Question 19.QE.4: Vibrating chair You want to design a bouncy seat for your ne......
Vibrating chair
You want to design a bouncy seat for your nephew using a bungee cord (see figure at right). The chair should allow a 12-kg baby (including the mass of the seat) to bounce up and down naturally at a frequency of about 0.40 Hz. Estimate the spring constant the cord should have.
Represent mathematically Model the cord as a linear spring and the child+seat as a point-like object. We know the requested vibration frequency and the proposed mass of the child and seat. We can use Eq. (19.8) to find k :
f=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}
Solve and evaluate Square both sides of the above
equation to get f² = (1/4π²)(k/m). Then, multiplying
both sides by 4π²m, we have
k=4 \pi^2 m f^2=4 \pi^2(12 \mathrm{~kg})\left(0.4 \mathrm{~s}^{-1}\right)^2
\simeq 80 \mathrm{~kg} / \mathrm{s}^2=80 \mathrm{~N} / \mathrm{m}
Try it yourself: If the total mass of the child+seat were increased to 24 kg without changing the spring constant, what would be the frequency of vibration? Describe any additional difficulties that will result from this change.
Answer: f=0.28 \mathrm{~s}^{-1} \text { or } T=3.5 \mathrm{~s} The cord will stretch twice as far—it may not fit in a doorway.