Question 19.8: Astronaut mass The BMMD chair (mass 32 kg) has a vibrational......
Astronaut mass
The BMMD chair (mass 32 kg) has a vibrational period of 1.2 s when empty. When an astronaut sits on the chair, the period changes to 2.1 s. Determine (a) the effective spring constant of the chair’s spring, (b) the mass of the astronaut, and (c) the maximum vibrational speed of the astronaut if the amplitude of vibration is 0.10 m.
Sketch and translate
■ Sketch the process described in the problem statement. Label physical quantities.
■ Choose an object (or objects) as the system of interest. Depending on the process you might need to analyze several systems.
We choose the chair and the spring as the system for (a) and the astronaut, chair, and spring for (b) and (c).
m_{\text {Chair }}=32 \mathrm{~kg}
T_{\text {Chair }}=1.2 \mathrm{~s}
T_{\text {Ast }+\text { Chair }}=2.1 \mathrm{~s}k = ?
v_{\max }=?
Simplify and diagram
■ Identify and evaluate assumptions and approximations.
■ Represent the process with force diagrams and/or bar charts if needed.
Assume that the chair’s spring obeys Hooke’s law and that the spring’s mass is much smaller than the mass of the chair plus astronaut. We determine the effective spring constant as though there were one spring pushing and pulling the chair. For (c) we represent the process using a bar chart starting from the state when the chair is pulled the farthest from equilibrium and ending as the chair and person pass equilibrium.
Represent mathematically
■ If necessary, use kinematics equations to describe the changing motion of the object.
■ If necessary, use force diagrams to apply the component form of Newton’s second law to the problem or use bar charts to apply work-energy principles.
■ If necessary, use the expressions for the period of an object attached to a spring or to a pendulum.
\text { (a) Use } T_{\text {Chair }}=2 \pi \sqrt{\frac{m}{k}} to determine the spring constant with the chair empty.
\text { (b) Use } T_{\text {Ast+Chair }}=2 \pi \sqrt{\frac{m+M}{k}} to determine the astronaut mass.
(c) Determine the energy in the spring-astronaut-chair system when at the extreme position x=+A: U=\frac{1}{2} k A^2 . Then use this energy to determine the maximum speed as the cart passes through equilibrium using U=\frac{1}{2} k A^2=\frac{1}{2}(m+M) v_{\max }^2 .
Solve and evaluate
■ Solve for the unknowns. Evaluate the solution—is it reasonable? Consider the magnitude of the answer, its units, limiting cases, etc.
(a) Square T_{\text {Chair }}=2 \pi \sqrt{\frac{m}{k}} \text { or } T_{\text {Chair }}^2=4 \pi^2 \frac{m}{k} . Rearrange to get
k=4 \pi^2 \frac{m}{T_{\text {Chair }}^2}=4 \pi^2 \frac{32 \mathrm{~kg}}{(1.2 \mathrm{~s})^2}=877 \mathrm{~N} / \mathrm{m}
(b) Square T_{\text {Ast }+\text { Chair }}=2 \pi \sqrt{\frac{m+M}{k}} \text { or } T_{\text {Ast }+\text { Chair }}^2=4 \pi^2 \frac{m+M}{k} \text {. }
Rearrange to get
m+M=\frac{T_{\text {Ast }+\text { Chair }}^2 k}{4 \pi^2} \text { r } M=\frac{(2.1 \mathrm{~s})^2(877 \mathrm{~N} / \mathrm{m})}{4 \pi^2}-32 \mathrm{~kg}=66 \mathrm{~kg}(c) From U=\frac{1}{2} k A^2=\frac{1}{2}(m+M) v_{\max }^2, k A^2=(m+M) v_{\max }^2 .
Thus v_{\max }=A \sqrt{\frac{k}{m+M}}=(0.10 \mathrm{~m}) \sqrt{\frac{(877 \mathrm{~N} / \mathrm{m})}{32 \mathrm{~kg}+66 \mathrm{~kg}}}=0.30 \mathrm{~m} / \mathrm{s} .
Check the units: \mathrm{m} \sqrt{\frac{\mathrm{N} / \mathrm{m}}{\mathrm{kg}}}=\sqrt{\frac{\mathrm{m}^2(\mathrm{~N} / \mathrm{m})}{\mathrm{kg}}}=\sqrt{\frac{\mathrm{m}(\mathrm{kg} \times \mathrm{m})}{\mathrm{kg} \times \mathrm{s}^2}}=\mathrm{m} / \mathrm{s}
The units are correct. The magnitudes are reasonable—a person can have a mass of 66 kg and the speed of 0.3 m/s is a realistic speed for such a process.
Try it yourself: What would be the period of vibration of the machine if used by an 80-kg astronaut?
Answer: About 2.25 s.