Lecture 10: Hooke's Law - Springs - Simple Harmonic Motion - Pendulum - Small Angle Approximation
recorded by: Massachusetts Institute of Technology, MIT
published: Oct. 10, 2008, recorded: October 1999, views: 14084
released under terms of: Creative Commons Attribution No Derivatives (CC-BY-ND)
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1. Restoring Force of a Spring:
The restoring force of a spring, described by Hooke's Law (F=-kx) is introduced. Professor Lewin discusses how to measure the spring constant, k, and he gives a brief demonstration.
2. Dynamic Equations of a Displaced Spring:
A differential equation is derived for a spring in the absence of damping forces. Using springs, spray paint and a moving target, a sketch of x(t) is created, suggesting a sine or cosine dependence of x on time. The angular frequency (and therefore the period) is shown to depend only on k and m (so you can measure k dynamically). The amplitude and phase depend on initial conditions (the displacement and velocity at t=0). An example is worked out to demonstrate this.
3. Measuring the Period of a Spring System:
The period of oscillation is measured for a mass on a spring system on an air track (to minimize friction). A measurement is made of 10 periods to reduce the relative error. Professor Lewin demonstrates that the period is independent of the amplitude. The mass is doubled, the new period is predicted and then empirically confirmed.
4. Dynamic Equations of a Pendulum:
A pair of differential equations is derived for a mass, m, suspended on a near massless string of length L. The small angle approximation is quantitatively justified and applied to arrive at a simple differential equation analogous to that for a spring. The period of oscillation is shown to be proportional to the square root of L/g; it is independent of m.
5. Comparing the Spring and Pendulum Periods:
Intuitive insights are presented as to why the period of an oscillating spring depends on the mass (attached to the spring) and the spring constant. Yet the period of a pendulum is independent of the mass hanging from the string. These insights are reinforced with several experiments with a very long pendulum. The uncertainties in the measurements are taken into account. To demonstrate that the period is independent of the mass of the bob, Professor Lewin places himself at the end of the 5 meter long cable and measures the period.
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