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# Oscillations

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## Springs

Consider a spring suspended from a point. Suppose that the length of the unstretched spring is ~l and that its mass is negligible.

_ #~{Hooke's Law} states that the force exerted by a spring is proportional to the amount it is stretched.

If we now attach a particle of mass ~m to the end of the spring, the force exerted by the spring on the particle is ~k&epsilon., where ~k is a constant called the #~{stiffness} of the spring, and &epsilon. is the distance the spring has been extended. This force will be in an upward (positive) direction, and will be counteracted by the force of gravity, ~m~g , in a downward direction.

So total force acting on particle will be

~F _ = _ ~k&epsilon. - ~m~g

The forces will balance exactly when the spring is extended a certain length, &epsilon._0 say. So

&epsilon._0 _ = _ ~m ~g ./ ~k

We will (arbitrarily) set the origin at this point of equilibrium, and let ~y measure the distance of the particle from this point (in an upward direction). So

~y _ = _ &epsilon._0 - &epsilon. _ _ => _ _ ~F _ = _ ~k ( &epsilon._0 - ~y ) - ~m~g _ = _ -~k~y _ _ => _ _ ~m ~y'' + ~k~y _ = _ 0

 ~y'' + &omega.^2~y _ = _ 0 , _ _ where _ &omega. = &sqrt.\${~k ./ ~m}

## Simple Harmonic Motion

The equation of motion

~y'' + &omega.^2~y _ = _ 0

has a solution

~y _ = _ ~C cos &omega.~t + ~D sin &omega.~t

(see Calculus notes on Orinary Differential Equations: Second Order Homogenous Equations .)

This can be written:

~y _ = _ ~A sin ( &omega.~t + &phi. ) , _ _ where _ ~A = &sqrt.\${ ~C&powtwo. + ~D&powtwo. } , _ and _ &phi. = sin^{-1} ( ~C ./ ~A ) ~A is called the #~{amplitude} , _ &omega. is known as the #~{angular frequency} , _ and &phi. is called the ~#{phase}. The graph for ~y is shown (right) with some of these quantities illustrated.

Note that this is (of course) a periodic function, which repeats every 2&pi./&omega. time units. This constitutes one #~{cycle} of the oscillation. This means that there are &omega./2&pi. cycles per time unit, this is known as the #~{frequency} of the oscillation. If the frequency is denoted by ~f, then

~f _ #:= _ &omega./2&pi. , _ _ _ ~y _ = _ ~A sin ( 2&pi.~f~t + &phi. )

There is a lot of information on the web about Simple Harmonic Motion. See for example, HyperPhysics , hosted by the Department of Physics and Astronomy at Georgia State University.

## Damped Oscillations

Suppose that in addition to the force towards the point of equilibrium, which is proportional to the distance from the point, there is a frictional force proportional to the velocity of the particle (and in the opposite direction).

~F _ = _ -~k~y - ~q~y'

Giving an equation of motion:

~y'' + 2&beta. ~y' + &gamma.^2 ~y _ = _ 0 , _ _ _ where _ &beta. = ~q ./ 2~m > and _ &gamma. = &sqrt.\${ ~k ./ ~m } If _ &gamma.&powtwo. > &beta.&powtwo. _ ( 4~m~k&powtwo. > ~q ) _ then this has a solution :

~y _ = _ ~e^{-&beta.~t} ~A sin ( ~t &sqrt.\${ &gamma.&powtwo. - &beta.&powtwo. } + &phi. )

where _ A _ and _ &phi. _ are arbitrary constants. _ Put &omega. = &sqrt.\${ &gamma.&powtwo. - &beta.&powtwo. } _ and the solution becomes:

~y _ = _ ~e^{-&beta.~t} ~A sin ( &omega.~t + &phi. )

So this has the same (initial) amplitude and angular frequency as the non-damped oscillation, but the amplitude decreases exponentially (see diagram right).

_

_ If _ &gamma.&powtwo. < &beta.&powtwo. _ ( 4~m~k&powtwo. < ~q ) _ then the solution is:

~y _ = _ ~A exp \{ ~t ( - &beta. + &sqrt.\${ &beta.&powtwo. - &gamma.&powtwo. } ) \} + ~B exp \{ ~t ( - &beta. - &sqrt.\${ &beta.&powtwo. - &gamma.&powtwo. } ) \}

Note that as _ &beta. > &sqrt.\${ &beta.&powtwo. - &gamma.&powtwo. } _ both exponents will be negative, and therefore the expression is the sum of two exponentially decreasing components.

There can be at most one turning point in such a function. Three examples are shown (left) where the arbitrary constants are the only difference in each case (same ~q and ~k etc.)

_ If _ &gamma.&powtwo. = &beta.&powtwo. _ ( 4~m~k&powtwo. = ~q ) _ then the solution is:

~y _ = _ ~A exp \{ - &beta. ~t \} + ~B ~t exp \{ - &beta. ~t \}

Motion is said to be ~#{critically damped} or ~#{dead beat}.

There can be at most one turning point. Examples with different arbitrary constants are shown in the graph on the right.

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