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Условие Отговори Решение

IPhO 1997, Теория — Задача 1. Scaling

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Всички материали от IPhO 1997

28th International Physics Olympiad, Sudbury, Canada — THEORETICAL COMPETITION · 17 юли 1997 г. · 10 т.

Read This First:

Use only the pen provided.
Use only the front side of the answer sheets and paper.
In your answers please use as little text as possible; express yourself primarily in equations, numbers and figures. Summarize your results on the answer sheet.
Please indicate on the first page the total number of pages you used.
At the end of the exam please put your answer sheets, pages and graphs in order.

This set of problems consists of 11 pages.

Условие

(a) A small mass hangs on the end of a massless ideal spring and oscillates up and down at its natural frequency $f$ . If the spring is cut in half and the mass reattached at the end, what is the new frequency $f'$ ? (1.5 marks) [1,5 т.]

(b) The radius of a hydrogen atom in its ground state is $a_0 = 0.0529\ \mathrm{nm}$ (the “Bohr radius”). What is the radius $a'$ of a “muonic-hydrogen” atom in which the electron is replaced by an identically charged muon, with mass 207 times that of the electron? Assume the proton mass is much larger than that of the muon and electron. (2 marks) [2 т.]

(c) The mean temperature of the earth is $T = 287\ \mathrm{K}$ . What would the new mean temperature $T'$ be if the mean distance between the earth and the sun was reduced by 1%?

(2 marks) [2 т.]

(d) On a given day, the air is dry and has a density $\rho = 1.2500\ \mathrm{kg/m^3}$ . The next day the humidity has increased and the air is 2% by mass water vapour. The pressure and temperature are the same as the day before. What is the air density $\rho'$ now? (2 marks)

Mean molecular weight of dry air: 28.8 (g/mol)
Molecular weight of water: 18 (g/mol)

Assume ideal-gas behaviour. [2 т.]

(e) A type of helicopter can hover if the mechanical power output of its engine is $P$ . If another helicopter is made which is an exact ½-scale replica (in all linear dimensions) of the first, what mechanical power $P'$ is required for it to hover? (2.5 marks) [2,5 т.]

Отговори

Решение

(a) Let the original spring have length $l$ and spring constant $k$ . The frequency $f$ of a mass $m$ oscillating on the end of this spring is given by:

f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}

The spring constant $k$ means that a force $F$ is required to produce an extension $\Delta x$ :

k = \frac{F}{\Delta x}

Consider the mid-point of the spring during such an extension; it has only moved a distance $\Delta x/2$ , while experiencing the same force $F$ . Therefore the spring constant of one half of the spring is given by:

k' = \frac{F}{\Delta x/2} = 2k

The frequency of the mass on the half-spring is:

f' = \frac{1}{2\pi}\sqrt{\frac{2k}{m}} = \underline{\sqrt{2}\, f}

Method 1 using angular momentum quantization:

The de Broglie wavelength of the particle is:

\lambda = \frac{h}{p}.

By de Broglie principle (ground state):

2\pi\, r = \lambda

Thus giving the result

pr = mvr = \hbar

(This can be stated directly as quantization of angular momentum.)

Using the Bohr model, we can consider a centripetal force due to electrostatic attraction:

\frac{mv^2}{r} = \frac{ke^2}{r^2}, \quad \therefore\ v = \sqrt{\frac{ke^2}{mr}}

\sqrt{k\, e^2 m r} = \hbar; \quad i.e.\ r \propto \frac{1}{m}

The radius of the muonic hydrogen atom is given by:

a_\mu = \frac{a_0}{207} = \underline{0.256\ \mathrm{pm}}

Method 2 using Dimensional Analysis:

The radius $r$ of the hydrogen atom in ground state depends on the following quantities:

Mass $m$ of the orbiting particle. (Since the mass of the nucleus is assumed to be much larger than the orbiting particle’s mass, the nucleus can be regarded as stationary, and thus the atomic radius does not depend on the nuclear mass.)
Electrical force between the orbiting particle and the nucleus. This depends on the charge of the nucleus $q_n$ , the charge of the orbiting particle $q$ and the constant $\varepsilon_0$ .
$\hbar$ . This is because the angular momentum is quantized as demonstrated above.

Thus:

r = A\hbar^{\alpha} m^{\beta} q_n^{\gamma_1} q^{\gamma_2} \varepsilon_0^{\delta}

where A, $\alpha$ , $\beta$ , $\gamma_1$ , $\gamma_2$ and $\delta$ are dimensionless constants. The dimensional equation is:

[D] = [M]^{\alpha+\beta-\delta}[D]^{2\alpha-3\delta}[Q]^{\gamma_1+\gamma_2+2\delta}[T]^{2\delta-\alpha}

where [D] are distance dimensions, [M] are mass dimensions, [Q] are charge dimensions and [T] are time dimensions.

Thus, letting $\gamma = \gamma_1 + \gamma_2$ :

\begin{aligned} \alpha + \beta - \delta &= 0 \\ 2\alpha - 3\delta &= 1 \\ \gamma + 2\delta &= 0 \\ 2\delta - \alpha &= 0 \end{aligned}

Solving:

\begin{aligned} \alpha &= 2 \\ \beta &= -1 \\ \gamma &= -2 \\ \delta &= 1 \end{aligned}

This indicates that the radius is inversely proportional to the mass of the orbiting particle:

r \propto \frac{1}{m}

The radius of the muonic hydrogen atom is given by:

a_\mu = \frac{a_0}{207} = \underline{0.256\ \mathrm{pm}}

(c) If the solar power output is $P$ and the radius of the earth’s orbit is $R$ , then $T$ is given by equating incoming and outgoing radiation:

(1-r)\frac{P}{4\pi R^2}\cdot \pi R_E^2 = 4\pi R_E^2\, \varepsilon \sigma T^4

Where $r$ is the reflectance of the earth with respect to solar radiation (albedo), $R_E$ is the earth’s radius, $\varepsilon$ its emissivity and $\sigma$ Stefan’s constant. The solar power output is $P$ and the mean orbital radius of the earth $R$ . The emissivity is a function of temperature (not known a priori) but the change in temperature is expected to be small.

Plainly, $T \propto \sqrt{\frac{1}{R}}$ , therefore a 1% reduction in $R$ gives a 0.5% rise in $T$ , i.e. 1.4 K

\underline{T' = 288.4\ \mathrm{K}}

(d) Ideal gas equation for N molecules: $pV = NkT$ . Two identical volumes of gas at the same pressure and temperature contain the same number of molecules; therefore the density of each is proportional to the mean molecular mass of the gas therein.

Here we use subscripts d, m and w to denote “dry”, “moist” and “water”.

For dry air, with mean molecular mass $m_d$ :

\rho = \rho_d = m_d \frac{N_d}{V} = \frac{m_d p}{kT}

For moist air, with mean molecular mass $m_m$ :

\rho_m = m_m \frac{N_m}{V} = \frac{m_m p}{kT}

For a mass $M$ of dry air:

N_d \propto \frac{M}{28.8}

For a mass $M'$ of moist air:

N_m \propto 0.02\frac{M'}{18} + 0.98\frac{M'}{28.8}

N_m = N_d

\frac{\rho_m}{\rho_d} = \frac{M'}{M} = \frac{1}{28.8\left(\frac{0.02}{18} + \frac{0.98}{28.8}\right)} = 0.9881

\rho' = \rho_m = 0.9925\rho_d = \underline{1.2352\ \mathrm{kg/m^3}}

(e) The mechanical power $P$ required for a helicopter to hover equals the downward thrust $T$ of the rotor blades (equal to its weight $W$ ) times the mean velocity $v$ of the downward moving column of air beneath its rotor blades:

P = Tv

The blades impart a velocity $v$ to the air flowing past at a rate of $dm/dt$ , and the swept area of the blades is $A$ :

T = v\frac{dm}{dt}; \quad \frac{dm}{dt} = \rho A v; \quad \therefore\ W = T = \rho A v^2

If the size of the helicopter is characterized by a linear dimension $L$ :

W \propto L^3;\ A \propto L^2

v \propto \sqrt{\frac{W}{A}} \propto \sqrt{L}, \quad \therefore\ P = Wv \propto L^{3.5}

Hence the power required for a half-scale helicopter is $\underline{P' = 0.0884\, P}$ .

Theory Question No.1: Mark Distribution

Smallest fractional mark allowed: 0.25

Marks allowed for errors consistently propagated only if physically reasonable.

Section		Marks	Subtotal
(a)	Relation between f and k	0.5
	Effect of halving spring	0.5
	Correct answer	0.5
			1.5
(b)	Quantization	0.5
	Expression for mvr	0.5
	Correct answer	1
			2
(c)	Correct proportionalities	1
	Correct answer	1
			2
(d)	Gas Law	0.5
	Numbers of molecules	0.5
	Correct answer	1
			2
(e)	Scaling of W, A	0.5
	F=v.dm/dt	0.5
	Right form for force	0.5
	Eliminate v	0.5
	Correct answer	0.5
			2.5
	Grand Total		10

Оригинал в Архива: IPhO_1997_Theoretical Questions.pdf · официални решения: IPhO_1997_Theoretical Solutions.pdf