## 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: 14319

released under terms of: Creative Commons Attribution No Derivatives (CC-BY-ND)

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# Description

**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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## Reviews and comments:

kaianna, October 28, 2008 at 4:44 a.m.:well explained, but much more than what i wanted to know

moni, January 6, 2009 at 6:05 p.m.:wo0o0o0ow nice lecture

plzz put some questions after the explanation so we can improve more and the solution also to check our answers >>> thxxx alot

JakotaSP, March 23, 2009 at 12:10 p.m.:Walter Lewin is awesome aha

I really enjoyed his lecture.

The experiments really helped me understand the the spring and pendulum equations.

Hope to see more like this.

Cheers

Dylan Quercia, December 8, 2009 at 5:06 a.m.:Awesome. Why do I not have a professor like this?

senthee, February 11, 2010 at 8:45 a.m.:He is really a physicist who explains the things with demo...vow! Great Prof. Ya, we too dont have like him to learn physics. Wonderful lec!

S.A.hadi, September 24, 2010 at 10:30 a.m.:very fantastic and interesting lecture

vidisha, January 9, 2011 at 7:57 p.m.:fantastic lecture...had hoped for some numericals and special cases in between.anyway,the explanation has helped a load!!

Mark, March 14, 2011 at 5:27 a.m.:Awesome, awesome lecture! My physics professor is never this clear. She skips over these fundamental relationships! I'm so glad I found this video.

Narasimha Murthy,K, July 20, 2011 at 2:04 p.m.:...wonderfull-breathh0lding pl include oscillations in air drag,in viscous medium and for larger anle swings

Atiq, October 11, 2011 at 6:19 p.m.:It is very helpful for me.

I relly want to see other lectures.

Tamirat Y., March 28, 2013 at 8:40 a.m.:I really enjoyed with your lecture and i got much knowledge from you. and he becomes my benchmark to teach my student 10q

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