We already know that the total energy (mechanical energy) for a system that is moving with SHM is the sum of the potential and kinetic energies.
When the object is at the extremes of its oscillation (x = A) it has no KE but the PE is at its maximum. When the object is mid way through its oscillation (when x = 0) the KE is at its maximum but there is no PE.
A metal strip is clamped to the edge of the table and has an object of mass 280g attached to the free end. The object is pulled down and released. The object vibrates with SHM with an amplitude of 8.0 cm and a period of 0.16 s.
Calculate the maximum acceleration of the object
Calculate the maximum force
State the position of the object when it has no KE.
Resonance is the tendency in a system to vibrate at its maximum amplitude at a certain frequency. This frequency is known as the system's resonance frequency. When damping is small, the resonance frequency is approximately equal to the natural frequency of the system, which is the frequency of free vibrations.
The natural or fundamental frequency is often written as f 0
Perhaps one of the more common examples of resonance is in musical instruments. For example in guitars it is possible to make other strings vibrate “sympathetically” when another is plucked, either at their fundamental or overtone frequencies.
The Tacoma narrows bridge is often used as an example of resonance, although it is not strictly scientifically accurate to do so. It does how ever give an example of what can happen if an object was to be kept at its resonant frequency for a long time.
All objects have a natural frequency of vibration or resonant frequency. If you force a system - in this case a set of pendulums - to oscillate, you get a maximum transfer of energy, i.e. maximum amplitude imparted.
When the driving frequency equals the resonant frequency of the driven system. The phase relationship between the driver and driven oscillator is also related by their relative frequencies of oscillation.
You also get a very clear illustration of the phase of oscillation relative to the driver. The pendulum at resonance is π/2 behind the driver, all the shorter pendulums are in phase with the driver and all the longer ones are π out of phase.
The amplitude of the forced oscillations depend on the forcing frequency of the driver and reach a maximum when forcing frequency = natural frequency of the driven cones.
If we change the period of oscillation of the driver by moving the mass (increasing L) the hacksaw blade will vibrate at different rates, if we get the driving frequency right the slave will reach resonant frequency and vibrate wildly.
If we move the masses on the blade it will have a similar effect.
Resonance driver applies forces that continually supply energy to oscillator increasing amplitude.
A increases indefinitely unless energy transferred away.
Severe case: A limit reached when oscillator destroys itself. E.g. wine glass shatters when opera singer reaches particular note.
How do we deal with unwanted resonance?
We could use damping, we could also change o of object by changing its mass, if we were to change the stiffness of supports ( moving resonant away from driving ) we could reduce the affect of resonance.
Set up a suspended mass-spring system with a ‘damper’ – a piece of card attached horizontally to the mass to increase the air drag. Alternatively, clamp a springy metal blade (e.g. hacksaw blade) firmly to the bench. Attach a mass to the free end, and add a damping card.