Chapter 1, Problem 93P

To determine

The chair’s frequency for a $75\text{kg}$ astronaut.

Answer to Problem 93P

The chair’s frequency for a $75\text{kg}$ astronaut is $\underset{\_}{0.46{\text{s}}^{-1}}$.

Explanation of Solution

Write the given expression for the frequency of the chair.

    $f\propto k^x{m}^y$        (I)

Here, $f$ is the frequency, $k$ is the spring constant, $m$ is the mass, $x$ is the power of $k$, $y$ is the power of $m$.

Use the given SI units in equation (I) to solve for the $f$.

    $\begin{array}{c}\left[{\text{M}}^{\text{0}}{\text{L}}^{\text{0}}{\text{T}}^{\text{-1}}\right]={\left[{\text{MT}}^{\text{-2}}\right]}^{\text{x}}{\left[\text{M}\right]}^{\text{y}}\\ =\left[{\text{M}}^{\text{x}+\text{y}}\right]\left[{\text{T}}^{-\text{2x}}\right]\end{array}$        (II)

Compare the powers of $\text{T}$ on LHS and RHS of equation (II) to solve for $x$.

    $\begin{array}{c}-1=-2x\\ x=1/2\end{array}$        (III)

Compare the powers of $\text{M}$ on LHS and RHS of equation (II) and use equation (III) to solve for $y$.

    $\begin{array}{c}x+y=0\\ y=0-x\\ =0-1/2\\ =-1/2\end{array}$        (IV)

Use equation (IV) and (III) in equation (I) to solve for $f$.

    $\begin{array}{c}f\propto \sqrt{\frac{k}{m}}\\ =2\pi \sqrt{\frac{k}{m}}\end{array}$        (V)

Use equation (V) to solve for $k$.

    $k=\frac{f^{2}m}{4{\pi }^2}$        (VI)

Write the expression for the $m$.

    $m=m_c+m_a$        (VII)

Here, $m_c$ is the mass of the chair, $m_a$ is the mass of the astronaut.

Use equation (VII) in (VI) to solve for $k$.

    $k=\frac{f^2\left(m_c+m_a\right)}{4{\pi }^2}$        (VIII)

Use equation (VII) in (V) to solve for $f$.

    $f=2\pi \sqrt{\frac{k}{m_c+m_a}}$        (IX)

Conclusion:

Substitute $0.50{\text{s}}^{-1}$ for $f$, $62\text{kg}$ for $m_a$, $10.0\text{kg}$ for $m_c$ in equation (VIII) to find $k$.

    $\begin{array}{c}k=\frac{{\left(0.50{\text{s}}^{-1}\right)}^2\left(62\text{kg}+10.0\text{kg}\right)}{4{\pi }^2}\\ =0.456\text{kg}/{\text{s}}^2\end{array}$

Substitute $75\text{kg}$ for $m_a$, $0.456\text{kg}/{\text{s}}^2$ for $k$, $10.0\text{kg}$ for $m_a$ in equation (IX) to find $f$.

    $\begin{array}{c}f=2\pi \sqrt{\frac{\left(0.456\text{kg}/{\text{s}}^2\right)}{\left(75\text{kg}+10.0\text{kg}\right)}}\\ =0.46{\text{s}}^{-1}\end{array}$

Therefore, the chair’s frequency for a $75\text{kg}$ astronaut is $\underset{\_}{0.46{\text{s}}^{-1}}$.