axes states that:

“The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.”

 

 

 

 

 

 

x and y axes are two perpendicular axes in the plane and the z-axis is perpendicular to the plane.

Iz = Ix + Iy

What if in place of a planar object, we have curved objects or something else? You can apply Theorem of Parallel Axes to objects having any shape. This theorem helps us to find the moment of inertia of a body about any axis. The moment of inertia of the body must be about a parallel axis through the centre of mass of the body.

 

 

 

 

 

 

 

 

 

The z and z′ axes are two parallel axes separated by a distance a; O is the centre of mass of the body, OO’ = a.

Theorem of Parallel Axes states that:

“The moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.”

I z′ = I z+ Ma2

Iz  = moments of inertia of the body about the z-axis

I z′ = moments of inertia of the body about z′ axis

M = Total mass of the body

a = Perpendicular distance between the two parallel axes

Q1. What is the system of particles?

A1. System of particles means a group of particles which are interrelated.

Q2. What is rotational motion? Explain with a diagram.

A2. In rotational motion, about a fixed axis every particle of the rigid body moves in a circle which lies in a plane perpendicular to the axis and has its centre on the axis.

 

 

 

 

 

 

 

 

 

Q3. What is the difference between linear and translational motion?

A3. In rotational motion, at every point in the rotating rigid body has the same angular velocity at any instant of time. In translation motion, every particle of the body moves with the same velocity at any instant of time.

Q4. What is the difference between linear and translational motion?

A4. In linear motion, the motion is one-dimensional along a straight line. In translation motion, every particle of the body moves with the same velocity at any instant of time.

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