Pendulum with Oscillating Fulcrum with Newton's Laws?

Question

I was given an honors project to solve for the equations of motions of a pendulum with an oscillating fulcrum. I (somewhat) understand the procedure on how to solve it with lagrangian mechanics and the Euler-Lagrange equation, but I am stuck at a question.

How do we know this cannot be solved with Newtonian mechanics? I tried to look up the answer on Google and even tried to solve for it myself but I obtained an equation with theta, acceleration in the x and y direction so Im not sure what to make of it.

Does anyone know? Or are there any resources to understand the precise limitations of Newtonian mechanics?

I am a physics undergrad who just finished electromagnetism and linear algebra, thank you!

Answer 1

With the Newton's laws.

We have two masses $M$ and $m$ linked by a holonomic constraint

$$ p_c = (x + R\sin\theta,-R\cos\theta)' $$

so according to Newton's laws

$$ \cases{ -k x + T\sin\theta = M\ddot x\\ T(-\sin\theta,\cos\theta)'+ m g(0,-1)'=m\ddot p_c } $$

Here $T$ is the tension along the fulcrum. Eliminating $T$ we arrive to the movement equations

$$ \left\{ \begin{array}{l} -k x-M \ddot x=m \left(R \ddot\theta \cos \theta-R \dot\theta^2 \sin \theta+\ddot x\right) \\ \cot \theta \left(k x+M \ddot x\right)-g m=m \left(R \ddot\theta \sin \theta+R \dot\theta^2 \cos \theta\right) \end{array} \right. $$

or

$$ \left\{ \begin{array}{rcl} \ddot x&=&\frac{g m \cos\theta-k x \csc \theta+m R \dot\theta^2}{m \sin \theta+M \csc \theta} \\ \ddot \theta&=&-\frac{\csc \theta \left(g (m+M)-k x \cot \theta+m R \dot\theta^2 \cos \theta\right)}{R\left(m+M \csc ^2\theta\right)} \\ \end{array} \right. $$

Follows a plot showing the mass $m$ path under the initial conditions: $R=1,m=1,M=10,k=1000,g=10,x(0)=0.2,\dot x(0)=0,\theta(0)=\frac{\pi}{2},\dot\theta(0)=0,t_{max}=10$.

Attached a MATHEMATICA script to build the shown plot.

Clear[M, x, theta, T, R, m, M]
solT = Solve[M x''[t] == - k x[t] + T Sin[theta[t]], T][[1]]
pe = {x[t] + R Sin[theta[t]], -R Cos[theta[t]]}
equsmov = Thread[T {-Sin[theta[t]], Cos[theta[t]]} + m g {0, -1} == m D[pe, t, t]] /. solT
parms = {R -> 1, m -> 1, M -> 10, k -> 1000, g -> 10};
tmax = 10;
equsmov0 = equsmov /. parms
cinits = {x[0] == 0.2, x'[0] == 0, theta[0] == Pi/2, theta'[0] == 0};
solx = NDSolve[Join[equsmov0, cinits], {x, theta}, {t, 0, tmax}];
ParametricPlot[Evaluate[pe /. parms /. solx], {t, 0, tmax}, PlotRange -> All, PlotStyle -> {Thick, Blue}, AspectRatio -> 1/2]