8. Dynamics of Multiple-Body System and Law of

YaleCourses
22 Sept 200872:10
EducationalLearning
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TLDRIn this lecture, Professor Ramamurti Shankar explores the dynamics of multiple bodies obeying Newton's laws, with a focus on the concept of the center of mass. He explains how the center of mass simplifies the analysis of systems with internal forces by responding only to external forces. Through examples such as the Solar System, a carriage with a horse, and a boat with a jumper, Shankar illustrates the application of the center of mass in various scenarios. He also delves into the conservation of momentum, particularly in collisions, distinguishing between totally elastic and totally inelastic collisions, and provides the equations for determining final velocities in each case.

Takeaways
  • ๐Ÿ“š The dynamics of multiple bodies can be studied using Newton's laws, focusing on the motion of two bodies in one dimension as a starting point.
  • ๐ŸŒŸ When analyzing systems of bodies, it's crucial to differentiate between internal and external forces, where internal forces cancel out according to Newton's Third Law.
  • ๐Ÿ” The center of mass coordinate is a weighted average that simplifies the analysis of systems with multiple bodies, responding only to external forces.
  • ๐ŸŽฏ The center of mass can be used to determine the effective motion of a system, such as in the case of an airplane with passengers fighting, where the center of mass continues to fall due to gravity despite the internal forces.
  • ๐Ÿ“Š The concept of momentum is introduced as the product of mass and velocity, and it's conserved when no external forces act on a system, leading to the Law of Conservation of Momentum.
  • ๐Ÿ”„ The conservation of momentum applies to various scenarios, including collisions, where the total momentum before and after the collision remains unchanged.
  • ๐Ÿš€ The rocket equation can be derived using the conservation of momentum, showing how the velocity of a rocket changes as it expels mass in the form of exhaust gases.
  • ๐Ÿ›ถ Problems involving the transfer of momentum, like a person jumping from a boat to the shore, require considering the change in momentum for both the person and the boat.
  • ๐Ÿคบ Inelastic and elastic collisions have different outcomes for the conservation of kinetic energy, with the latter conserving energy and the former resulting in some energy conversion to other forms like heat.
  • ๐ŸŽฑ The Ballistic Pendulum is an example of a practical application where conservation of momentum is used to determine the speed of a bullet by observing the motion of a wooden block it embeds into.
  • ๐Ÿ“ When solving problems involving collisions, it's essential to apply the correct principles, such as using conservation of momentum in all cases, but only using conservation of energy in elastic collisions.
Q & A
  • What is the main focus of the lecture?

    -The main focus of the lecture is the study of the dynamics of more than one body obeying Newton's laws, specifically the concept and application of the center of mass in various scenarios.

  • How does the notation with dots represent derivatives in the lecture?

    -In the lecture, one dot represents the first derivative, two dots represent the second derivative, and three dots represent the third derivative.

  • What is the significance of the center of mass in the context of Newton's laws?

    -The center of mass is significant because it responds only to the total external force and does not care about internal forces, making it a useful concept for simplifying the analysis of systems with multiple interacting bodies.

  • How does the center of mass behave when there are no external forces acting on a system?

    -When there are no external forces, the center of mass will either remain at rest or maintain a constant velocity if it was already in motion, as the internal forces between the bodies cancel each other out.

  • What is the relationship between the external forces and the motion of the center of mass?

    -The motion of the center of mass is directly influenced by the external forces acting on the system. If there are external forces, the center of mass will accelerate according to Newton's laws, while internal forces between the bodies cancel out and do not affect the center of mass's motion.

  • How does the concept of momentum relate to the conservation of momentum in a collision?

    -The concept of momentum relates to the conservation of momentum in a collision because the total momentum of the system before the collision must equal the total momentum after the collision, regardless of how the individual momenta of the objects involved change.

  • What is the Ballistic Pendulum method and how is it used to measure the speed of a bullet?

    -The Ballistic Pendulum method is a technique used to measure the speed of a bullet by firing it into a block of wood that is suspended from the ceiling, causing the block to swing like a pendulum. The speed of the bullet is determined by analyzing the motion of the pendulum and applying the conservation of momentum principle.

  • Why is it incorrect to use the Law of Conservation of Energy during a totally inelastic collision?

    -It is incorrect to use the Law of Conservation of Energy during a totally inelastic collision because some of the energy is converted into other forms, such as heat or sound, and is not conserved in the form of kinetic energy for the system as a whole.

  • What are the two extreme cases for collisions and how are they defined?

    -The two extreme cases for collisions are totally inelastic and totally elastic collisions. A totally inelastic collision is when the two bodies stick together after the collision and move as a single unit, while a totally elastic collision is when the kinetic energy of the system is conserved throughout the collision.

  • How does the rocket equation derived in the lecture relate to the conservation of momentum?

    -The rocket equation derived in the lecture demonstrates the conservation of momentum in the context of a rocket expelling exhaust gases. The equation shows how the change in the rocket's velocity is related to the mass of the rocket, the mass of the expelled gases, and the exhaust velocity, in accordance with the conservation of momentum principle.

  • What is the significance of the boat and person jumping example in understanding the conservation of momentum?

    -The boat and person jumping example illustrates the conservation of momentum in a practical scenario. When the person jumps from the boat, both the person and the boat experience a change in momentum in opposite directions to maintain the total momentum of the system constant, as no external forces are acting on the system.

Outlines
00:00
๐Ÿ“š Introduction to Multi-Body Dynamics

The paragraph introduces the concept of studying the dynamics of more than one body, using Newton's laws. It explains the difference between the dynamics of the Solar System, where the Sun and a planet are considered, and the new scenario where multiple bodies are involved, all obeying Newton's laws. The focus is on the simplest case of two bodies moving in one dimension, with an emphasis on the importance of understanding the forces acting on each body and how they relate to the center of mass.

05:02
๐Ÿ”„ Newton's Third Law and Center of Mass

This section delves into Newton's Third Law, which states that forces between two bodies are equal and opposite, leading to the cancellation of internal forces when considering the motion of a system as a whole. The introduction of the center of mass as a concept is highlighted, explaining its role as a weighted average position of the system's mass. The center of mass is described as responding only to external forces, with internal forces canceling out, and its significance in simplifying the analysis of systems with multiple bodies is emphasized.

10:03
๐Ÿš€ Center of Mass in Motion and at Rest

The discussion continues with scenarios where the center of mass is either in motion or at rest, emphasizing the conservation of momentum. It is explained that if the center of mass was initially at rest, it will remain at rest, and if it was moving, it will continue to move at a constant velocity in the absence of external forces. The concept is illustrated with examples such as a person jumping from a boat and the resulting motion of the boat, and the movement of the center of mass in a carriage with a horse.

15:03
๐ŸŒŒ Application of Center of Mass in Astronomy

The application of the center of mass concept in astronomy is discussed, specifically in the context of planetary motion. The example of the Earth orbiting the Sun is used to illustrate that both bodies actually revolve around their mutual center of gravity. The paragraph clarifies that the traditional view of the Sun being stationary while the Earth orbits is not accurate according to the principles of center of mass and momentum conservation. The correct model involves both bodies moving in their respective orbits around the center of mass.

20:06
๐Ÿšค Rocket Propulsion and the Conservation of Momentum

This section introduces the concept of rocket propulsion and how it relates to the conservation of momentum. The process of a rocket emitting gases at a high exhaust velocity is described, and the resulting change in the rocket's velocity is explained. The rocket equation is derived, showing the relationship between the change in velocity of the rocket, the mass of the exhaust gases, and the rocket's initial and final mass. The importance of considering the direction and magnitude of velocities relative to the rocket and the ground is emphasized.

25:07
๐ŸฅŠ Collision Dynamics and Energy Conservation

The dynamics of collisions are explored, with a focus on the conservation of momentum and the conditions under which energy conservation applies. The differences between totally inelastic and totally elastic collisions are explained, with formulas provided for calculating the final velocities in each case. The example of a ballistic pendulum is used to illustrate how to apply these principles, emphasizing the importance of using the correct conservation laws based on the nature of the collision and the system being considered.

Mindmap
Keywords
๐Ÿ’กNewton's Laws
Newton's Laws are fundamental principles that describe the motion of objects. In the video, these laws are applied to analyze the dynamics of multiple bodies, such as the interaction between two masses and their response to external forces. The first law, for example, states that an object will remain at rest or in uniform motion unless acted upon by an external force, which is illustrated when discussing the center of mass and its motion in the absence of external forces.
๐Ÿ’กCenter of Mass
The center of mass is a concept used in physics to describe the average location of the mass of an object or a system of objects. It is a weighted average of the positions of all the particles in the system, with each particle's position weighted by its mass. In the video, the center of mass is used to simplify the analysis of systems with multiple bodies, as it responds only to external forces and not to internal forces. The example of a rod with uniform mass distribution is used to illustrate the calculation of the center of mass.
๐Ÿ’กMomentum
Momentum is a measure of the quantity of motion of an object or a system of objects, defined as the product of mass and velocity. In the video, the concept of momentum is introduced as a key quantity that is conserved in the absence of external forces. The conservation of momentum is a fundamental principle used to analyze collisions and interactions between bodies. The example of two people on ice pushing against each other demonstrates the conservation of momentum, as their momenta before and after interaction must sum to the same total.
๐Ÿ’กConservation of Momentum
The law of conservation of momentum states that the total momentum of a closed system of objects remains constant if no external forces act on it. In the video, this principle is used to analyze scenarios where bodies interact with each other, such as in collisions or when objects are connected by a spring. The example of a bullet embedded in a block illustrates the application of this law to find the velocity of the bullet before impact.
๐Ÿ’กCollision
A collision is an event in which two or more objects exert forces on each other for a very short period of time. In the video, collisions are discussed in the context of conservation of momentum and energy, with different types of collisions such as elastic and inelastic being considered. The outcomes of collisions depend on the conservation laws applied, and these principles are used to calculate final velocities and energies of the colliding bodies.
๐Ÿ’กProjectile Motion
Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity alone. In the video, the concept is used to describe the motion of the center of mass in a complex system when external forces are present, such as the motion of a center of mass in a projectile-like trajectory under the influence of gravity.
๐Ÿ’กExternal Forces
External forces are forces that are applied to a system from the outside. In the context of the video, external forces are those that act on the system of bodies from sources not included within the system itself, such as gravity or the force from a spring. The response of the center of mass to external forces is a key focus, with the video explaining that the center of mass will only accelerate if there are external forces acting on the system.
๐Ÿ’กInelastic Collision
An inelastic collision is a type of collision where the colliding objects stick together after the impact, resulting in a loss of kinetic energy due to the deformation of the objects or the conversion of energy into other forms like heat or sound. In the video, the concept is used to discuss the outcomes of collisions where momentum is conserved but kinetic energy is not.
๐Ÿ’กElastic Collision
An elastic collision is a type of collision where the colliding objects retain their original shape after the impact and the total kinetic energy of the system is conserved. In the video, this concept is used to analyze collisions where both momentum and kinetic energy are conserved, allowing for the calculation of the final velocities of the objects involved.
๐Ÿ’กBallistic Pendulum
A ballistic pendulum is an experimental setup used to measure the velocity of a projectile, such as a bullet, by observing the pendulum-like motion of a block into which the projectile embeds itself. In the video, the ballistic pendulum is described as a method to determine the speed of a bullet by analyzing the motion of the bullet-block combination after the collision.
Highlights

Professor Ramamurti Shankar introduces the study of dynamics involving more than one body, emphasizing the importance of understanding Newton's laws in this context.

The lecture clarifies the concept of the center of mass, explaining its relevance in physics and how it simplifies the analysis of systems with multiple bodies.

The cancellation of internal forces between two bodies is discussed, highlighting that the center of mass only responds to external forces.

An example of two masses connected by a spring is used to illustrate the transmission of force and the concept of external forces.

The center of mass is defined as a weighted average location of multiple bodies, with a detailed explanation of how it is calculated.

The concept of momentum is introduced, explaining its significance in the context of the center of mass and the conservation of momentum.

A detailed explanation of how to calculate the center of mass for a uniform rod is provided, including the integral calculation and the use of symmetry.

The lecture discusses the application of the center of mass concept in complex scenarios, such as the motion of an airplane during a fight on board.

The conservation of momentum is used to explain the dynamics of a system when external forces are absent, emphasizing the importance of this principle in physics.

A method for calculating the center of mass of a triangle is presented, incorporating the principles of calculus and the importance of symmetry in the analysis.

The lecture provides a clear explanation of the Law of Conservation of Momentum, including its application in scenarios involving collisions.

The concept of the rocket equation is introduced, explaining how the conservation of momentum principle applies to the operation of rockets.

A detailed discussion on the dynamics of a system involving a carriage, horse, and the center of mass is provided, illustrating the application of the center of mass concept in practical scenarios.

The lecture concludes with a discussion on the Ballistic Pendulum, explaining how the conservation of momentum can be used to determine the speed of a bullet.

The importance of distinguishing between the conservation of momentum and the conservation of mechanical energy is emphasized, particularly in scenarios involving inelastic collisions.

The lecture provides a comprehensive overview of the dynamics of multiple bodies, center of mass, momentum, and energy conservation, offering valuable insights for students of physics.

Transcripts
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