Съдържание

Условие Section I (2 т.)Section II Решение

APhO 2009, Експеримент — Задача 2. Oscillation of Water-Filled Vessel

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от APhO 2009

Experimental competition, 28 April 2009

Условие

The student is required to perform non-destructive measurements in order to determine the thickness $t$ of an aluminium vessel whose cavity is completely filled with water. The aluminium vessel is composed of a cylinder and two end plates. The cylinder is of length $L$ and outer radius $R$ . The total length of the vessel is $h$ . The thickness of both end plates is 0.60 cm (see Figure 1). You can neglect the error of this thickness. In this problem, please use gramme and centimetre as units for mass and length, respectively.

Cross-section of the water-filled vessel with axes X and Y through centre O: outer radius R, cylinder length L, total length h, end plates of thickness 0.60 cm at top and bottom, wall thickness t cm, cavity labelled water. — Figure 1

Bifilar suspension: a mass M (ellipse with centre of mass CM) hung from a ceiling by two vertical strings of length ℓ attached at distance a on either side of the vertical axis; gravity arrow g and a rotation arrow about the axis. — Figure 2

Figure 2 shows the so-called bifilar suspension of mass $M$ . The two strings are each of equal length $\ell$ . The period $T$ of a small-amplitude oscillation of $M$ is

T = 2\pi\sqrt{\frac{\ell}{g}\cdot\frac{I}{Ma^2}} \qquad (i)

where $I$ is the effective moment of inertia about the vertical axis through the centre of mass of $M$ and $g$ is the acceleration due to gravity at Bangkok $\left(g = 978\ \mathrm{cm\ s^{-2}}\right)$ .

This experiment consists of two parts. Section I concerns a derivation of formulae and Section II concerns the actual experimentation.

Apparatus

Each student is provided with:

a water-filled vessel
a stand
a stop watch
a ruler
a nylon string
a protractor
masking tapes
a knife (not shown in the figure below)

Photograph of the apparatus with numbered labels: 1 the aluminium vessel, 2 the stand, 3 the stop watch, 4 the ruler, 5 the nylon string, 6 the protractor, 7 the masking tape.

Section I (2 т.)

The student is to derive expressions in terms of $R, L, t$ and the density $\rho$ of aluminium of the following quantities, [see Figure 1]

Thin rod of length L (double-headed arrow labelled L) and mass m, with a vertical rotation axis through its middle and a rotation arrow. — Figure 3

Thin cylinder (ring) of mass m with inner radius R1 and outer radius R2, centred at O, with a vertical Y axis carrying a rotation arrow and a horizontal X axis. — Figure 3

i) mass $(m_1)$ of the cylindrical body of the vessel,

ii) mass $(m_2)$ of each end plate,

iii) mass $(m_3)$ of water in the whole cavity,

iv) the total mass $(M)$ of the water-filled vessel, and

v) the effective moment of inertia, $I_{\mathrm{y}}$ , about the Y-axis, of this water-filled vessel (see Figure 1), assuming that the water is ideal fluid.

Then perform measurements of $R, h, L$ . By substituting the values, derive expressions in terms of $t$ for the quantities i)-v) above. The aluminium density $\rho = 2.70\ \mathrm{g/cm^3}$ and the water density is 1.00 g/cm $^3$ .

Hint:

I = m\frac{L^2}{12}

Thin rod of length $L$

I_{\mathrm{y}} = \frac{1}{2}m\left(R_2^2+R_1^2\right),

I_{\mathrm{x}} = \frac{1}{4}m\left(R_2^2+R_1^2\right)

Thin cylinder of inner radius $R_1$ and outer radius $R_2$

Section II

The vessel hung upright by two strings from the top plate of a stand, oscillating about its vertical axis of symmetry (dotted axis with rotation arrow). — Figure 4

The vessel hung horizontally by two strings attached at its two ends from the top plate of a stand, oscillating about the vertical central axis perpendicular to its length (dotted axis with rotation arrow). — Figure 5

a) Angular oscillation about the axis of symmetry

For one fixed value of $\ell$ , perform precise measurements of the period $T_{\mathrm{y}}$ for a small-amplitude oscillation as in Figure 4. Then compute the value of the thickness $(t)$ of the cylindrical wall.

Estimate the experimental error $\Delta t$ for the thickness.

Compute also the values of $m_1, m_2, m_3$ , and $M$ using this value of $t$ . [4 т.]

b) Angular oscillation about the central axis perpendicular to the length

Change the bifilar suspension of the vessel to that of Figure 5 and make similar measurements as in (a).

Then use the value of the period of oscillation just found together with the values of $t, m_1, m_3, M$ found in (a) to compute the value of the effective moment of inertia $I_{\mathrm{x}}^{\mathrm{Exp}}$ of the vessel about the X-axis (see Figure 2 and Figure 5).

Compute also the theoretical estimate of the value of $I_{\mathrm{x}}^{\mathrm{Theo}}$ based on the value of $t$ found in (a) assuming that the whole of the computed mass of water found in (a) is now constrained to take part in the oscillatory motion of the vessel. [2,8 т.]

c) Comparing experimental and theoretical values of the moment of inertia

What is the difference $(\Delta I_{\mathrm{x}})$ between the values of $I_{\mathrm{x}}^{\mathrm{Theo}}$ and $I_{\mathrm{x}}^{\mathrm{Exp}}$ ?

Do you consider this difference statistically significant?

Estimate the percentage of the mass of water that takes part in the oscillatory motion in (b), assuming this water to be circular discs adhering to the end plates. [1,2 т.]

Hint:

I_{\mathrm{x}}^{\mathrm{Theo}} = m_1\left[\frac{L^2}{12}+\frac{R^2+(R-t)^2}{4}\right]+2m_2\left[\frac{(0.6\ \mathrm{cm})^2}{12}+\frac{R^2}{4}+\left(\frac{L}{2}+\frac{0.6\mathrm{cm}}{2}\right)^2\right]+m_3\left[\frac{L^2}{12}+\frac{(R-t)^2}{4}\right]

Решение

Внимание: Непълно решение

В архива няма официално решение на тази задача.

Оригинал в Архива: exp.pdf