Small oscillations problem 
A weight W is suspended from a rigid support by a hard spring
  with stiffness constant K . The spring is enclosed in a- hard plastic
  sleeve, which prevents horizontal motion, but allows vertical
  oscillations (see figure). A simple pendulum of length $l$ with a
  bob of mass m($mg<<W$) is suspended from the weight W and is
  set oscillating in a horizontal line with a small amplitude. After
  some time has passed, the weight W is observed to be oscillating
  up and down with a large amplitude (but not hitting the sleeve). It
  follows that the stiffness constant K must be?
  

I dont even know where to begin. I know the problem is of harmonic motion and small oscillations. What i dont understand is after the weight $W$ starts bouncing up and down with large amplitudes, is that to be understood as the weight $W$ usurping energy from the oscillation of the mass $m$? If so, then how to find the amplitude?
 A: The problem isn't 100% clear, and a full treatment would probably require the use of coupled oscillation techniques that you may or may not have learned yet.  But if this is meant to be solved with "basic" techniques, here's how I would think about it:


*

*For a normal pendulum, the tension in the string is largest when the string is passing through the horizontal (since its angular speed is largest there.)  

*Thus, if the pendulum has a frequency $f$, the tension in the string will oscillate with a frequency $2f$.

*The pendulum string is therefore acting as a driving force with frequency $2f$ on the system consisting of the weight $W$ and the spring $k$.  What's more, this driving force, at this frequency, causes the weight-spring system to oscillate with a "large amplitude".
Take it from there.

To sketch out a more formal technique, we can use Lagrangian mechanics.  Let $Y$ denote the displacement of the upper mass from its equilibrium position, let $\theta$ denote the angle between the string and the vertical, and let $M = W/g$ denote the mass of the upper block.  After some geometry, we can show that the Lagrangian for this system is
$$
\mathcal{L} = \frac{1}{2} (m+M) \dot{Y}^2 - m \ell \sin \theta \dot{\theta} \dot{Y} + \frac{m}{2} \ell^2 \dot{\theta}^2 + m g \ell \cos \theta - \frac{1}{2} k Y^2,
$$ 
and taking the associated Euler-Lagrange equations, we conclude that
\begin{align*}
M \ddot{Y} - m \ell \left(\cos \theta \dot{\theta}^2 + \sin \theta \ddot{\theta} \right) &= - k Y \\
- m \ell \sin \theta \ddot{Y} + m \ell^2 \ddot{\theta} &= - m g \ell \sin \theta. 
\end{align*}
We now can look for a formal power series solution to these equations:
\begin{align}
Y(t) &= \epsilon Y^{(1)}(t) + \epsilon^2 Y^{(2)}(t) + \dots  \\
\theta(t) &= \epsilon \theta^{(1)}(t) + \epsilon^2 \theta^{(2)}(t) + \dots 
\end{align}
We now want to plug these in to the Euler-Lagrange equations and expand them out order by order in $\epsilon$.  At $\mathcal{O}(\epsilon)$, we find that
$$
M \ddot{Y}^{(1)} = -k Y^{(1)}, \qquad m \ell^2 \ddot{\theta}^{(1)} = - m g \ell \theta^{(1)},
$$
from which we conclude that for small oscillations, we have simple harmonic motion in both $\theta$ and $Y$.  Moreover, these oscillations are uncoupled;  at this level of approximation, we would not see the behavior described in the problem.
To see the coupling effects between the two coordinates, we have to expand the Euler-Lagrange equations to $\mathcal{O}(\epsilon^2)$;  if we do this, we get (after some algebra)
\begin{align}
M \ddot{Y}^{(2)} = m \ell \left( \left( \dot{\theta}^{(1)} \right)^2 + \theta^{(1)} \ddot{\theta}^{(1)} \right) -k Y^{(2)}
\end{align}
along with a similar equation for $\theta^{(2)}$.  This latter equation can be rearranged to yield an undamped driven oscillator equation, where the function $\theta^{(1)}$ and its derivatives act as the "driving force" for the second-order perturbations $Y^{(2)}$.  The fact that these oscillations become "large" allows us to say something about the values of $k$ and $M$.
A: I agree with Michael Seifert's 'basic' solution. 
The context of the question usually indicates how much effort is required. A multiple choice question usually requires only simple reasoning based on insight (such as resonance and 2:1 ratio of frequencies), and should take no more than a couple of minutes to solve. 
