Lecture 2 | Modern Physics: Classical Mechanics (Stanford)

Transcript

this program is brought to you by Stanford University please visit us at stanford.edu review first of all the idea of initial conditions and why we need both position and velocity let's just take the case of one dimensional motion as an example and let me begin with a false law of physics I think I called it last time Aristotle's law Aristotle thou... Read More

Summary

This video discusses the concept of initial conditions in physics and the importance of both position and velocity. It starts with the false law of physics proposed by Aristotle, which states that continuous force is required to keep an object in motion. However, it is proven that without force, an object in motion will continue moving due to inertia. The video then goes on to explain how the knowledge of initial position can determine velocity, and how differentiation of force with respect to time can determine acceleration. It also introduces the concept of jerk, which is the derivative of acceleration with respect to time. The video then transitions to Newton's laws of physics and explains how knowing both position and velocity are necessary to determine acceleration. The conservation of energy and momentum are also discussed, with derivations and explanations provided. Finally, the video concludes by introducing the principle of least action and discussing the mathematics behind minimizing functions and trajectories.

Questions & Answers

Q: What was Aristotle's law of physics and why was it proven to be false?

Aristotle's law stated that continuous force was necessary to keep an object in motion. However, it was proven to be false because an object in motion will continue moving without an external force due to inertia.

Q: What is the equation for force according to Aristotle's law?

According to Aristotle's law, force is equal to the product of mass and velocity. This equation assumes that no force means no velocity, and velocity at rest means no force.

Q: What is the correct equation for force according to Newton's laws of physics?

According to Newton's laws of physics, force is equal to mass times acceleration. This equation is formulated as F = ma, where F is force, m is mass, and a is acceleration.

Q: How can the acceleration of an object be determined using force and velocity?

Differentiating the equation for force with respect to time yields acceleration. This is done by taking the time derivative of F = ma, which gives the relationship between force and acceleration.

Q: What is jerk and how can it be determined using derivatives of force and acceleration?

Jerk is the derivative of acceleration with respect to time. It can be calculated by differentiating the force equation with respect to time twice, giving the relationship between force, acceleration, and jerk.

Q: What are the assumptions for following a system and determining its motion according to Newton's laws?

In order to determine the motion of a system using Newton's laws, the initial conditions (position and velocity) of each particle in the system must be known, as well as the force law that defines the relationship between position and force.

Q: How is conservation of energy related to Newton's laws of physics?

Conservation of energy is a more general principle than Newton's laws of physics. While Newton's laws focus on determining the motion of objects, conservation of energy applies to various physical phenomena. In the case of Newton's laws, if the forces acting on a system are conservative, then the total energy of the system is conserved.

Q: What is the equation for kinetic energy and how is it related to the mass and velocity of an object?

Kinetic energy is defined as one-half the mass of an object times the velocity squared. It represents the energy associated with the motion of an object, and the equation illustrates that kinetic energy increases with both mass and velocity.

Q: How can the time derivative of total energy be calculated using Newton's laws?

The time derivative of total energy can be calculated by summing the time derivatives of kinetic and potential energy. By applying Newton's laws and simplifying the expression, it can be shown that the total time derivative of energy is zero, indicating the conservation of energy.

Q: What are the conditions for determining stationary points or minima of functions?

In order to determine stationary points or minima of functions, the derivative of the function with respect to each independent variable should be equal to zero. This condition implies that if the function is varied a little bit, it doesn't change significantly in the first order.

Q: How can the principle of stationary action be applied to determining minima of functions?

The principle of stationary action can be applied to finding minima of functions by considering the variation of the function with respect to a whole trajectory or set of variables. The goal is to find the points where the function achieves a minimum, which implies that any small variation in the trajectory increases the function.

Takeaways

In classical physics, initial conditions play a crucial role in determining the motion of objects. The false law proposed by Aristotle, which stated that continuous force was needed to keep an object in motion, was proven to be incorrect. Newton's laws of physics and the conservation of energy and momentum provide a more accurate description of motion. Newton's laws allow us to calculate the motion of objects by knowing the initial positions and velocities, as well as the force law that relates position and force. Conservation of energy ensures that the total energy of a system remains constant if the forces acting on it are conservative. The principle of stationary action, which involves finding minima or stationary points of functions or trajectories, provides a powerful framework for understanding motion and determining the most probable paths of objects.