Let the mass of the astronaut be m_a and the mass of the chair be m_c; given m_c = 10 kg. From the hint, we know that the frequency f depends on the mass attached to the spring (m_a + m_c) and the spring constant k. We can write:
f = C * (m_a + m_c) ^ n1 * (k) ^ n2
where C is a constant, and n1 and n2 are the powers of dependence on mass and spring constant.
Now we apply dimensional analysis. Let M be the dimension of mass, L be the dimension of length, and T be the dimension of time. The dimensions of f, k, and mass are [T^-1], [M*T^-2], and [M], respectively. So the equation becomes:
[T^-1] = [M ^ n1 * T ^ (-2n2)]
Comparing the dimensions of M and T, we get:
n1 = 0
and
-2n2 = -1
which gives n2 = 1/2, so the equation becomes:
f = C * sqrt(k)
Now, we are given that for a 62 kg astronaut, f = 0.5 s^-1. We can use this information to find the constant C:
0.5 s^-1 = C * sqrt(k)
Also, we know that m_a = 62 kg and m_c = 10 kg, so the mass attached to the spring is m_a + m_c = 72 kg. Thus, the spring constant k = 72 kg * g, where g is acceleration due to gravity (approximately 9.8 m/s^2).
k = 72 kg * 9.8 m/s^2 ≈ 705.6 kg/s^2
Now, we can solve for C:
C = (0.5 s^-1) / sqrt(705.6 kg/s^2)
Let's find chair's frequency for a 75 kg astronaut. Given the mass of the astronaut m_a2 = 75 kg, the total mass attached to the spring is m_a2 + m_c = 85 kg. The spring constant k2 = 85 kg * g ≈ 833 kg/s^2.
Now we can calculate the chair's frequency for a 75 kg astronaut using the equation:
f2 = C * sqrt(k2) = (0.5 s^-1) / sqrt(705.6 kg/s^2) * sqrt(833 kg/s^2)
f2 ≈ 0.54 s^-1
The chair's frequency for a 75 kg astronaut is approximately 0.54 s^-1.
The Space Shuttle astronauts use a massing chair to measure their mass. The chair is attached to a spring and is free to oscillate back and forth. The frequency of the oscillation is measured and that is used to calculate the total mass m attached to the spring. If the spring constant of the spring k is measured in kg/s^2 and the chair's frequency f is 0.50 s-^1 for a 62-kg astronaut, what is the chair's frequency for a 75-kg astronaut? The chair itself has a mass of 10.0 kg. [Hint: use dimensional analysis to find out how f depends on m and k.]
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1 answer