Month: July 2014

047: Coupled Pendulum

You should have noticed that the two pendulums take turns to grow and shrink in amplitude. This is because they take turns to be the driver.

The driver always leads the driven by a quarter of a cycle. As driver does positive work on the driven, it passes its energy to the driven. When the driver’s amplitude decays completely, the other pendulum takes over to be the driver, and the cycle continues.

046 Lenz’s Law Jumping Ring

When the solenoid was energized, it produces a magnetic field. The ring thus experienced an increase in magnetic flux linkage. This results in an induced emf around the ring. (Faraday’s Law: emf=dΦ/dt)

The magnetic flux captured by all three rings are exactly the same. (It’s is the same solenoid, so B is the same. The rings have the same circular area, so A is the same. And Φ=BA) Clearly, all three rings experienced the same change in magnetic flux linkage. This leads to the conclusion that the induced emf in all three rings are exactly the same!

So why do they jump to different heights?

Firstly, they jump because the induced current (caused by the induced emf) produces its own magnetic field. The interaction of the magnetic field of the ring and the magnetic field of the solenoid leads to a mutually repulsive force. (This outcome is predicted by Lenz’s Law). Note that the effect of the induced emf is felt only when it results in an induced current.

The solid ring presents the largest cross sectional area, and thus smallest resistance to the induced emf. (R=ρL/A) The induced current is the largest, hence it jumped the highest. The holes in the second ring confines the current to the edges of the ring. With a larger resistance, the induced current is less (despite the same induced emf), hence it jumped less high. As for the broken ring, there was practically zero induced current (despite the same induced emf) thanks to the infinite resistance presented by the air gaps.

045 Natural Frequency

Many people do not realize that the amplitude of oscillation does not affect the period of oscillation. Some believe that with larger amplitude, the oscillator moves at higher speed and thus should have a smaller period. Others believe in the contrary, that with larger amplitude, the oscillator must travel a longer distance and hence should have a longer period.

editsimplependulum

It turns out that for a simple harmonic motion, the amplitude has no effect on the period of oscillation. For a pendulum, it is the length of the pendulum* (and the gravitational acceleration) that determine the period of oscillation.( ebb8ac66b94598631b8fd035f3b0c6c5)

shm8

For a (vertical) spring-mass system, it is the stiffness of the spring and the mass that decides the period of oscillation. (Period-mass-spring) This period does not change with the amplitude of oscillation.

* In the video, the longer pendulum has 4 times the length, but only double the period compared to the shorter one. This hints that period is proportional to the square root of pendulum length.