2021 Live Review 4 | AP Physics C: Mechanics | Everything You Need to Know about Nonlinear Springs

The video begins with an introduction to the topic of non-linear springs, contrasting them with the more familiar Hooke's law for elastic springs. The instructor outlines the session's logistics, including the recording of the session for future review and the availability of materials at a provided URL. The main focus is on calculating work done to compress or stretch a non-linear spring, which requires the use of calculus. The instructor emphasizes that non-linear springs are not simple harmonic oscillators, leading to deviations in stored energy and oscillation period. Examples of free response and multiple-choice questions from past exams are promised to illustrate the concepts.

In this segment, the instructor delves into the specifics of calculating work done on a non-linear spring, using a 1986 exam question as an example. The spring's force-displacement relationship is cubic, and the goal is to find the potential energy stored and the maximum speed that a mass attached to the spring could achieve. The process involves using calculus to integrate the force function and equating potential and kinetic energy to solve for velocity. The instructor also discusses how to handle changes in amplitude for non-linear springs and the implications for oscillation periods.

The instructor continues the discussion on non-linear springs, focusing on how non-linearity affects the amplitude and period of oscillation. Using a multiple-choice question from a past exam, the concept of energy conservation is applied to determine the displacement at which potential and kinetic energies are equal. The segment also touches on the acceleration of an object attached to a non-linear spring, explaining why the period of oscillation will decrease due to the increased acceleration as the object moves back towards equilibrium.

This part of the video addresses the concept of linearization when dealing with non-linear springs. The instructor uses a 2006 exam question to demonstrate how to transform a non-linear force-displacement relationship into a linear one by squaring the displacement values. The process involves graphing force against the square of displacement, finding the slope of the best-fit line, and using this to calculate work and speed. The importance of units and significant figures in calculations is also highlighted.

The focus shifts to applying the work-energy theorem to a scenario involving a cart and a non-linear spring. The instructor explains how to calculate the work done by the cart to compress the spring and how this work translates into kinetic energy. The problem involves determining the speed of the cart before it compresses the spring and the amount of work done during compression, emphasizing the conservation of energy throughout the process.

In this section, the video script describes a collision problem involving two blocks and a non-linear spring. The instructor guides the viewer through drawing a free body diagram, applying the conservation of momentum to find the velocity of the blocks after the collision, and using the work-energy principle to calculate the maximum compression distance of the spring. The complexity of the problem is managed by breaking it down into smaller, manageable parts.

The final part of the script deals with the dynamics following a collision involving a non-linear spring. The instructor explains how to determine the net force on one of the blocks when the spring is at maximum compression and the magnitude of the force on each block. The segment concludes with a discussion on whether the system will exhibit simple harmonic motion, which is not the case for non-linear springs due to the cubic force-displacement relationship.

The video concludes with a summary of key takeaways, emphasizing the importance of understanding work calculations for non-linear springs, the conservation of energy and momentum, and the distinction between linear and non-linear springs in terms of harmonic motion. The instructor encourages students to practice with past exam questions and provides a link for feedback. The session ends with a reminder of upcoming review topics and a motivational note to help students aim for a top score.