System of Particles and Rotational Dynamics
We know dynamics is a branch of physics that deals with the physics associated with the objects or particles in motion. The dynamics have been subdivided into various categories depending on the subject of interest. For example, the motion of the fluids is studied under fluid dynamics, the motion of electrical charges are studied under electrodynamics, similarly one of these categories is rotational dynamics. Rotational dynamics is a branch of physics that deals with the forces and motion about an axis of rotation.
In this article, we will study the system of particles and rotational dynamics with a brief introduction to the system of particles and rotational motion notes. Let’s begin!!
Archimedes of Syracuse
Since Archimedes' time, the concept of rotational dynamics, and specifically the moment of force or torque associated with rotational dynamics, has been researched. You may recall Archimedes' famous phrase, "Give me a lever, and I shall move the globe," from your school days. A lever is used to apply torque. The lever is employed to put rotational force on a threaded fastener in this situation.
Archimedes of Syracuse (c. 287 – c. 212 BC) was a Greek mathematician, physicist, engineer, astronomer, and inventor who lived in the ancient Sicilian city of Syracuse. Despite the fact that nothing is known about his life, he is recognised as one of classical antiquity's foremost scientists.
System of Particles and Rotational Motion Notes
In rotational dynamics physics, Newton’s Second Law of motion is used to develop a formalism to describe how objects rotate. In other words, rotational dynamics introduce the concept of torque which represents a comparable role to that of force in non-rotational dynamics. In rotational dynamics, we will study the concept of torque, a moment of inertia to describe how objects oppose the rotational motion.
When a particle or an object is subjected to a motion in a straight line then the motion is known as translational motion, at the same time when the same particle moves along an axis in a circular path then it is known as Rotational Motion and the branch of physics that deals with the rotational motion are known as rotational dynamics. The main concepts of rotational dynamics are the system of particles and rotational motion. For example, consider a ring rolling without slipping in a straight line. The forth displacement of the ring is equal to the linear displacement of a point fixed on the rim.
We often refer to an object as an extended body in rotational dynamics, it implies that we are referring to it as a system of particles. Rigid bodies are those bodies with definite shape and size. In rigid bodies, the distance between the constituting pairs of particles does not change.
Torque: Rotational Force
During the linear motion, an object which is subjected to linear motion will move till it feels any external force acting on it. The external force acting will change the direction of motion of objects. However, when a body is moving about a fixed axis in any circular path or curved path then, it will also feel the force and the force acting on the bodies under rotational motion is known as the Torque. It is always better to understand with an example, we notice that it is always easy to open a door by pushing on the edge farthest from the joints than by pushing in the middle. Therefore, torque is a force that plays an important role in rotational dynamics.
To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force F on a point mass m that is at a distance r from a pivot point, as shown in Figure.
Because the force is perpendicular to r, an acceleration a is obtained in the direction of F According to newton’s laws of motion force is equal to the product of mass and acceleration i.e.,F = ma. We know that the relation between the linear acceleration(a) and rotational acceleration(α) is given by a = rα, and we substitute this value of a into F=ma, we get:
$\Rightarrow F = mr\alpha$……(1)
Where,
a - linear acceleration
α-rotational acceleration
Now, we know that the physical quantity torque is the changing effectiveness of a force. Therefore, since the force F is perpendicular to r, torque (τ) can be simply written as
τ = Fr.
Therefore, multiplying equation (1) on both sides by r, we get:
$\Rightarrow F=mr\alpha$
$\Rightarrow rF=mr^2\alpha$
$\Rightarrow \tau=mr^2\alpha$….(2)
Equation (2) is known as Newton’s second law of motion in rotational dynamics, where the torque is analogues to force and angular acceleration is analogous to linear or translational acceleration and the term mr2 is analogous to mass. The product of mass and square of the distance from the axis of the rotation is known as the moment of inertia.
The best examples of rotational motion are rigid bodies. The rigid bodies experience both translational and rotational motion. Thus, for rigid bodies, both the linear velocity (corresponding to translational motion) and the angular velocity (corresponding to rotational motion) need to be analyzed. These problems can be simplified by defining the translational and rotational motion of any rigid body separately.
The System Of Particles And Rotational Dynamics is completely analogous to linear or translational dynamics. Most of the equations that are derived for the mechanics of rotating objects are quite similar to the motion equations derived for linear motion. In System Of Particles And Rotational Motion, particularly rigid bodies are considered. A rigid body is an object with a mass that holds a rigid shape.
For a system of particles, the centre of mass is described as that point at which the entire mass of the system is imagined to be concentrated, for consideration of its translational motion. If all the external forces acting on the system of particles were to be applied at the centre of mass, the state of rest or motion of the system of particles and the rotational motion shall remain unaffected.
The centre of mass of a body is a balancing point for any system of particles. The centre of mass of a two-particle system is always found to be on the line joining the two particles and it is often located in between the particles.
Moment of Inertia and Rotational Motion
Now, let us understand the important concept of rotational motion and rotational dynamics, i.e., Moment of inertia. The moment of inertia of a rigid body is often defined with respect to a fixed rotation axis, it is a physical quantity that determines the amount of torque (Force acting on bodies under rotational motion) required for obtaining required angular acceleration or a physical property of a rigid body due to which it resists angular acceleration.
The formula for the moment of inertia of a rigid body is given by the sum of the product of the mass of each particle with the square of rigid body distance from the axis of the rotation. Mathematically, the moment of inertia formula is given by:
$I=\sum_{i=1}^nm_ir_i^2$
Where,
m -The mass of the rigid body
r - The distance from the axis of the rotation
i - The number of particles
Similarly, we study the rolling motion, angular acceleration, angular velocity, theorems related to the moment of inertia, etc… in the system of particles and rotational motion notes or rotational dynamics notes.
Solved Example
1. In a car garage loosens a bolt by applying a force of 500 N to a wrench. Further, the man applies the force such that the applied force will be perpendicular to the arm of the wrench. The total distance of the bolt to the man's hand is about 0.50 m. Calculate the magnitude of the torque produced?
Sol:
Given,
The amount of force applied to the bolt = F = 500 N
The angle between the applied force and the wrench θ = 900
The distance between the wrench and the arm of man = r = 0.50 m
Now, we are asked to determine the magnitude of the torque produced while loosening the wrench. We know that the Torque can be calculated using the following equation:
$\tau=F\times r \times \sin \theta$
$\tau=500\times 0.5 \times \sin 90^o=250 Nm$
Therefore, the magnitude of the torque produced is 250 Nm.
Interesting Facts:
Torque is an important factor in fidget spinners!! Yes, the technology of fidget spinners evolved earlier than they ever began spinning. Look at it. It simply sits there. Even while it seems to be doing not anything at all, the spinners are exemplifying essential physics concepts. If you need the item to rotate you need to exert a torque at the spinner.
Torque relies upon the importance of the carried out pressure, how a way from the rotation axis the pressure is carried out, and what sort of the pressure is perpendicular to the arm of the spinner. You'll want to push it with loads of oomph as far from the middle as possible, and at the least barely withinside the path you need it to spin.
Conclusion
In this article we understood the dynamics of rotational motion and the importance of the Torque in rotational motion.
FAQs on Rotational Dynamics
1. What is rotational dynamics, and how does it differ from rotational motion?
Rotational dynamics is the branch of physics that studies the causes of rotational motion. It focuses on the relationship between torque, moment of inertia, and the resulting angular acceleration. In contrast, rotational motion (often studied under kinematics) simply describes the motion of an object rotating about an axis, using variables like angular velocity and displacement, without explaining what causes the rotation.
2. What are the key concepts studied in rotational dynamics?
The core concepts in rotational dynamics, as covered in the NCERT syllabus, include:
- Torque (τ): The rotational equivalent of force that causes an object to experience angular acceleration.
- Moment of Inertia (I): A measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion.
- Angular Momentum (L): The rotational equivalent of linear momentum. It is a conserved quantity when no external torque acts on the system.
- Centre of Mass: A point where the entire mass of a body can be assumed to be concentrated for analysing its translational motion.
- Radius of Gyration: The distance from the axis of rotation to a point where the entire mass of the body could be concentrated to give the same moment of inertia.
3. Why is moment of inertia considered the rotational equivalent of mass?
Moment of inertia is called the rotational equivalent of mass because it plays the same role in rotation as mass does in linear motion. Mass measures an object's inertia, or its resistance to being linearly accelerated by a force (F = ma). Similarly, the moment of inertia measures an object's resistance to being angularly accelerated by a torque (τ = Iα). An object with a larger mass is harder to push, and an object with a larger moment of inertia is harder to spin.
4. What are the SI units for torque and moment of inertia as per the NCERT syllabus?
The SI units for these fundamental quantities in rotational dynamics are:
- The SI unit for torque (τ) is the Newton-meter (N·m).
- The SI unit for moment of inertia (I) is kilogram-meter squared (kg·m²).
5. How is the concept of torque applied in everyday situations?
Torque is applied constantly in daily life whenever we need to create a turning or twisting effect. Common examples include:
- Opening a door: Pushing on the handle far from the hinges creates a larger torque for the same amount of force, making it easier to open.
- Using a wrench: A longer wrench allows you to apply the same torque with less force to tighten or loosen a bolt.
- Turning a steering wheel: Your hands apply a pair of forces to create a net torque that rotates the wheel.
- Pedalling a bicycle: Pushing down on the pedals creates a torque that turns the crankset and propels the bike forward.
6. What are the fundamental equations that govern rotational dynamics?
The key equation in rotational dynamics is Newton's second law for rotation, which is analogous to F = ma. It is expressed as: τ = Iα, where 'τ' is the net external torque, 'I' is the moment of inertia, and 'α' is the angular acceleration. Other important relations include the definition of torque (τ = r × F) and angular momentum (L = Iω).
7. How does the principle of conservation of angular momentum work? Provide an example.
The principle of conservation of angular momentum states that if the net external torque acting on a system is zero, its total angular momentum (L) remains constant. Since angular momentum is the product of moment of inertia and angular velocity (L = Iω), if the moment of inertia changes, the angular velocity must change inversely to keep the product constant. A classic example is an ice skater spinning. When she pulls her arms in, her moment of inertia (I) decreases, causing her angular velocity (ω) to increase dramatically to conserve L.
8. Why is the choice of axis of rotation crucial when calculating the moment of inertia?
The choice of axis is crucial because the moment of inertia is not just a property of an object's mass, but also of how that mass is distributed relative to the axis of rotation. The formula for moment of inertia involves the term r² (the square of the distance of mass particles from the axis). A long rod has a very small moment of inertia when rotated about its long axis, but a much larger moment of inertia when rotated about an axis through its centre and perpendicular to its length, because more of its mass is farther from the axis.
9. Can an object have both translational and rotational motion? Explain with an example.
Yes, an object can experience both translational (moving from one point to another) and rotational (spinning) motion simultaneously. This is called general plane motion. A perfect example is a rolling wheel of a car. The wheel's centre of mass moves forward along the road (translational motion) while the wheel itself spins around its axle (rotational motion). The Earth also exhibits both motions, translating in orbit around the Sun while rotating on its own axis.
10. How do the parallel and perpendicular axis theorems simplify calculations for moment of inertia?
These theorems are powerful tools for finding the moment of inertia without complex integration.
- The Parallel Axis Theorem helps find the moment of inertia (I) about any axis, provided you know the moment of inertia (I_cm) about a parallel axis passing through the body's centre of mass. The formula is I = I_cm + Md², where M is the total mass and d is the perpendicular distance between the two axes.
- The Perpendicular Axis Theorem applies to planar objects (laminae). It states that the moment of inertia about an axis perpendicular to the plane (the z-axis) is the sum of the moments of inertia about two perpendicular axes in the plane that intersect at the same point (I_z = I_x + I_y).
11. In which class is the chapter on System of Particles and Rotational Motion covered under the CBSE syllabus for the year 2025-26?
The chapter 'System of Particles and Rotational Motion', which covers the principles of rotational dynamics, is a core part of the Class 11 Physics syllabus as per the CBSE curriculum for the academic year 2025-26. It is typically Chapter 7 in the NCERT Part 1 textbook.
