Physics 9702 · AS & A Level · Simple harmonic oscillations

Simple harmonic oscillations — practice question

A tube with a circular cross-section, closed at one end, has area $A$ and holds sand. The combined mass of the tube and the sand is $M$. As shown in Fig. 3.1, the tube floats vertically in a liquid of density $\rho$. It is pressed down a small distance into the liquid and then let go.
(a(i))[1]

State the two forces acting on the tube immediately after release.

(a(ii))[2]

State and explain the direction of the resultant force on the tube immediately after release.

(b)[2]

The acceleration $a$ of the tube is given by the expression $a = -\left(\frac{A\rho g}{M}\right)x$, where $x$ is the vertical displacement of the tube from its equilibrium position. Use the expression to explain why the tube performs simple harmonic oscillations in the liquid.

(c(i))[2]

A tube with cross-sectional area $A$ equal to $4.5\,\text{cm}^2$ and total mass $M$ equal to $0.17\,\text{kg}$ oscillates with period $1.3\,\text{s}$. Determine the angular frequency $\omega$ of the oscillations.

(c(ii))[3]

Use your answer in (i) and the expression in (b) to find the density $\rho$ of the liquid that the tube is floating in.

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