Dynamics of Rotational Motion and System of Particles

The study of the system of particles and rotational motion is the fundamental aspect to classical mechanics that seamlessly blends the gap between the elegance of various tools of mathematics along with the fascinating concepts of physics. This topic delves deeper into the dynamics of how objects move and interact under the influence of various forces. It covers the concepts of both rotational motion (describes how objects rotate around a point) and linear motion ( explains how objects move along a straight path).  

By analyzing and understanding the system of particles, you can develop an idea how the centre of mass acts and their momentum is conserved. Rotational motion introduces you to various other concepts like torque, conservation of angular momentum and moment of inertia. 

Once you are well equipped with these concepts, you can clearly understand the reason behind why various objects spin in specific ways. 

Rigid body:

In order to grasp the concept of rotational motion, you need to understand what a rigid body is!
A rigid body is said to be one in which the distance between the particles remains constant in spite of any constant force acting upon it.
If the rigid body is not fixed in place, it has the ability to move in any direction or even multiple directions at the same time.
When a rigid body is rotating, its angular velocity remains consistent at all times.

In basic terms, what is the meaning of Centre Of Mass?
In a system with two molecules, the centre of mass lies on the line that connects the two particles and is located midway between them.
For a complex object such as a moving car, the centre of mass is located at the point where all the mass is concentrated.
The centre of mass of a group of particles acts as if all the mass is concentrated at that point, and all the external forces are applied to it. If no external forces are acting on the body, the centre of mass will maintain a constant energy.

What are the cases of mechanical equilibrium in rigid bodies?

To find a mechanical equilibrium of a rigid body we need to follow two conditions.

• Translational Equilibrium: All external forces acting on an object need to cancel each other out, resulting in no net force on the body.
• Rotational Equilibrium: All external torques acting on an object need to balance each other out, so there is no net torque on the body.

Torque is calculated by multiplying the magnitude of the force by the perpendicular distance from the point of application of the force to the axis of rotation.

Draw the relationship between Angular velocity and Linear Velocity.
Angular velocity: This is the rate at which an object rotates around an axis. 
Linear velocity: This is the rate at which an object moves through a straight line path Typically it is measured in metres per second.

The relationship between these two is explained by:

v = ωr
v is the linear velocity
r is the radius of distance from the axis of rotation
ω is the angular velocity

Define Torque and Centre of Gravity

Torque refers to the tendency of a force to cause an object to rotate around an axis. It is calculated by taking the component of the force that lies in the plane perpendicular to the axis of rotation and multiplying it by the shortest distance between the axis and this component. 

In simple words, torque is the amount of force applied multiplied by the distance from the axis of rotation to the point where this force occurs perpendicularly to the axis of rotation. 

The centre of gravity is the point on an object where its entire weight is thought to be concentrated.

What do you understand by the term Moment of Inertia?

When an object rotates around an axis, it naturally resists changes to its rotational motion. 
I=mr2.
There are two main theorems that govern the concept of the system of particles and rotational motion. Grasping these key theorems is essential to understand these topics. They are:

Perpendicular Axis Theorem: The moment of inertia (I) of a body with respect to its specific perpendicular axis is the magnitude of the moments of inertia with respect to two axes, lying in the plane of the body. These two axes must be perpendicular to each other and intersect at the point where the perpendicular axis goes through.

The formula is: Iz=Ix+Iy
Where Ix is the moment of inertia along the x-axis and Iy is the moment of inertia along the y-axis.
Parallel Axis Theorem: This theorem states that the moment of inertia of an object around any axis is parallel to one passing through its center of mass can be calculated by taking a sum of the moment of inertia around the center of mass to the product of the object’s mass and the square of the distance between the two axes.

I=Icm+Md2

Where:
I is the moment of inertia about the new axis
Icm is the moment of inertia about the centre of mass axis
M is the mass of the object
D is the distance between the centre of mass axis and new axis.

Understanding Angular Momentum and its Conservation Law

Angular momentum is determined by multiplying an object's momentum by the perpendicular distance from the axis of rotation. This results in a measure of how much rotational motion an object has around that axis, with the direction of angular momentum being perpendicular to the plane of motion.

According to the law of conservation of angular momentum, the total angular momentum of a rigid body or a system of particles remains constant if no external forces act on it. 

When can you say that a rigid body is in an equilibrium state?

A rigid body is said to be in an equilibrium state when it is in translational equilibrium and rotational equilibrium.

FAQS:

1. Work done is zero in two cases. What are they?

There are two situations when the work done is zero. They are: 
When there is no displacement 
When the displacement is normal to the direction of force

2. What methods can be used to alter the rotational direction of an object?

When a body is fixed at a pivot point, applying force at the right spot makes it rotate around that pivoted point. To change the direction in which it spins, you need to exert the force at a different point on the body. This alteration in the point of application of force shifts the rotational direction accordingly.

3. What factors lead to the rise of rotational motion?

When you apply torque to an object around a specific axis, it causes the object to spin. This spinning motion is similar to how a regular force causes an object to move in a straight line. In both cases, the torque leads to motion. But in case of rotational motion, the object turns around an axis, instead of moving linearly.

4. What do you understand about angular velocity and angular acceleration?

Angular velocity refers to how quickly an object or a particle rotates around a central point or axis within a specific time frame. Also known as rotational velocity, it is measured in angles per unit time, typically in radians per second (rad/s). 

When this angular velocity changes, this change rate is called angular acceleration.