\begin{problem}
A frictionless block-spring system oscillates with amplitude $A$.  If
the mass of the block is doubled without changing the amplitude,
\Part{a} does the total energy change?
\Part{b} does the frequency of oscillation change?
\begin{center}
\begin{asy}
import Mechanics;
real u = 1cm;

Surface table = Surface(pFrom=(0,0), pTo=(2u,0));
Surface wall = Surface(pFrom=(0,0.5u), pTo=(0,0), thickness=0);

real blockside = 0.3u;
pair blockpos = (1.5u,blockside/2);
path block = shift(blockpos-blockside*(.5,.5))*scale(blockside)*unitsquare;
Spring s = Spring(pFrom=(0,blockpos.y), pTo=blockpos, width=0.6blockside);

table.draw();
wall.draw();
s.draw();
filldraw(block);
\end{asy}
\end{center}
\end{problem} % Derived from Ch. 12, Question 7.

\begin{solution}
\Part{a}
The total energy is equal to the spring potential energy at maximum
extension (when the kinetic energy is zero), so $E_\text{T} =
\frac{1}{2}kA^2$.  Neither the spring constant, nor the amplitude
changed, so \ans{the total energy is unaffected}.

\Part{b}
The heavier mass will move more slowly under the influnce of the same
spring, so the frequency is smaller for the bigger mass.  Quantitatively
\begin{equation}
  f = \frac{1}{2\pi}\omega = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\;,
\end{equation}
so doubling the mass reduces the frequency to $\ans{f' = \frac{f}{\sqrt{2}}}$.
\end{solution}
