State what is meant by simple harmonic motion.
A tube, sealed at one end, has a circular cross-sectional area $A$ of $4.9 \times 10^{-4}\,\text{m}^2$. Sand is added so that the combined mass $M$ of the tube and its contents is $70\,\text{g}$. The tube floats upright in a liquid, as shown in Fig. 4.1. The liquid has a density $\rho$ of $0.79\,\text{g cm}^{-3}$. Using the liquid pressure acting on the base of the tube, show that the depth $h$ of the tube base below the liquid surface is $18\,\text{cm}$. Show your working.
The tube in (b) is moved vertically and then let go. The way the distance $h$ changes with time $t$ is shown in Fig. 4.2. The system oscillates with simple harmonic motion of angular frequency $\omega$ given by $\omega^{2} = \frac{\rho A g}{M}$, where $g$ is the acceleration of free fall. Use the data from (b) to find the time $t_{1}$.
Use the data from (b) to find the time $t_{3}$.
Use the data from (b) to find the time $t_1$.
Use the data from (b) to find the time $t_3$.
Find the loss in total energy of the oscillating system between $t = 0$ and $t = t_4$.