Classical Mechanics - Newtonian Dynamics & Lagrangian Mechanics Flashcards

Question 1

What is Newton's First Law of Motion?

Accepted Answer

An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This is also known as the Law of Inertia.

Question 2

State Newton's Second Law of Motion and its formula.

Accepted Answer

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The formula is $\vec{F} = m\vec{a}$, where $\vec{F}$ is the net force, $m$ is mass, and $\vec{a}$ is acceleration.

Question 3

Explain Newton's Third Law of Motion.

Accepted Answer

For every action, there is an equal and opposite reaction. If object A exerts a force $\vec{F}_{AB}$ on object B, then object B simultaneously exerts a force $\vec{F}_{BA} = -\vec{F}_{AB}$ on object A.

Question 4

Define linear momentum and provide its formula.

Accepted Answer

Linear momentum, $\vec{p}$, is the product of an object's mass and its velocity. Its formula is $\vec{p} = m\vec{v}$, where $m$ is mass and $\vec{v}$ is velocity. It is a vector quantity.

Question 5

What is impulse, and how is it related to momentum?

Accepted Answer

Impulse, $\vec{J}$, is the change in momentum of an object. It is also defined as the integral of force over the time interval during which the force acts: $\vec{J} = \int_{t_1}^{t_2} \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$.

Question 6

Define work done by a force and provide its general formula.

Accepted Answer

Work, $W$, is done when a force causes a displacement of an object. For a constant force, $W = \vec{F} \cdot \Delta \vec{r}$. For a variable force, $W = \int \vec{F} \cdot d\vec{r}$. The SI unit of work is the Joule (J).

Question 7

What is kinetic energy, and what is its formula?

Accepted Answer

Kinetic energy, $E_k$ or $T$, is the energy an object possesses due to its motion. Its formula is $E_k = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is speed. The SI unit is the Joule (J).

Question 8

Define potential energy and give examples.

Accepted Answer

Potential energy, $U$ or $V$, is the energy stored in an object due to its position or configuration. Examples include gravitational potential energy ($U_g = mgh$) and elastic potential energy of a spring ($U_s = \frac{1}{2}kx^2$).

Question 9

State the Work-Energy Theorem.

Accepted Answer

The net work done on an object is equal to the change in its kinetic energy: $W_{net} = \Delta E_k = E_{k,f} - E_{k,i}$.

Question 10

When is mechanical energy conserved, and what is its formula?

Accepted Answer

Mechanical energy ($E = E_k + U$) is conserved if only conservative forces (like gravity or spring force) do work. If non-conservative forces (like friction) are absent or do no work, then $E_i = E_f$ or $\Delta E = 0$.

Question 11

State the Law of Conservation of Linear Momentum.

Accepted Answer

In an isolated system (where no external net force acts), the total linear momentum remains constant: $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$. This is a direct consequence of Newton's Third Law.

Question 12

When is angular momentum conserved, and what is its formula?

Accepted Answer

Angular momentum, $\vec{L}$, is conserved if the net external torque acting on a system is zero. For a particle, $\vec{L} = \vec{r} 	imes \vec{p}$. For a rigid body, $\vec{L} = I\vec{\omega}$ (for rotation about a principal axis). If $\sum \vec{	au}_{ext} = 0$, then $\vec{L}_{initial} = \vec{L}_{final}$.

Question 13

Define power and provide its formulas.

Accepted Answer

Power, $P$, is the rate at which work is done or energy is transferred. It can be expressed as $P = \frac{dW}{dt}$ or $P = \vec{F} \cdot \vec{v}$. The SI unit of power is the Watt (W).

Question 14

What is a central force?

Accepted Answer

A central force is a force that always points towards or away from a fixed point (the center of force) and whose magnitude depends only on the distance from that point. Mathematically, $\vec{F}(\vec{r}) = f(r)\hat{r}$.

Question 15

State Newton's Law of Universal Gravitation.

Accepted Answer

Every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is $F = G\frac{m_1m_2}{r^2}$, where $G$ is the gravitational constant ($6.674 	imes 10^{-11} 	ext{ N m}^2/	ext{kg}^2$).

Question 16

What are generalized coordinates in Lagrangian mechanics?

Accepted Answer

Generalized coordinates, $q_i$, are a set of independent coordinates that completely describe the configuration of a system. They are not necessarily Cartesian coordinates and can be chosen to simplify the problem, often reducing the number of coordinates needed by incorporating constraints.

Question 17

Define degrees of freedom (DOF) in a mechanical system.

Accepted Answer

The degrees of freedom of a mechanical system is the minimum number of independent generalized coordinates required to completely specify its configuration. For $N$ particles with $k$ constraints, the DOF is $3N - k$ (in 3D).

Question 18

What is the Lagrangian of a system, and what is its formula?

Accepted Answer

The Lagrangian, $L$, is a function that summarizes the dynamics of a system in terms of its kinetic and potential energies. It is defined as $L = T - V$, where $T$ is the total kinetic energy and $V$ is the total potential energy of the system.

Question 19

Explain Hamilton's Principle (Principle of Least Action).

Accepted Answer

Hamilton's Principle states that the path taken by a system between two states at specified times $t_1$ and $t_2$ is the one for which the action integral, $S = \int_{t_1}^{t_2} L dt$, is stationary (often a minimum). This principle leads directly to the Euler-Lagrange equations.

Question 20

State the Euler-Lagrange equation for a generalized coordinate $q_i$.

Accepted Answer

The Euler-Lagrange equation is the fundamental equation of motion in Lagrangian mechanics, derived from Hamilton's Principle. For each generalized coordinate $q_i$, it is given by $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$, where $\dot{q}_i = \frac{dq_i}{dt}$ is the generalized velocity.

Question 21

Define canonical (or generalized) momentum.

Accepted Answer

The canonical momentum, $p_i$, conjugate to a generalized coordinate $q_i$, is defined as $p_i = \frac{\partial L}{\partial \dot{q}_i}$. It is a conserved quantity if its conjugate coordinate $q_i$ is cyclic.

Question 22

What is a cyclic (or ignorable) coordinate in Lagrangian mechanics?

Accepted Answer

A generalized coordinate $q_i$ is cyclic (or ignorable) if the Lagrangian $L$ does not explicitly depend on $q_i$, i.e., $\frac{\partial L}{\partial q_i} = 0$. For such a coordinate, the Euler-Lagrange equation simplifies to $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) = 0$, implying that its conjugate canonical momentum $p_i = \frac{\partial L}{\partial \dot{q}_i}$ is conserved.

Question 23

Briefly explain Noether's Theorem in the context of classical mechanics.

Accepted Answer

Noether's Theorem states that for every continuous symmetry of the Lagrangian (or action), there exists a corresponding conserved quantity. For example, translational symmetry implies conservation of linear momentum, rotational symmetry implies conservation of angular momentum, and time-translation symmetry implies conservation of energy.

Question 24

List key advantages of Lagrangian Mechanics over Newtonian Mechanics.

Accepted Answer

1. It uses scalar quantities (energy) instead of vector forces, simplifying calculations. 2. Constraints can be easily incorporated using generalized coordinates, often eliminating the need for constraint forces. 3. It provides a systematic approach to derive equations of motion. 4. It naturally leads to conservation laws via cyclic coordinates and Noether's Theorem. 5. It is more easily generalized to advanced theories like quantum mechanics and field theory.

Classical Mechanics - Newtonian Dynamics & Lagrangian Mechanics

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