A pendulum is oscillating with small amplitude when its string is slowly started to be pulled thus decreasing the pendulum length, but at a vanishingly small rate. By what factor does the amplitude change when the string becomes half of its original length?

Here's a hint: consider the energy in the system when the pendulum is at max displacement. The PE is a function of the string length and the displacement angle. Can you see why the shorter the string the larger the displacement angle has to be, in order to conserve energy? Post back with what you think the answer is.