Mastering Classical Mechanics and Fluid Dynamics

10 Questions

Which of the following correctly describes Newton's first law?

An object will remain at rest or in uniform motion in a straight line unless acted upon by a force

What is the relationship between force and acceleration according to Newton's second law?

Force is directly proportional to acceleration.

What is the principle behind Newton's third law?

For every action, there is an equal and opposite reaction.

What is the work-energy theorem?

Work is equal to the change in kinetic energy.

What is the centripetal force required for uniform circular motion?

Directed towards the center of the circle

What is the Hamiltonian function used for in Hamiltonian mechanics?

To represent dynamics.

What is the Euler-Lagrange equation used for in Lagrangian mechanics?

To determine the entire trajectory of an object.

Which equation becomes the Navier-Stokes equation when viscosity is incorporated?

The Euler equation.

What is the Kepler problem used to solve?

The shape of orbits.

What is the principle behind harmonic motion?

The force upon the body is proportional to the displacement from the equilibrium point.

Study Notes

Newton's laws of motion describe the relationship between the motion of an object and the forces acting on it.
These laws were first stated by Isaac Newton in his book "Philosophiæ Naturalis Principia Mathematica" in 1687.
Newton's laws are often stated in terms of point or particle masses.
The mathematical description of motion is based on the idea of specifying positions using numerical coordinates.
Velocity and acceleration are vector quantities, and force is a vector quantity as well.
Newton's first law expresses the principle of inertia.
Newton's second law states that the time derivative of the momentum is the force.
The forces acting on a body add as vectors, and the total force on a body depends upon both the magnitudes and the directions of the individual forces.
Newton's third law relates to the conservation of momentum.
The conservation of momentum remains true even in cases where Newton's statement does not, for instance when force fields as well as material bodies carry momentum.1. Newton's laws of motion explain the behavior of objects at rest or in motion.

The first law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by a force.
The second law explains how the acceleration of an object depends on the force applied to it and its mass.
The third law states that for every action, there is an equal and opposite reaction.
Energy can be classified into kinetic and potential energy.
The work-energy theorem states that when a force acts upon a body while that body moves along the line of the force, the force does work upon the body, and the amount of work done is equal to the change in the body's kinetic energy.
Uniformly accelerated motion follows a constant rate of acceleration, and the speed attained during free fall is proportional to the elapsed time.
Uniform circular motion is when the force on an object changes the direction of its motion but not its speed.
The centripetal force required to sustain this acceleration is directed towards the center of the circle.
Harmonic motion occurs when the force upon the body is proportional to the displacement from the equilibrium point and directed to the equilibrium point.1. Newtonian physics treats matter as neither created nor destroyed.
A pendulum has a stable equilibrium in the vertical position.
Neglecting air resistance and friction, the force upon the pendulum is gravity.
A harmonic oscillator can be damped or driven by an applied force.
Rigid body motion is often understood by separating it into movement of the body's center of mass and movement around the center of mass.
Newton's law of universal gravitation states that any body attracts any other body along the straight line connecting them.
The Kepler problem can be solved to find the shape of orbits.
The three-body problem has no exact solution in closed form.
Newton's laws allow the possibility of chaos.
Fluid dynamics can be described by a velocity field.- Fluid flow can change velocity due to changes in velocity field or location.

The acceleration of an infinitesimal portion of fluid has two terms.
The Euler equation becomes a Navier-Stokes equation when viscosity is incorporated.
The Euler and Navier-Stokes equations may exhibit unphysical behavior known as a "noncollision singularity."
Classical mechanics can be formulated in multiple ways, including Lagrangian and Hamiltonian.
Lagrangian mechanics considers entire trajectories and uses the Euler-Lagrange equation.
Hamiltonian mechanics uses a Hamiltonian function and Hamilton's equations to represent dynamics.
Hamilton-Jacobi equation is another formulation of classical mechanics that uses a function of positions and time.
Bodies move over time in such a way that their trajectories are perpendicular to the surfaces of constant S.
The Hamiltonian is incorporated into the Hamilton-Jacobi equation.