Vision is blurred if the head is vibrated at 29 Hz because the vibrations are resonant with the natural frequency of the eyeball in its socket. If the mass of the eyeball is 7.5 g, a typical value, what is the effective spring constant of the musculature that holds the eyeball in the socket?

Verified step by step guidance

1. First, recognize that the problem involves simple harmonic motion and can be approached using the formula for the natural frequency of a spring-mass system, which is given by \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \), where \( f \) is the frequency, \( k \) is the spring constant, and \( m \) is the mass.
2. Convert the mass of the eyeball from grams to kilograms for consistency in SI units. Since 1 gram = 0.001 kilograms, multiply the mass value by 0.001.
3. Rearrange the formula to solve for the spring constant \( k \). The formula becomes \( k = 4\pi^2 f^2 m \).
4. Substitute the given values for the frequency \( f \) and the mass \( m \) into the rearranged formula.
5. Calculate the value of \( k \) using the substituted values to find the effective spring constant of the musculature that holds the eyeball in its socket.

Key Concepts

Natural Frequency

Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving force. In the context of the eyeball, it refers to the specific frequency at which the eyeball vibrates when subjected to external forces, such as vibrations from head movement. When the external frequency matches this natural frequency, resonance occurs, leading to amplified oscillations and potential blurriness in vision.

Spring Constant

The spring constant, denoted as 'k', is a measure of a spring's stiffness, defined by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement. In this scenario, the musculature holding the eyeball can be modeled as a spring, where the spring constant reflects how much force is needed to displace the eyeball from its equilibrium position. A higher spring constant indicates a stiffer system that resists displacement.

Resonance

Resonance occurs when a system is driven at its natural frequency, resulting in large amplitude oscillations. In the case of the eyeball, when the head vibrates at 29 Hz, which matches the natural frequency of the eyeball, the vibrations cause the eyeball to oscillate more than it would at other frequencies. This phenomenon can lead to visual disturbances, as the eye's position becomes unstable during these amplified movements.