Question 5.TC.1: Measure the Mass in the Weightless State In the spacecraft o......
Measure the Mass in the Weightless State
In the spacecraft orbiting the Earth, there is weightless state, so that one cannot use ordinary instrument to measure the weight and then to deduce the mass of 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 N/m.
(1) When the craft is fixed on the pad, the chair (without person) oscillates with the period T_{\mathrm{0}}=1.281\,95\ \mathrm{s}.
Calculate the mass m_{\mathrm{0}} of the chair.
(2) When the craft orbits the Earth the astronaut straps himself into the chair and measures the period {\boldsymbol{T}}^{\prime} of the chair oscillation. He obtains {\boldsymbol{T}}^{\prime} = 2.33044 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 {\boldsymbol{T}}^{\prime}_{0} = 1.273 95 s.
What is the true value of the astronaut’s mass and the craft’s mass?
Note: Mass of spring is negligible and the astronaut is floating.
(1) Formula for period of oscillation:
T_{0}={\frac{2\pi}{\omega_{0}}}=2\pi\sqrt{{\frac{m_{0}}{k}}}\,. (1)
One can deduce
m_{0}={\frac{k}{4\pi^{2}}}T_{0}^{2}=2521\,{\mathrm{kg}}. (2)
(2) When the craft orbits the Earth, oscillating system is the spring with one end attached to the chair of mass m_{\mathrm{0}} and the other end attached to the craft with mass M. This system oscillates like an object with mass
m_{\mathrm{0}}^{\prime}=\frac{m_{\mathrm{0}}M}{m_{\mathrm{0}}+M} (3)
attached to an end of the spring, the other end of the spring is fixed ({\boldsymbol{m}}^{\prime}_{0} is the reduced mass of the system craft-chair). The period {\boldsymbol{T}}^{\prime}_{0} of the system is also given by (1) and (2). We can deduce
\frac{m_{\mathrm{0}}}{m_{\mathrm{0}}^{\prime}}=\left(\frac{T_{\mathrm{0}}}{T_{\mathrm{0}}^{\prime}}\right)^{2}=\left(\frac{1.281\,95}{1.273\,95}\right)^{2}. (4)
The mass M of the craft can be calculated from (3)
M={\frac{m_{0}}{{\frac{m_{0}}{m_{0}^{\prime}}}-1}}={\frac{25.21}{\left({\frac{T_{0}}{T_{0}^{\prime}}}\right)^{2}-1}}
= \frac{25.21}{\left(\frac{1.281\:95}{1.273\:95}\right)^{2}-1}=2001\:\mathrm{kg}. (5)
Let m be the mass of astronaut and chair, the coressponding reduce mass is m′ :
m^{\prime}=\frac{m M}{m+M}, (6)
the expression of m is then
m=\frac{m^{\prime}}{1-\frac{m^{\prime}}{M}}.The reduce mass m′ can be calculated from the oscillation period T′ by using formula (2) :
m^{\prime}=\frac{605.6}{4}.\left(\frac{2.330\,44}{3.\,141\,6}\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 kg.