Podcast
Questions and Answers
In classical dynamics, when is a system considered 'discrete'?
In classical dynamics, when is a system considered 'discrete'?
- When the particles of a system are distinctly separated from each other. (correct)
- When the system's particles interact through continuous fields.
- When the system consists of a single, indivisible particle.
- When the system's energy is quantized.
What does the 'degree of freedom' of a system of particles refer to?
What does the 'degree of freedom' of a system of particles refer to?
- The number of independent coordinates needed to fully specify the position of a system. (correct)
- The amount of external force required to move the system.
- The number of particles in the system.
- The total kinetic energy of the system.
For a system of N particles moving freely in space, how many independent coordinates or degrees of freedom does it have?
For a system of N particles moving freely in space, how many independent coordinates or degrees of freedom does it have?
- 2N
- N^2
- 3N (correct)
- N
Given the position vectors $r_i$ and constant masses $m_i$ for a system of N particles, what is the formula for the position vector R of the center of mass?
Given the position vectors $r_i$ and constant masses $m_i$ for a system of N particles, what is the formula for the position vector R of the center of mass?
If $M$ represents the total mass of a system of particles, which of the following expressions correctly relates $M$ to the individual particle masses $m_i$?
If $M$ represents the total mass of a system of particles, which of the following expressions correctly relates $M$ to the individual particle masses $m_i$?
How is the total momentum $P$ of a system of particles with masses $m_i$ and velocities $v_i$ expressed?
How is the total momentum $P$ of a system of particles with masses $m_i$ and velocities $v_i$ expressed?
Given that $R$ is the position vector of the center of mass and $V = \frac{dR}{dt}$ is its velocity, how is the total momentum $P$ of the system related to the total mass $M$?
Given that $R$ is the position vector of the center of mass and $V = \frac{dR}{dt}$ is its velocity, how is the total momentum $P$ of the system related to the total mass $M$?
What is the formula for the total kinetic energy ($T$) of a system of N particles, given their masses $m_i$ and velocities $v_i$?
What is the formula for the total kinetic energy ($T$) of a system of N particles, given their masses $m_i$ and velocities $v_i$?
According to Newton's second law applied to the $i^{th}$ particle in a system, which equation correctly represents the relationship between the forces acting on the particle ($F_{ji}$ internal forces, $F_i^{(e)}$ external forces) and its momentum ($P_i$)?
According to Newton's second law applied to the $i^{th}$ particle in a system, which equation correctly represents the relationship between the forces acting on the particle ($F_{ji}$ internal forces, $F_i^{(e)}$ external forces) and its momentum ($P_i$)?
Assuming a system obeys Newton's third law of motion, what is the implication for the sum of internal forces ($F_{ji}$) acting between particles in the system?
Assuming a system obeys Newton's third law of motion, what is the implication for the sum of internal forces ($F_{ji}$) acting between particles in the system?
If the total external force acting on a system is zero, what can be said about the motion of the center of mass?
If the total external force acting on a system is zero, what can be said about the motion of the center of mass?
What does the conservation theorem imply for the total linear momentum of a system of particles?
What does the conservation theorem imply for the total linear momentum of a system of particles?
How is the total angular momentum $L$ of a system, comprised of individual angular momenta $L_i$, defined?
How is the total angular momentum $L$ of a system, comprised of individual angular momenta $L_i$, defined?
If $r_i$ is the position vector and $p_i$ is the linear momentum of the $i^{th}$ particle, how is its angular momentum ($L_i$) defined?
If $r_i$ is the position vector and $p_i$ is the linear momentum of the $i^{th}$ particle, how is its angular momentum ($L_i$) defined?
Considering the equation $\frac{dL}{dt} = \sum_{i=1}^{N} (r_i \times F_i^{(e)})$, what does this equation represent?
Considering the equation $\frac{dL}{dt} = \sum_{i=1}^{N} (r_i \times F_i^{(e)})$, what does this equation represent?
In the analysis of angular momentum, one term involves $\sum (r_i \times F_{ji})$. If the system obeys Newton's third law, what is the value of this term?
In the analysis of angular momentum, one term involves $\sum (r_i \times F_{ji})$. If the system obeys Newton's third law, what is the value of this term?
What does $N^{(e)}$ represent in the context of the total angular momentum of a system?
What does $N^{(e)}$ represent in the context of the total angular momentum of a system?
If $\frac{dL}{dt} = N^{(e)}$, and $N^{(e)} = 0$, what does this indicate about the total angular momentum $L$ of the system?
If $\frac{dL}{dt} = N^{(e)}$, and $N^{(e)} = 0$, what does this indicate about the total angular momentum $L$ of the system?
How does the motion of the center of mass of a system respond to exclusively internal forces?
How does the motion of the center of mass of a system respond to exclusively internal forces?
Two particles exert forces on each other. If the vector from particle j to particle i is $r_{ij}$ and the force exerted by j on i is $F_{ji}$, what does $r_{ij} \times F_{ji} = 0$ imply?
Two particles exert forces on each other. If the vector from particle j to particle i is $r_{ij}$ and the force exerted by j on i is $F_{ji}$, what does $r_{ij} \times F_{ji} = 0$ imply?
Flashcards
Dynamics
Dynamics
Branch of mathematics dealing with the effect of forces on the motion of an object.
Mechanics
Mechanics
Branch of mathematics dealing with the motion of particles, objects, or bodies with forces.
Degree of Freedom
Degree of Freedom
Number of coordinates required to specify the position of a system of particles.
Position Vectors
Position Vectors
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Center of Mass
Center of Mass
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Total Momentum
Total Momentum
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Kinetic Energy
Kinetic Energy
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Center of Mass Motion
Center of Mass Motion
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Newton's Second Law
Newton's Second Law
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Total External Force
Total External Force
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Angular Momentum
Angular Momentum
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Study Notes
- Elementary classical dynamics studies the dynamics of objects viewed as particles or point masses.
- Practical situations involve studying the dynamics of collections of particle systems.
- If particles of a system are separated (distinct), the system is discrete.
- Otherwise, the system is called a continuous system.
Dynamics
- Dynamics is the branch of mathematics dealing with the effect of forces on the motion of an object.
Mechanics
- Mechanics is the branch of mathematics dealing with the motion of particles, objects, or bodies with forces.
Degree of Freedom of a System
- The number of coordinates required to specify the position of a system of particles is the degree of freedom of that system.
- A system of N particles moving freely in space has 3N independent coordinates or degrees of freedom.
- r₁, r₂, ..., rN denote the position vectors of a system of N particles of constant masses.
- The vector to the center of mass is R
- R = (Σ mᵢrᵢ) / M, where M = Σ mᵢ
Total Mass
- M is the total mass of the system of particles.
Total Momentum
- The total momentum P of the system is given by P = Σ mᵢvᵢ
- vᵢ = drᵢ/dt.
- P = d/dt (Σ mᵢrᵢ) = d/dt (MR) = MV
- V = dR/dt, the velocity of the center of mass.
- P = MV
Kinetic Energy
- Total kinetic energy (T) of a system of N particles is given by T = (1/2) Σ mᵢ|vᵢ|²
Motion of the Center of Mass
- Considers the motion of the mass center of mass due to forces acting on the particles.
- Considers the influence outside the system and internal forces on some particle i due to other particles.
- By Newton's second law, the equation of motion for the ith particle is pᵢ = Σ Fⱼᵢ + Fᵢ⁽ᵉ⁾
- Fᵢ⁽ᵉ⁾ is the resultant external force on the ith particle.
- Fⱼᵢ is the internal force on the ith particle due to the jth particle (Fᵢᵢ = 0).
- Assume system obeys Newton's third law: forces two particles exert on each other are equal and opposite, acting along the line joining them.
- Σ Fⱼᵢ = 0
Equation of Motion becomes
- Σ pᵢ = d²/dt² (Σ mᵢrᵢ) = Σ Fᵢ⁽ᵉ⁾
- Md²R/dt² = Σ Fᵢ⁽ᵉ⁾ = F⁽ᵉ⁾
- F⁽ᵉ⁾ is the total external force.
- The center of mass moves as if the total external force F⁽ᵉ⁾ were acting on the entire mass.
- Internal forces have no effect on the motion of the center of mass.
- If F⁽ᵉ⁾ = 0, either at rest or moving object, then d/dt (MdR/dt) = d/dt (MV) = 0.
Total Linear Momentum Conversation
- Total linear momentum is conserved. This is conservation of linear momentum for a system of particles.
Total Angular Momentum
- Angular momentum is represented as Lᵢ of the ith particle.
- Lᵢ = rᵢ x pᵢ
- Total angular momentum of the system is L = Σ Lᵢ = Σ rᵢ x pᵢ
- Σ rᵢ x pᵢ = r₁ x p₁ + r₂ x p₂ + ... + rN x pN
- dL/dt = d/dt (Σ rᵢ x pᵢ) = Σ (rᵢ x pᵢ)
- pᵢ = Σ Fⱼᵢ + Fᵢ⁽ᵉ⁾
- L = Σ rᵢ x (Σ Fⱼᵢ + Fᵢ⁽ᵉ⁾)
- L = Σ (rᵢ x Fᵢ⁽ᵉ⁾) + Σ (rᵢ x Fⱼᵢ)
- Consider the last term Σ (rᵢ x Fⱼᵢ) = (rᵢ x Fⱼᵢ) + (rⱼ x Fᵢⱼ) = (rᵢ - rⱼ) x Fⱼᵢ
Reaction
- By the equality of action and reaction, if rᵢ - rⱼ = rⱼᵢ is the vector from j to i, the law of action and reaction gives Fᵢⱼ x Fⱼ,ᵢ = 0.
Particles
- Since Fⱼᵢ is along the line between the two particles, Σ (rⱼ x Fⱼᵢ) = 0
- L = dL/dt = Σ rᵢ x Fᵢ⁽ᵉ⁾ = N⁽ᵉ⁾
- Can be represented as N⁽ᵉ⁾
- N⁽ᵉ⁾ is the applied (i.e., external) torque on the system.
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