6 Most SAQ’s of System of Particles & Rotational Motion Chapter in Inter 1st Year Physics (TS/AP)

4 Marks

SAQ-1 : Distinguish between center of mass and center of gravity.

Introduction

In physics, two important concepts related to the distribution of mass and weight within a body are the “center of mass” and the “center of gravity.” These concepts are crucial for understanding the motion and stability of objects.

Center of Mass

1. Definition: The center of mass is a point inside a body where the entire mass of the body is considered to be concentrated. It represents the point around which the body’s mass is evenly distributed.
2. Pertains to: The center of mass is associated with the body’s mass distribution.
3. Coincidence: In small bodies or when subjected to a uniform gravitational field, the center of mass and center of gravity coincide, meaning they are at the same point within the body.
4. Sum of Moments: The algebraic sum of moments (torque) of masses about the center of mass is zero, indicating a balanced mass distribution around this point.
5. Use: The center of mass is used to analyze the translatory motion of a body during complex movements.

Center of Gravity

1. Definition: The center of gravity is a point inside a body where the total weight of the body acts. It is where the force of gravity can be considered to be concentrated.
2. Pertains to: The center of gravity relates to the distribution of weight acting on all particles of the body and is influenced by the gravitational field.
3. Coincidence: For large bodies or in non-uniform gravitational fields, the center of mass and center of gravity may not coincide, especially when the gravitational field varies across the body.
4. Sum of Moments: The algebraic sum of moments (torque) of weights about the center of gravity is zero, implying a balanced weight distribution around this point.
5. Use: The center of gravity is crucial for determining the stability of a body and identifying where support is needed to prevent tipping or falling over.

Summary

In summary, the center of mass is where a body’s mass is considered concentrated, whereas the center of gravity is where its total weight acts. These points may coincide in small bodies or under uniform gravitational fields but can differ in larger bodies or non-uniform fields. The center of mass is significant for studying a body’s motion, while the center of gravity is essential for understanding its stability and support requirements.

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SAQ-2 : Define vector product. Explain the properties of a vector product with two examples.

Introduction

In vector mathematics, the vector product, also known as the cross product, is an operation between two vectors that results in a third vector. The vector product is a fundamental concept used in physics, engineering, and other fields to describe quantities with both magnitude and direction.

Definition of Vector Product

1. Vector Product (Cross Product): A vector product, also known as the cross product, is an operation on two vectors in three-dimensional space. It results in a new vector that is perpendicular to the plane containing the original vectors.
2. Mathematical Representation: For two vectors A and B, the vector product is denoted as A × B. The magnitude of this product is given by ∥A×B∥ = ∥A∥ ∥B∥ sin(θ), where θ is the angle between A and B, and the direction is given by the right-hand rule.

Properties of Vector Product

1. Perpendicularity: The vector product of two vectors is always perpendicular to the plane containing these vectors.
2. Anticommutativity: The cross product is anticommutative, meaning A × B = −(B×A).
3. Distribution over Addition: The cross product distributes over vector addition, i.e.,
   A × (B+C) = A × B + A × C.
4. Scalar Multiplication: Scalar multiplication is distributive over the cross product, i.e.,
   k (A×B) = (kA) × B = A × (kB), where k is a scalar.

Examples of Vector Product

1. Perpendicularity:
   • Consider two vectors A (in the x-direction) and B (in the y-direction). Their cross product A × B will be in the z-direction, perpendicular to the plane containing A and B.
   • This illustrates the property of perpendicularity.
2. Anticommutativity:
   • If A is in the x-direction and B is in the y-direction, then A × B is in the positive z-direction.
   • Conversely, B × A will be in the negative z-direction, showing that A × B = −(B×A).
   • This example demonstrates the property of anticommutativity.

Summary

In summary, the vector product or cross product is an operation in three-dimensional space producing a vector perpendicular to the plane of the original vectors. Its magnitude is the product of the magnitudes of the vectors and the sine of the angle between them, directed according to the right-hand rule. The key properties include perpendicularity, anticommutativity, and distribution over addition and scalar multiplication. The provided examples illustrate these properties, highlighting the distinctive features of vector products in vector algebra.

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SAQ-3 : Define angular acceleration and torque establish the relation between angular acceleration and torque.

Introduction

Angular acceleration and torque are important concepts in rotational dynamics, which describe the motion of objects that are rotating about an axis. Understanding these concepts is essential in various fields such as engineering, physics, and astronomy.

Definition of Angular Acceleration

1. Angular Acceleration: Angular acceleration is a vector quantity that describes the rate of change of angular velocity of an object in rotational motion. It is denoted by the symbol α and is measured in radians per second squared (rad/s²).
2. Mathematical Expression: Angular acceleration is defined as the derivative of angular velocity (ω) with respect to time (t), expressed as $$\alpha = \frac{d\omega}{dt}$$

Definition of Torque

1. Torque: Torque, also known as the moment of force, is a measure of the force causing an object to rotate about an axis. It is a vector quantity and is denoted by the symbol τ.
2. Mathematical Expression: Torque is defined as the product of the force (F) and the lever arm distance (r), which is the perpendicular distance from the axis of rotation to the line of action of the force. The formula is τ = r×F, where ×× denotes the vector cross product.

Relationship Between Angular Acceleration and Torque

1. Newton’s Second Law for Rotation: The relationship between angular acceleration and torque is analogous to Newton’s second law of motion for linear motion. It is given by the equation τ = Iα, where I is the moment of inertia of the object.
2. Moment of Inertia: The moment of inertia (I) is a scalar quantity that depends on the mass distribution of the object relative to the axis of rotation. It is a measure of an object’s resistance to changes in its rotational motion.
3. Interdependence: The equation τ = Iα establishes that the torque applied to an object is directly proportional to its angular acceleration. The greater the torque, the greater the angular acceleration, given a constant moment of inertia.

Summary

In summary, angular acceleration describes the rate of change of angular velocity, while torque is the force causing rotation about an axis. The relationship between these two quantities is established by an adaptation of Newton’s second law to rotational motion, stating that the torque applied to an object is equal to the product of its moment of inertia and its angular acceleration. This relationship highlights the interplay between force, mass distribution, and the resulting rotational behavior of an object.

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SAQ-4 : Define angular velocity (ω). Derive v=rω

Definition of Angular Velocity

1. Angular Velocity (ω): Angular velocity is a vector quantity that represents the rate of rotation of an object around a specific axis. It is defined as the angle rotated per unit time and is usually measured in radians per second (rad/s).
2. Mathematical Expression: Angular velocity is given by $$\omega = \frac{\Delta \theta}{\Delta t}$$​ where Δθ is the change in angular position (in radians), and Δt is the time taken for this change.

Derivation of the Relationship v = rω

1. Starting Point: Consider an object moving in a circular path with radius r and linear velocity v.
2. Linear Displacement: In a small time interval Δt, the object covers a linear distance Δs along the circumference of the circle.
3. Angular Displacement: The corresponding angular displacement is $$\Delta \theta = \frac{\Delta s}{r}$$​ (since s = rθ for a circle).
4. Linear Velocity: The linear velocity v is defined as $$v = \frac{\Delta s}{\Delta t}$$
5. Angular Velocity: Angular velocity ω is defined as $$\omega = \frac{\Delta \theta}{\Delta t}$$
6. Combining the Expressions: Substituting $$\Delta \theta = \frac{\Delta s}{r}$$ into the angular velocity equation gives $$\omega = \frac{\Delta s}{r\Delta t}$$
7. Final Equation: Rearranging the terms, we obtain Δs = rωΔt, and since $$v = \frac{\Delta s}{\Delta t}$$​ it follows that v = rω.

Summary

In summary, angular velocity (ω) is the rate of rotation of an object around an axis, and the relationship v = rω links the linear velocity v of an object moving in a circular path with its angular velocity ω and the radius r of the path. This equation is fundamental in understanding the connection between linear and rotational motion.

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SAQ-5 : Show that a system of particles moving under the influence of an external force, moves as if the force is applied at its center of mass.

Introduction

Understanding how a system of particles behaves under the influence of an external force is crucial in classical mechanics. This concept is pivotal in analyzing the motion of complex systems, from rigid bodies to celestial objects.

1. Concept of Center of Mass: The Center of Mass of a system is a point where the entire mass of the system can be assumed to be concentrated. It is a crucial concept when considering the motion of a system of particles under an external force.
2. Dynamics of the System Under External Force
   • When an external force is applied to a system, each particle experiences a force and consequently an acceleration.
   • The total external force acting on the system is the vector sum of the forces acting on each particle.
   • Likewise, the total torque acting on the system can be calculated about any point, but it is most conveniently analyzed about the center of mass.
3. Motion of the Center of Mass
   • According to Newton’s second law, the acceleration of the center of mass of a system is directly proportional to the total external force acting on the system and is inversely proportional to the total mass of the system.
   • Mathematically, this is expressed as where a_cm​ is the acceleration of the center of mass, Fext​ is the total external force, and m_total​ is the total mass of the system.
   • This equation shows that the system’s center of mass moves as if the total external force is applied at that point.

Summary

When a system of particles is subject to an external force, its center of mass behaves as if all the external forces are applied at that point. The system’s motion can be effectively analyzed by considering the dynamics of its center of mass, simplifying complex systems into a more manageable form. This principle is fundamental in various fields of physics, including mechanics and astrophysics.

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SAQ-6 : State the principle of conservation of angular momentum. Give two examples.

Principle of Conservation of Angular Momentum

1. The Principle of Conservation of Angular Momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant. This principle is a fundamental concept in physics, analogous to the conservation of linear momentum.
2. Mathematical Expression: The conservation of angular momentum is expressed as $$L_{\text{initial}} = L_{\text{final}}​$$ where L represents the angular momentum of the system.

Examples Demonstrating Conservation of Angular Momentum

1. Figure Skater Performing a Spin:
   • A figure skater begins a spin with arms extended and then pulls the arms inwards. Initially, with arms extended, the skater has a larger moment of inertia and spins slowly. Upon pulling the arms in, the moment of inertia decreases, causing the skater to spin faster.
   • This change in speed without an external torque demonstrates the conservation of angular momentum, as the product of the moment of inertia and angular velocity (angular momentum) remains constant.
2. Planetary Motion:
   • Planets moving around the sun provide a cosmic example of the conservation of angular momentum. As a planet moves closer to the sun in its elliptical orbit, its speed increases; when it moves away, its speed decreases.
   • This variation in speed, while the angular momentum remains constant, illustrates the principle of conservation of angular momentum. It implies that no significant external torque acts on the planet-sun system.

Summary

The Principle of Conservation of Angular Momentum is a key concept in rotational dynamics, stating that the total angular momentum of a system remains constant in the absence of external torques. This principle is widely observable in various physical phenomena, from the intricate movements of a figure skater to the grand orbits of celestial bodies. The examples of a spinning figure skater and planetary motion clearly illustrate how this principle operates in different contexts.