BPO 2026, Теория — Задача 1. Hydrodynamic drag

EIGHTH BALKAN PHYSICS OLYMPIAD (BPO-8) · 29 юни 2026 г. · 25 т.

Условие

Any object immersed in a fluid experiences a buoyancy (Archimedes) force $\vec{F}_\mathrm{b}$. If the object, however, moves in the fluid, additional drag (resistance) force $\vec{F}_\mathrm{d}$ acts on it, having the direction opposite to the direction of its velocity $\vec{v}$. For the conditions of the present problem, the magnitude of the drag force could be represented as:

where $\rho$ is the density of the fluid, $A$ is the area of the widest cross section of the object, oriented perpendicular to the velocity (Fig. 1.1), and $C_\mathrm{d}$ is a dimensionless parameter, called drag coefficient. The drag coefficient is a characteristic of any particular shape of the object. The ultimate goal of this problem is to estimate the drag coefficient for a spherical object on the basis of experimental data regarding the motion of a metal ball in water.

The experimental setup is shown schematically in Fig. 1.2. A metal ball (1) of unknown radius $R$ and mass $m = 0{,}5\ \mathrm{kg}$ is held under water in a swimming pool by means of electronically controlled clips (2). The clips open by an electric signal from a control block (3), which activates a digital camera (4), making a series of photos of the ball falling through water, with a repetition rate of $f = 5$ frames per second. There is an unknown time delay between dropping the ball and the camera activation. Based on the analysis of images, a series of ball positions $z_i$ has been estimated and recorded in Table 1 on the Answer sheet. The coordinate axis $Z$ points vertically downwards with an origin $z = 0$ corresponding to the first recorded position of the ball.

You are given also a two sheets of Graph paper for data plots. The used graph paper and the answer sheet with Table 1 must be supplied together with your solutions.

A) Make a sketch of all forces acting on the ball during its fall in water. Derive a theoretical formula, relating the instantaneous acceleration $a$ of the ball to its instantaneous velocity $v$ in terms of parameters and constants defined previously. [2 т.]

B) Suppose that $z_{i-1}$, $z_i$, and $z_{i+1}$ are the ball positions recorded at three equidistant moments of time, $t_{i-1} = t_i - \Delta t$, $t_i$ and $t_{i+1} = t_i + \Delta t$, respectively. If the time interval $\Delta t$ between the frames is small enough, the ball motion from $t_{i-1}$ to $t_{i+1}$ can be considered approximately as uniformly accelerated. Derive formulas, giving approximate values for the ball velocity $v_i$ and acceleration $a_i$ at the middle moment $t_i$. Append Table 1 with columns containing calculated approximate values of the velocity and the acceleration at the corresponding moments of time. [8,5 т.]

C) Propose suitable auxiliary variables $y$ and $x$, which linearize the acceleration–velocity relationship: $y = px + q$. Represent the available experimental data in these variables on a graph. By using a graphical approach estimate the parameters $p$ and $q$ of the suggested linear dependence. [3 т.]

D) Estimate the density $\rho_\mathrm{M}$ of the metal ball and the drag coefficient $C_\mathrm{d}$ for an object of spherical form. [11,5 т.]

Some useful constants:

• Acceleration due to gravity, $g = 9{,}81\ \mathrm{m/s^2}$;
• Density of water, $\rho_\mathrm{w} = 1000\ \mathrm{kg/m^3}$.

Answer sheet

Table 1. Fill-in calculation data in the empty cell, where appropriate. Supply this sheet together with your solution.

Supply this paper together with your solution

Отговори

Решение

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