## Lecture 13: Potential Energy - Energy Considerations to Derive Simple Harmonic Motion

recorded by: Massachusetts Institute of Technology, MIT

published: Oct. 10, 2008, recorded: October 1999, views: 3508

released under terms of: Creative Commons Attribution Non-Commercial Share Alike (CC-BY-NC-SA)

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# Description

**1. Gravitational Potential Energy:**

A review is given of the spatial dependence of the gravitational potential energy both close to the Earth's surface and at large distances from Earth. The gravitational force pulls objects in the direction of decreasing potential energy.

**2. Calculating U(x) from F(x) and Vice Versa:**

The potential energy, U(x), of a spring system is derived and sketched as a function of displacement x. The force can be derived if the function U(x) is known: F(x)=-dU(x)/dx.

**3. Equilibrium Points:**

The minima and maxima of potential energy are positions where the net force is zero. At the stable equilibrium points the 2nd derivative of U(x) is positive, at the unstable equilibrium points the 2nd derivative is negative.

**4. Parabolic Potential Energy Well ==> SHO:**

Using the parabolic shape of the potential energy for a spring, and the conservation of mechanical energy, it is shown that the mass on the spring oscillates as a simple harmonic oscillator (SHO).

**5. Circular Potential Energy Well ==> SHO:**

Using a circular potential energy well and the conservation of mechanical energy, it is shown that for SMALL ANGLES, the oscillations are simple harmonic. A circular track with very large radius is used to demonstrate this.

**6. Sliding on a Circular Track ==> SHO:**

The known radius of a circular air track is used to predict the period of oscillation of a sliding object (small angles!), and a measurement is made to confirm this. The process is repeated for a ball bearing rolling in another circular track. The period of oscillation can now not be predicted in a similar way as was possible in the case of the air track. Why? ==> No, it has nothing to do with friction!

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## Reviews and comments:

Arihant Sogani, November 10, 2008 at 3:04 p.m.:could have done with a few more demonstrations

Li, March 14, 2010 at 4:46 p.m.:i could not figure out the problem of the second air track demonstration, anybody has any idea?

Charles, March 1, 2014 at 4:27 a.m.:To someone who commented in 2010, Li, asking about the second demonstration... The ball slowed down, therefore the timing was more than the predicted. That is because the motion of the ball. It is rolling, therefore it 'spends' some energy into stoping the motion going up and rolling motion.

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