Question

An infant is bouncing in a spring chair. the infant has a mass of 8 kg, and the chair functions as a spring with spring constant 72 n/m. the bouncing of the infant applies a force of the form 6 cos(!t) for some frequency !. assume that the infant starts at rest at the equilibrium position of the chair. 168 chapter 2. higher order linear odes if there is no dampening coeffcient.

What frequency would the infant need to force at in order to generate pure resonance?
A) 6 Hz
B) 12 Hz
C) 18 Hz
D) 24 Hz

Answer 1

The frequency for pure resonance in a spring system is determined by the formula for natural frequency, √(k/m) / (2π). For an infant of 8 kg and a spring constant of 72 N/m, the natural frequency is around 0.48 Hz, which does not match any of the provided answer choices.

Step-by-step explanation:

To determine the frequency needed to generate pure resonance in the system, we can use the formula for the natural frequency of a mass-spring system, which is given by √(k/m) / (2π), where 'k' is the spring constant and 'm' is the mass of the object. In this case, the infant has a mass of 8 kg and the spring has a spring constant of 72 N/m.

Let's calculate the frequency (ω) for resonance:

ω = √(72 N/m / 8 kg) / (2π) = √(9 s⁻²) / (2π) = √(0.5 Hz) / (2π) = (3 Hz) / (2π)

Since 3 Hz/2π is approximately 0.48 Hz (after calculation), none of the answer choices A, B, C, or D exactly matches this value. This may indicate a problem with the answer choices provided or may require reevaluating the initial conditions mentioned in the question. For pure resonance, the system would have to oscillate at its natural frequency, which is around 0.48 Hz, and this option is not listed among the answer choices A, B, C, or D.