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APhO 2004, Теория — Задача 1

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Условие

Theoretical competition – Problem No.1

Measure the mass in the weightless state

In the spacecraft orbiting the Earth, there is weightless state, so that one cannot use ordinary instruments to measure the weight and then to deduce the mass of the astronaut. Skylab 2 and some other spacecrafts are supplied with a Body Mass Measurement Device which consists of a chair attached to one end of a spring. The other end of the spring is attached to a fixed point of the spacecraft. The axis of the spring passes through the center of mass of the craft. The force constant (the hardness) of the spring is $k = 605.6\ \mathrm{N/m}$ .

When the craft is fixed on the pad, the chair (without person) oscillates with the period

T_0 = 1.28195\ \mathrm{s}.

Calculate the mass $m_0$ of the chair [2 pts].

When the craft orbits the Earth the astronaut straps himself into the chair and measures the period $T'$ of the chair oscillations. He obtains $T' = 2.33044\ \mathrm{s}$ , then calculates roughly his mass. He feels some doubt and tries to find the true value of his mass. He measures again the period of oscillation of the chair (without person), and find $T'_0 = 1.27395\ \mathrm{s}$ .

What is the true value of the astronaut’s mass and the craft’s mass? [4pts]

Note: The mass of the spring is negligible and the astronaut is floating.

Решение

Solution of Problem No. 1

Measure the mass in the weightless state

1/ Formula for period of oscillation :

T_0=\frac{2\pi}{\omega_0}=2\pi\sqrt{\frac{m_0}{k}}\qquad (1)

One can deduce :

m_0=\frac{k}{4\pi^2}T_0^2=25,21\ \mathrm{kg}\qquad (2)

2/ When the craft orbits the Earth, oscillating system is the spring with one end attached to the chair of mass m₀ and other end attached to he craft with mass M. This system oscillates like an object with mass

m'_0=\frac{m_0M}{m_0+M}\qquad (3)

attached to an end of the spring, other end of the spring is fixed (m₀’ is the reduced mass of the system craft-chair). The period T₀’ of the system is also given by (1) and (2). We can deduce :

\frac{m_0}{m'_0}=\left(\frac{T_0}{T'_0}\right)^2=\left(\frac{1,28195}{1,27395}\right)^2\qquad (4)

The mass M of the craft can be calculate from (3) :

M=\frac{m_0}{\frac{m_0}{m'_0}-1}=\frac{25,21}{\left(\frac{T_0}{T'_0}\right)^2-1}=\frac{25,21}{\left(\frac{1,28195}{1,27395}\right)^2-1}=2001\ \mathrm{kg}\qquad (5)

Let m be the mass of astronaut and chair, the corresponding reduce mass is m’:

m'=\frac{mM}{m+M}\qquad (6)

the expression of m is then :

m=\frac{m'}{1-\frac{m'}{M}}

The reduce mass m’ can be calculated from the oscillation period T’ by using formula (2) :

m'=\frac{605,6}{4}\cdot\left(\frac{2,33044}{3,1416}\right)^2=83,31\ \mathrm{kg}

the true value of the mass m is :

m=\frac{83,31}{1-\frac{83,31}{2001}}=86,93\mathrm{kg}

the true value of the mass of astronaut :

86,93-25,21=61,72\ \mathrm{kg}

Теми и трудност

Трудност: неоценена.

Механика · Механични трептения · Връзки и свързани тела

Оригинал в Архива: 1.pdf · официални решения: 1_sol.pdf