Angular Momentum about Center of Mass of a Rigid Body

Lecture Notes - Gyrostatic Applications - Adopted From D. S.Mathur and P. S.Hemne, 2000, Mechanics

Lecture Notes - Gyroscopic Precession

Non-conservative Forces and General Law of Conservation of Energy

Derivation of Moments of Inertia of A Circular Disk, Annular Disk, Solid Sphere and Hollow Sphere

Study Techniques, Strategies and Exam Tips - Mechanics 23BPH3C1

Dear Student-Friends,

Guten Tag!

Here are some study techniques and strategies you can follow for optimal performance, tailored to the university exam's structure, specifically in Mechanics. I request you to use these techniques to your advantage.

Part A (2 Marks - Short Answers)

• Core Concepts: Focus on understanding definitions, fundamental laws, and equations for quick recall, as these are likely to form the basis of the short-answer questions.

• Flashcards: Create flashcards for each unit’s key terms, equations, and principles. This is helpful for reinforcing knowledge quickly.

• Practice Concise Answers: Practice writing brief, accurate answers for potential two-mark questions on each topic.

Part B (5 Marks - Either/Or Choice)

• Understand Key Applications: Since this part allows an either/or choice, focus on grasping applications in each unit (e.g., conservation laws, types of forces, moments of inertia).

• Review Sample Problems: Review previous assignments or sample questions that reflect the type of questions that may appear here.

Part C (10 Marks - Long Answers)

• In-Depth Study of Each Unit: Since these questions require detailed responses, focus on understanding the larger principles and their derivations (e.g., Newton’s Laws, Lagrangian mechanics) and practice writing well-structured answers.

• Focus on 3 Units for Depth: Prioritize three units for deeper mastery, ensuring you can answer any long-form question that may arise from them. This approach ensures focus and depth in your study time. Also note that the first unit is lengthy and may consume a lot of your preparation time. So don't focus on the first unit too much.

• Explain Concepts in Your Own Words: Practicing how to explain these concepts clearly will prepare you to tackle questions that require detailed discussion or derivation.

General Study Tips

• Create a Study Schedule: Divide your time equally among the units, focusing a little extra on complex concepts or areas that require practice with calculations.

• Teach-Back Method: Teach the main concepts to someone else or explain them aloud to yourself. This reinforces understanding and exposes any gaps in knowledge.

• Past Exam Papers: Practice with internal and model exam question papers to understand the format, timing, and types of questions asked. This is invaluable for both content and exam-day readiness.

Best Wishes For Your Grand Success,

Yours Teacher,

KAZ

Newton's Laws of Motion - Significance and Examples :

Newton's Laws :
First Law (Law of Inertia): An object will stay at rest or keep moving at a constant velocity unless acted upon by an external force.
Second Law (F = ma): The force acting on an object is proportional to the rate of change of its momentum  mv. In the case of constant mass and with the choice of suitable units F = ma .
Third Law (Action-Reaction Law): For every action, there is an equal and opposite reaction.

Significance : Newton's Laws are important because they give us a clear way to describe and predict how objects move when forces act on them. These laws are the foundation of classical mechanics and are essential in fields like engineering, astronomy, and everyday situations involving force and motion.

Examples
First Law : If we are in a moving car that suddenly stops, our body tends to keep moving forward even though the car has stopped, which is why seat belts are essential—they provide the external force needed to stop your body safely. This tendency to keep moving in the same direction until an outside force acts is an illustration of inertia.
Second Law : When we hit cricket ball with a bat, the more force we apply to the ball the more momentum the ball gains. 
Third Law : The flapping of a bird's wings is a good example of Newton's Third Law. When a bird pushes its wings downward, it exerts a force on the air. According to Newton's Third Law, the air pushes back with an equal and opposite force, propelling the bird upward and forward. This action-reaction pair allows the bird to stay in the air and control its direction while flying.

Degrees of Freedom - Definition and Examples

Definition : Degrees of freedom of a system the minimum number of independent coordinates needed to uniquely define the position or configuration of the system

Examples : 

Single Particle in 3D Space : A single particle in three-dimensional space has 3 degrees of freedom. It can move independently in the x, y and z directions.

Rigid body in  3D Space : A rigid body  has 6 degrees of freedom. Since it has 3 translational (movement along x, y and z axes) degrees of freedom and can rotate freely about three perpendicular axes, hence has 3 rotational degrees of freedom in addition.

Simple Pendulum : The bob of a simple pendulum has only one degree of freedom  as it can only swing around a fixed point in a plane with fixed orientation in space. This degree of freedom is indicated by the angle θ of the deflection of the pendulum string from the normal from the support point.

Double Pendulum :  A double pendulum consisting of two pendula connected in sequence, has 2 degrees of freedom. One degree of freedom corresponds to the angle of the first pendulum, and the second corresponds to the angle of the second pendulum relative to the first.

Diatomic Molecule : A diatomic molecule (like O₂) in a three-dimensional space has 5 degrees of freedom at room temperature: 3 translational (motion along x, y, and z axes) and 2 rotational (rotation about two perpendicular axes). The third rotation axis along the bond length is negligible for most diatomic gases due to quantum constraints.

Consolidated Question Bank - Mechanics - 23BPH3C1

 

 UNIT - I
PART A - 2 MARK QUESTIONS

1. State Newton's First Law of Motion.
2. What is the frictional force? [Nov 2024]
3. Define gravitational potential.

4. State Newton's Law of Gravitation.[Nov 2024]
4. What is the escape velocity from the Earth's surface?
5. State Kepler's First Law and its significance in planetary motion.

6. State Kepler's Second Law.

8. State Kepler's Third Law.
9. What is the Principle of Equivalence?
10. Explain gravitational redshift.

PART B - 5 MARK QUESTIONS

1.Discuss the types of everyday forces in physics with examples.
2. State and Explain Newton's Laws of Motion. [Nov 2024]
3. Derive the equations of motion for a particle moving under a uniform gravitational field.
4.Describe the concept of escape velocity and derive the formula for escape velocity from the Earth.
5.State and explain Kepler’s three laws of planetary motion.
6. Write short notes on earth satellites. [Nov 2024]
7.Explain the Earth-Moon system and discuss its influence on tides and orbital motion.
8.Describe the gravitational potential energy of a satellite in orbit.
9. Describe Einstein’s Theory of Gravitation
10.Explain the phenomenon of the perihelion shift of Mercury.

PART C - 10  MARK QUESTIONS

1. Explain Newton's laws of motion in detail. 

2. Derive and explain the equation for the gravitational potential.

3.Explain the determination of the universal gravitational constant (G) using Boys' method.

4. Discuss Kepler’s laws of planetary motion in detail. Derive Kepler’s Third Law. ; Describe the classical theory of gravitation. State and explain Kepler's laws of motion. [Nov 2024]

5. Explain Einstein’s Theory of Gravitation with reference to the Principle of Equivalence. Discuss the experimental tests supporting the theory, including gravitational redshift and light bending.

 UNIT - II
PART A - 2 MARK QUESTIONS

1.State the law of conservation of linear momentum. [Nov 2024]
2.What is the center of mass of a system?
3.Define torque.
4.What is an elastic collision? Give one example.
5.State the principle of conservation of angular momentum.
6.Explain what is meant by "system with variable mass" with an example.; What is meant by the system of variable mass? [Nov 2024]
7.What happens to the angular momentum in a proton scattering event with a heavy nucleus?

PART B - 5 MARK QUESTIONS

1.Describe the concept of a center of mass and explain its importance in the motion of a  system of particles.

2.Explain the law of conservation of linear momentum with an example involving internal and external forces.

3.State and explain the conservation of linear momentum. [Nov 2024]

4.What is angular momentum? Derive the expression for the angular momentum of a rigid body about its center of mass.

5.Explain torque due to internal forces. [Nov 2024]

6.Write Note on Torque Due to Gravity.

PART C - 10 MARK QUESTIONS

1.Derive the Expressions for Velocities of Two Particles Elastically Colliding with Each Other Along Their Line of Sight, After Impact.

2.Describe the mechanics of proton scattering by a heavy nucleus.; Derive an expression for the distance of closest approach of a proton in the Coulomb potential. [Nov 2024]

 

 UNIT - III
PART A - 2 MARK QUESTIONS

1.What is the significance of conservation laws in physics?
2.Define work and provide its SI unit

3.What are conservative forces? Give one example. [Nov 2024]
4.State the law of conservation of energy. [Nov 2024]
5.What is potential energy in a gravitational field?
6.Differentiate between conservative and non-conservative forces.
7.Explain the concept of power and provide its SI unit.

PART B - 5 MARK QUESTIONS

1.Explain the law of conservation of energy with an example in a gravitational field.

2.Describe the relationship between work, power, and energy.

3.Explain the concepts of work and power. [Nov 2024]

4.Explain the concept of potential energy in an electric field and how it relates to conservative forces.; Explain the Conservation of Energy in electric field.[Nov 2024]

5.Discuss the difference between conservative and non-conservative forces.

6.Explain the concept of work done by a force and derive the formula for work done in moving an object over a distance.

PART C - 10 MARK QUESTIONS

1.Explain the law of conservation of energy in detail. Derive the general law of conservation of energy, including the concepts of work done by conservative and non-conservative forces.

2.Discuss the concepts of work, power, and energy in physics. Derive the formulas for work done by a constant force, power, and kinetic energy.

3.Explain the concept of mechanical energy and its conservation in a closed system. Derive the principle of conservation of mechanical energy, including the roles of kinetic and potential energy.

4.Describe the relationship between work and energy using the work-energy theorem. Derive the theorem and discuss its implications for kinetic energy.

5.Explain the significance of conservation laws. Write short notes on non-conservative forces. [Nov 2024]

 UNIT - IV
PART A - 2 MARK QUESTIONS

1.Define angular momentum and state its SI unit.
2.What is moment of inertia? State its significance in rotational motion. ; What is the physical significance of moment of inertia? [Nov 2024]

3.State the parallel axis theorem for the moment of inertia.[Nov 2024]
4.State the perpendicular axis theorem for the moment of inertia.
5.What is kinetic energy of rotation? Write its formula.
6.What is gyroscopic precession?
7.Differentiate between translational and rotational motion.

8.Write expression for the acceleration of of a body rolling down an inclined plane.

9.What is gyrostatic effect? Provide one example of its application.

PART B - 5 MARK QUESTIONS

1.Derive the formula for rotational kinetic energy in terms of moment of inertia and angular velocity.

2.Define the moment of inertia of a body and discuss its' physical significance. [Nov 2024]

3.State and prove the parallel axis theorem. 

4.State and prove the perpendicular axis theorem. [Nov 2024]

5.Derive the expression for the moment of inertia of a solid cylinder about its central axis.

6.Describe gyroscopic precession and derive an expression for the precession speed. 

PART C - 10 MARK QUESTIONS

1.Explain the dynamics of a rigid body rotating about a fixed axis. Derive the equations of rotational motion and discuss the relationship between torque, angular momentum, and angular acceleration.

2.Discuss the concept of a body rolling without slipping along a plane surface. Derive the expression for the total kinetic energy of the rolling body in terms of its translational and rotational kinetic energy.

3.Explain the motion of a body rolling down an inclined plane. Derive an expression for its acceleration in terms of its moment of inertia and radius. ; Obtain an expression for the acceleration of a body rolling down an inclined plane [Nov 2024].

4.Explain gyroscopic precession in detail. Derive the precession rate for a gyroscope,

5.Describe the general theorems of moment of inertia (parallel axis and perpendicular axis theorems) and provide proofs for each.

 UNIT - V
PART A - 2 MARK QUESTIONS

1.What are generalized coordinates? Provide one example.
2.Define degrees of freedom in a mechanical system.

3.What is a constraint in mechanics? Give an example of a constraint in a physical system.; What are constraints? [Nov 2024]
4.What is the principle of virtual work?
5.State D’Alembert’s Principle in mechanics. [Nov 2024]
6.Write down Lagrange’s equation of motion and briefly explain its significance.

PART B - 5 MARK QUESTIONS

1.Describe degrees of freedom in a mechanical system and explain how constraints affect the degrees of freedom.

2.State and explain principle of virtual work. [Nov 2024]

3.Explain the different types of constrains with examples. [Nov 2024]

4.Explain the concept of generalized coordinates and illustrate their use with an example involving a pendulum.

5.State and explain D'Alembert's Principle. 

6.Apply Lagrange’s equation to derive the equation of motion for a simple pendulum.

7.Explain the working of Atwood’s Machine and derive its equation of motion using Lagrange’s method.

PART C - 10 MARK QUESTIONS

1.Derive Lagrange’s equation of motion from D’Alembert’s principle.  [Nov 2024]

Question Bank - Mechanics - 23BPH3C1 - UNIT V

   

 UNIT - V
PART A - 2 MARK QUESTIONS

1.What are generalized coordinates? Provide one example.
2.Define degrees of freedom in a mechanical system.

3.What is a constraint in mechanics? Give an example of a constraint in a physical system.
4.What is the principle of virtual work?
5.State D’Alembert’s Principle in mechanics.
6.Write down Lagrange’s equation of motion and briefly explain its significance.

PART B - 5 MARK QUESTIONS

1.Describe degrees of freedom in a mechanical system and explain how constraints affect the degrees of freedom.

2.Explain the concept of generalized coordinates and illustrate their use with an example involving a pendulum.

3.State and explain D'Alembert's Principle. 

4.Apply Lagrange’s equation to derive the equation of motion for a simple pendulum.

5.Explain the working of Atwood’s Machine and derive its equation of motion using Lagrange’s method.

PART C - 10 MARK QUESTIONS

1.Derive Lagrange’s equation of motion from D’Alembert’s 

Question Bank - Mechanics - 23BPH3C1 - UNIT IV

  

 UNIT - IV
PART A - 2 MARK QUESTIONS

1.Define angular momentum and state its SI unit.
2.What is moment of inertia? State its significance in rotational motion.

3.State the parallel axis theorem for the moment of inertia.
4.State the perpendicular axis theorem for the moment of inertia.
5.What is kinetic energy of rotation? Write its formula.
6.What is gyroscopic precession?
7.Differentiate between translational and rotational motion.

8.Write expression for the acceleration of of a body rolling down an inclined plane.

9.What is gyrostatic effect? Provide one example of its application.

PART B - 5 MARK QUESTIONS

1.Derive the formula for rotational kinetic energy in terms of moment of inertia and angular velocity.

2.State and prove the parallel axis theorem. 

3.State and prove the perpendicular axis theorem. 

4.Derive the expression for the moment of inertia of a solid cylinder about its central axis.

5.Describe gyroscopic precession and derive an expression for the precession speed. 

6.Explain the motion of a body rolling down an inclined plane. Derive an expression for its acceleration in terms of its moment of inertia and radius.

PART C - 10 MARK QUESTIONS

1.Explain the dynamics of a rigid body rotating about a fixed axis. Derive the equations of rotational motion and discuss the relationship between torque, angular momentum, and angular acceleration.

2.Discuss the concept of a body rolling without slipping along a plane surface. Derive the expression for the total kinetic energy of the rolling body in terms of its translational and rotational kinetic energy.

3.Explain gyroscopic precession in detail. Derive the precession rate for a gyroscope,

4.Describe the general theorems of moment of inertia (parallel axis and perpendicular axis theorems) and provide proofs for each.

Question Bank - Mechanics - 23BPH3C1 - UNIT III

 

 UNIT - III
PART A - 2 MARK QUESTIONS

1.What is the significance of conservation laws in physics?
2.Define work and provide its SI unit

3.What are conservative forces? Give one example.
4.State the law of conservation of energy.
5.What is potential energy in a gravitational field?
6.Differentiate between conservative and non-conservative forces.
7.Explain the concept of power and provide its SI unit.

PART B - 5 MARK QUESTIONS

1.Explain the law of conservation of energy with an example in a gravitational field.

2.Describe the relationship between work, power, and energy.

3.Explain the concept of potential energy in an electric field and how it relates to conservative forces.

4.Discuss the difference between conservative and non-conservative forces.

5.Explain the concept of work done by a force and derive the formula for work done in moving an object over a distance.

PART C - 10 MARK QUESTIONS

1.Explain the law of conservation of energy in detail. Derive the general law of conservation of energy, including the concepts of work done by conservative and non-conservative forces.

2.Discuss the concepts of work, power, and energy in physics. Derive the formulas for work done by a constant force, power, and kinetic energy.

3.Explain the concept of mechanical energy and its conservation in a closed system. Derive the principle of conservation of mechanical energy, including the roles of kinetic and potential energy.

4.Describe the relationship between work and energy using the work-energy theorem. Derive the theorem and discuss its implications for kinetic energy.

Mechanics - Internal and Model Exam Questions - For the Semester Nov 2024

Question Bank - Mechanics - 23BPH3C1 - UNIT II

 

 UNIT - II
PART A - 2 MARK QUESTIONS

1.State the law of conservation of linear momentum.
2.What is the center of mass of a system?
3.Define torque.
4.What is an elastic collision? Give one example.
5.State the principle of conservation of angular momentum.
6.Explain what is meant by "system with variable mass" with an example.
7.What happens to the angular momentum in a proton scattering event with a heavy nucleus?

PART B - 5 MARK QUESTIONS

1.Describe the concept of a center of mass and explain its importance in the motion of a  system of particles.

2.Explain the law of conservation of linear momentum with an example involving internal and external forces.

3.What is angular momentum? Derive the expression for the angular momentum of a rigid body about its center of mass.

4.Write Note on Torque Due to Gravity.

PART C - 10 MARK QUESTIONS

1.Derive the Expressions for Velocities of Two Particles Elastically Colliding with Each Other Along Their Line of Sight, After Impact.

2.Describe the mechanics of proton scattering by a heavy nucleus. Explain how conservation laws apply to the process and discuss the implications for angular momentum and energy transfer.

Question Bank - Mechanics - 23BPH3C1 - UNIT I

 UNIT - I
PART A - 2 MARK QUESTIONS

1. State Newton's First Law of Motion.
2. What is frictional force?
3. Define gravitational potential.

4. State Newton's Law of Gravitation.
4. What is the escape velocity from the Earth's surface?
5. State Kepler's First Law and its significance in planetary motion.

6. State Kepler's Second Law.

8. State Kepler's Third Law.
9. What is the Principle of Equivalence?
10. Explain gravitational redshift.

PART B - 5 MARK QUESTIONS

1.Discuss the types of everyday forces in physics with examples.

2. Derive the equations of motion for a particle moving under a uniform gravitational field.

3. Derive the equations of motion for a particle moving under a uniform gravitational field.

4.Describe the concept of escape velocity and derive the formula for escape velocity from the Earth.

5.State and explain Kepler’s three laws of planetary motion.

6.Explain the Earth-Moon system and discuss its influence on tides and orbital motion.

7.Describe the gravitational potential energy of a satellite in orbit.

8. Describe Einstein’s Theory of Gravitation

9.Explain the phenomenon of the perihelion shift of Mercury.

PART C - 10  MARK QUESTIONS

1. Explain Newton's laws of motion in detail. Discuss their significance and provide examples to illustrate each law.

2. Derive and explain the equation for the gravitational potential.

3.Explain the determination of the universal gravitational constant (G) using Boys' method.

4. Discuss Kepler’s laws of planetary motion in detail. Derive Kepler’s Third Law.

5. Explain Einstein’s Theory of Gravitation with reference to the Principle of Equivalence. Discuss the experimental tests supporting the theory, including gravitational redshift and light bending.

Derivation of Lagrange Equations From D' Alembert's Principle

Derivation of Lagrange Equations From D' Alembert's Principle

Mechanics : Unit III : Part I - Introduction – Significance of Conservation Laws – Law of Conservation of Energy Concepts of Work - Power – Energy – Conservative Forces – Potential Energy and Conservation of Energy in Gravitational and Electric Field - Non-Conservative Forces

CONSERVATION OF ENERGY - CONCEPTS OF WORK AND POWER

Significance of Conservation Laws

In Physics, we come across various conservation laws, as mentioned above, and even though all of them may not be equally exact or accurate, they nevertheless prove helpful in many ways. Thus, for instance,

1. Without going into details of the trajectories or the forces involved in any particular case, they give us a broad and generalised picture of the significant facts that emerge in consequence of the equations of motion.

2. Even in cases where the nature of the forces involved is not clearly known, the conservation laws have been successfully invoked, particularly in the realm of what are called fundamental or elementary particles and have, indeed, helped predict the existence of quite a few more of them viz, conservation of parity.

3. They forewarn us of the impossibility of the occurrence of certain types of phenomena (like, for example, a perpetual motion machine) and thus prevent wastage of time and effort that we might otherwise feel tempted to devote in tackling such problems.

4. They seem to have an intimate relationship with the concept of invariance and we may often use them, with success, in exploring and unraveling new and hitherto not well understood phenomena. For an example, the principle of conservation of linear momentum can be obtained more or less as a direct consequence of Galilean invariance.

[From Mathur's Mechanics, Page 226-227]

INTERNAL FORCES AND MOMENTUM CONSERVATION, TORQUE DUE TO INTERNAL FORCES - LECTURE NOTES

INTERNAL FORCES AND  MOMENTUM CONSERVATION,TORQUE DUE TO INTERNAL FORCES LECTURE NOTES

Here is a p5.js Simulation I've made to illustrate the point that internal forces can not change the linear or angular momentum of a body. Here a fire cracker is thrown up in the air where it explodes. The center of mass of the whole system of the cracker is shown by a blinking point in its trajectory. You can see that the trajectory of the CoM is unaffected by the explosion!. Also, please note that I have applied equal and opposite forces between random parts of the fire cracker to make it explode as shown in the code snippet below :

Also note how I have directly implemented the CoM formula in Coding :

Click Here :   Go Bang!

Types of Everyday Forces in Physics

Credit : https://earthhow.com

In our daily lives, we encounter a variety of forces that influence how objects move and interact. These forces can be divided into several categories:

Gravitational Force:

Description: This is the force of attraction between any two masses. On Earth, it pulls objects toward the center of the planet, giving them weight.

Example: The force that keeps us grounded and causes objects to fall when dropped.

Normal Force:

Description: This is the support force exerted by a surface when an object is placed on it. It acts perpendicular to the surface.

Example: The force that stops a book from falling through a table.

Frictional Force:

Description: Friction is the force that opposes the motion of objects sliding against each other. It acts parallel to the surface of contact.

Types:

Static Friction: Prevents motion when a force is applied.

Kinetic Friction: Opposes motion once an object is moving.

Example: The resistance we feel when trying to push a heavy box on the floor.

Tension Force:

Description: Tension is the pulling force transmitted through a string, rope, or cable when it is pulled tight by forces acting at each end.

Example: The force in a rope holding up a hanging object or a cable used to pull an elevator.

Air Resistance (Drag Force):

Description: This is a type of frictional force that acts against the motion of objects as they travel through air. It increases with speed and surface area.

Example: The force that slows down a parachute when it's deployed.'

Applied Force:

Description: Any force that is applied to an object by a person or another object.

Example: Pushing a shopping cart or pulling a door open.

Spring Force:

Description: The force exerted by a compressed or stretched spring on any object attached to it, described by Hooke's law (F = -kx, where k is the spring constant and x is the displacement).

Example: The force we feel when compressing a spring or a mattress.

Centripetal Force:

Description: The force that acts on an object moving in a circular path, directed toward the center of the circle. This force keeps the object moving in a curve.

Example: The force that keeps a car on a curved road or a planet in orbit.

Electromagnetic Forces:

Description: This is a fundamental force that includes both electric and magnetic forces. It can act between charged particles or magnets.

Example: The force that causes magnets to attract or repel each other, or the force between electrically charged objects.

These everyday forces are important in understanding how objects move and interact in the physical world.