Physics 243 ... Vibrations and Waves ...

Phys 243: Vibrations and Waves Main Ideas Hooke’s Law Energy Conservation applied to a spring Spring as an exemplar of other systems Simple Harmonic Motion Damped versus Free Motion Phys 243: Vibrations and Waves

Phys 243: Vibrations and Waves Hooke’s Law becomes: The function x(t) is the displacement-time function for the spring-mass oscillator Phys 243: Vibrations and Waves

Phys 243: Vibrations and Waves Our intuition tells us that x(t) looks like: Which can be described by the equation... angular frequency amplitude How can we show that this really is so? Phys 243: Vibrations and Waves

Phys 243: Vibrations and Waves Feed into How can we get the left and right hand sides of the equation to match? Phys 243: Vibrations and Waves

“grind out the derivatives and equate both sides” Phys 243: Vibrations and Waves

Velocity and acceleration How can we derive the following? Phys 243: Vibrations and Waves

Velocity and acceleration Phys 243: Vibrations and Waves

The Simple (and not so simple) Pendulum A pendulum can, under appropriate circumstances act as a simple harmonic oscillator The sine component of W provides a restoring torque Phys 243: Vibrations and Waves

Phys 243: Vibrations and Waves This can be expressed... Recall but, if L = length of string then: Phys 243: Vibrations and Waves

Phys 243: Vibrations and Waves Re-arrange to give: Now, notice the similarity with Phys 243: Vibrations and Waves

Phys 243: Vibrations and Waves Important step... If the angle is small - less than about 0.1 radians (7 degrees) then We can re-write to get: Phys 243: Vibrations and Waves

Phys 243: Vibrations and Waves From this we can Find an expression for the period of a simple pendulum... Try it! Phys 243: Vibrations and Waves

General equation for SHM... Whenever the acceleration in the system is directly proportional and opposed to the displacement, SHM will result: Acceleration = - (Displacement) Some constant “opposed to” Phys 243: Vibrations and Waves