8.01x - Lect 10 - Hooke's Law, Springs, Pendulums, Simple Harmonic Motion

The video begins with an introduction to springs, pendulums, and simple harmonic oscillators, emphasizing their importance in the study of physics. The lecturer explains the concept of a spring's relaxed length and how extending it creates a restoring force that pushes the spring back to equilibrium. This force is proportional to the displacement and is described by Hooke's Law, where the force is equal to the spring constant (k) times the displacement (x). The spring constant can be measured using gravity and a hanging mass, allowing for the calculation of the spring's force based on the mass and displacement. The video also touches on the limitations of Hooke's Law and the potential for permanent deformation of the spring if stretched too far.

The lecturer discusses alternative methods for measuring the spring constant, focusing on a dynamic approach that involves oscillating a mass attached to the spring on a frictionless surface. The period of oscillation is introduced as a key factor, revealing that it is dependent on the mass and the spring constant. The relationship between the period, mass, and spring constant is derived, and it is shown that the period is independent of the amplitude of oscillation. The lecturer then demonstrates this concept with a practical experiment involving a spray paint can suspended by springs and oscillating vertically, visually illustrating the sinusoidal motion of the mass as a function of time.

The lecturer presents a detailed derivation of the differential equation governing the motion of a simple harmonic oscillator. Starting with Newton's Second Law, the讲师 simplifies the equation by considering a massless spring and no external forces other than the spring force. The resulting second-order differential equation is then solved by assuming a sinusoidal function for the position of the mass as a function of time. The讲师 substitutes this trial solution into the differential equation and shows that the angular frequency (ω) of the oscillation is related to the spring constant (k) and the mass (m). The period of oscillation is derived and shown to be independent of the amplitude and phase angle, highlighting the fundamental characteristics of simple harmonic motion.

The video explores how initial conditions, such as the initial position and velocity of the mass, determine the amplitude (A) and phase angle (φ) in the equation of motion for a simple harmonic oscillator. The lecturer provides a specific example where a mass is released from equilibrium with a given velocity, and the resulting motion is analyzed. By applying the initial conditions to the sinusoidal function, the讲师 calculates the amplitude and phase angle, demonstrating how they are influenced by the initial state of the system. The讲师 also emphasizes the importance of understanding initial conditions when solving physical problems involving oscillatory motion.

The lecturer conducts an experiment to verify the nonintuitive concept that the period of oscillation for a simple harmonic oscillator is independent of its amplitude. Using an air track, the讲师 oscillates an object with different amplitudes and measures the periods for each case. Despite varying the amplitude, the measured periods remain consistent, confirming the theoretical prediction. The讲师 also discusses the impact of the object's mass on the period, explaining that the period is inversely proportional to the square root of the mass. This is demonstrated by combining two objects of equal mass and showing that the period for the combined system is the same as for a single object with half the mass.

The lecturer introduces the concept of small-angle approximations and their application to the motion of a pendulum. By assuming that the angle of oscillation (θ) is small, the讲师 simplifies the forces acting on the pendulum and derives the differential equations of motion. Using these equations, the讲师 shows that the motion of the pendulum can be approximated as simple harmonic oscillation, with a period that depends only on the length of the pendulum (l) and the acceleration due to gravity (g), and is independent of the mass of the pendulum bob. The lecturer emphasizes the importance of these approximations in making the mathematical analysis tractable and in understanding the fundamental behavior of the pendulum.

The lecturer compares the periods of oscillation for a spring and a pendulum, highlighting the differences in how mass and length affect each system. For a spring, the period is dependent on the mass attached to it, while for a pendulum, the period is independent of the mass and only related to the length of the string and the acceleration due to gravity. The讲师 also discusses the implications of these differences, such as the ability to measure mass in a weightless environment using a spring, and the lack of pendulum motion in the absence of gravity. The video concludes with a demonstration of a large pendulum with a period of approximately 4.5 seconds, reinforcing the principles of simple harmonic motion.

The lecturer physically demonstrates the period of a pendulum by oscillating it at different angles and measuring the time for ten oscillations. Despite varying the angle of release, the measured period remains consistent, confirming the theoretical prediction that the period is independent of the amplitude of oscillation. The lecturer also humorously involves the audience in the counting process, creating an interactive and engaging learning experience. The video concludes with a dramatic demonstration of the讲师 swinging the pendulum while sitting on it, showing that the period remains unchanged even with the讲师's added mass and motion, further illustrating the principles discussed.