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.]
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.
To determine how the frequency (f) depends on the mass (m) and spring constant (k), we can use dimensional analysis.
Starting with the dimensions of frequency (f), it is measured in units of 1/time (s^-1).
The mass (m) is measured in kilograms (kg), and the spring constant (k) is measured in kilograms per second squared (kg/s^2).
Based on these units, we know that the frequency (f) is inversely related to the square root of the mass (m) and directly related to the square root of the spring constant (k).
Mathematically, this relationship can be described as:
f ∝ sqrt(k/m)
Now, let's apply this relationship to the given information:
Given:
Initial frequency (f1) = 0.50 s^-1
Initial mass (m1) = 62 kg
Spring constant (k) = kg/s^2
Chair mass (m_chair) = 10.0 kg
We need to find the final frequency (f2) for a 75-kg astronaut.
Substituting the given values into the equation, we have:
f1 = sqrt(k/m1)
Solving for k:
k = f1^2 * m1
Substituting the values:
k = (0.50 s^-1)^2 * 62 kg
Next, let's calculate the spring constant (k):
k = (0.25 s^-2) * 62 kg = 15.5 kg/s^2
Now we can use the derived relationship to calculate the final frequency (f2) for a 75-kg astronaut:
f2 = sqrt(k/m2)
Since we already know the mass of the chair (m_chair = 10.0 kg), we need to calculate the effective mass (m2) attached to the spring for the 75-kg astronaut:
m2 = m_chair + m_astronaut
m2 = 10.0 kg + 75.0 kg = 85.0 kg
Now, substitute the values into the equation:
f2 = sqrt(15.5 kg/s^2 / 85.0 kg)
Using a calculator, the final frequency can be calculated as follows:
f2 = sqrt(0.18235 s^-2) ≈ 0.427 s^-1
Therefore, the chair's frequency for a 75-kg astronaut is approximately 0.427 s^-1.
To begin, let's analyze the problem using dimensional analysis. We want to determine how the chair's frequency (f) depends on the total mass (m) and the spring constant (k).
We have the following information:
Frequency measured for a 62-kg astronaut (f1) = 0.50 s^(-1)
Total mass for the 62-kg astronaut (m1) = 62 kg
Spring constant (k)
Using dimensional analysis, we can write the relationship between the frequency (f), mass (m), and spring constant (k) as follows:
f ∝ (m^a)(k^b)
where 'a' and 'b' are the exponents to be determined.
Let's substitute the given values and solve for the exponents 'a' and 'b':
f1 = m1^a * k^b
0.50 s^(-1) = (62 kg)^a * k^b
To simplify the equation, let's convert the time unit to the second (s) by taking the reciprocal:
2.00 s = (62 kg)^a * k^b
At this point, we can see that the units are consistent. The left side of the equation represents time (s), while the right side is a combination of mass (kg) and the spring constant (kg/s^2).
Equating the exponents on both sides, we get two equations:
a = 0
b = 1
Therefore, the relationship between frequency (f), mass (m), and spring constant (k) is:
f ∝ k
Now that we know the relationship, let's solve the problem for a 75-kg astronaut.
Given:
Total mass for the 75-kg astronaut (m2) = 75 kg
Using the relationship we found earlier, we can write:
f2 ∝ k
Since the spring constant (k) remains the same, the frequency (f2) for the 75-kg astronaut will be the same as the frequency (f1) for the 62-kg astronaut:
f2 = f1 = 0.50 s^(-1)
Therefore, the chair's frequency for a 75-kg astronaut is also 0.50 s^(-1).