Lecture 5: Equilibrium and Oscillatins
Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if E < U This represents a frbidden regin fr the particle
With this knwledge, we can quickly determine whether a particle s mtin is bunded r nt: E U(X) E E 3 E 4 E 5 0 5 3 4 6 7 8
Equilibrium If several frces are acting n a system, it is ften imprtant t understand the situatin in which all f them cancel t zer this is called equilibrium If all the frces are cnservative, equilibria ccur when F = U = s we can determine the equilibrium psitins frm the ptential alne Fr eample, cnsider the ne-dimensinal ptential belw: 0 U(X) = equilibrium pint
There are three distinct varieties f equilibria. The tw that ccur mst ften are:. Stable: in this case, when an bject is displaced frm the equilibrium psitin, it will tend t mve back twards equilibrium. Unstable: In this case, an bject displaced frm equilibrium will accelerate away frm the equilibrium psitin T determine whether and equilibrium is stable r unstable, we Taylr-epand the ptential near the equilibrium pint: du d U U ( ) = U ( ) + ( ) + ( ) 3 3 d U + ( ) 3 6 +
T simplify this further, nte that:. We can chse t set the rigin f the crdinate system at the equilibrium pint. We are free t set the value f the ptential t zer there 3. Since we re epanding abut an equilibrium pint, the first derivative must be zer. With all this, we have: 3 d U 3 d U U ( ) = 3 + + 6 Stability is determined by the sign f If it s psitive, equilibrium is at a lcal minimum f U Stable If it s negative, equilibrium is at a lcal maimum f U Unstable d U
What abut the case where d U = 0? Need t lk at leading nn-zer term in the epansin 3 d U 0 : Unstable 3 4 d U > 0 : Stable 4 4 d U < 0 : Unstable 4 If all terms in the epansin are 0, the equilibrium is neutral
Oscillatins Yu prbably remember studying the mtin f a mass attached t a spring that beys Hke s law: F = k where is the displacement frm the equilibrium length f the spring T understand why yu spent s much time n this, cnsider a general frce near a stable equilibrium pint Can be Taylr-epanded as: ( ) df d F = + + + F F 0 S, near the equilibrium pint, df F ( )
This is f the same frm as Hke s Law, with k df = S almst all small scillatins behave like a mass n a spring Thus it s imprtant t understand this type f mtin We start frm Newtn s secnd law: F = m = k k + ω = 0; ω m This is a straightfrward differential equatin a sine r csine functin fr (t) will wrk. General frm f result is: ( ) = Asin ( ω t δ ) t Amplitude Angular frequency Phase A and δ depend n initial cnditins; ω des nt
Energy f Oscillatr We can directly find the energy f the scillatr: ( E = T + U = m + k ) = + + + ( ( ) ( ) ) = ka cs ωt + δ + sin ωt + δ = ka ( mω cs( ) sin ( ) ) A ωt δ ka ωt δ Nte that this desn t depend n time Cnservatin f energy really wrks! The frequency and perid f scillatin are: ω k ν = = τ = = π π π m ν m k
Oscillatins in Multiple Dimensins The previus discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where: F = kr N big deal we can cnsider ne cmpnent at a time: F = m = k F = m = k S we just need t slve the same equatin as befre, nce fr each dimensin Slutins are: ( ) = sin ( ω + δ ) ( ) = sin ( ω + δ ) t A t t A t Nte that frequency is the same in each directin
The Tw-Dimensinal Case We can find the path taken by a particle underging twdimensinal scillatin Easiest if we start with a little trick rewrite as: ( ω δ δ δ ) ( ω δ ) ( δ δ ) ( ω δ ) ( δ δ ) = A sin t + + = A sin t + cs + cs t + sin = A cs + sin A A ( δ δ ) ( δ δ ) ( δ δ ) sin ( δ δ ) ( δ δ ) ( δ δ ) ( δ δ ) A sin ( δ δ ) A A cs = A A A A A cs + A cs = A A sin
Finally, we have: A A A cs δ δ + A = A A sin δ δ δ This is the equatin fr an ellipse ( ) ( ) Nte that the shape f the ellipse depends nly n the relative amplitudes and difference in phase between the mtin in each directin δ = δ δ = 0 30 δ δ = 90 δ δ 0 = δ δ = 80
Lissajus Curves The tw-dimensinal mtin becmes even mre interesting if the frequencies are different in each dimensin As lng as the rati f frequencies is a ratinal number, the mtin is still peridic: 7 ω = ω 5 δ δ = 90 Nte that there are 7 maima in and 5 maima in
If the rati f the tw frequencies is nt ratinal, the mtin wn t be peridic Eventually, every pint between in the rectangle defined by A and A will be reached: