Ch 15: Oscillations

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 15: Oscillations

Problem 35

Chapter 15, Problem 35

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

Step 1: Recognize that this problem involves the concept of resonance in a mass-spring system. The natural frequency of a system is given by the formula:

f = \frac{1}{2} π \sqrt{\frac{k}{m}}

, where

f

is the frequency,

k

is the spring constant, and

m

is the mass.

Step 2: Rearrange the formula to solve for the spring constant

k

. The rearranged formula is:

k = 4 π 2^{2} f^{2} m

Step 3: Convert the mass of the eyeball from grams to kilograms, as SI units are required for calculations. The mass is given as 7.5 g, which is equivalent to

7.5 \times 10^{- 3}

kg.

Step 4: Substitute the given values into the formula. The frequency

f

is 29 Hz, and the mass

m

7.5 \times 10^{- 3}

kg. Plug these values into the formula for

k

k = 4 π 2^{2} 29^{2} \times 7.5 \times 10^{- 3}

Step 5: Perform the calculation to determine the effective spring constant

k

. Ensure all units are consistent and verify the result for accuracy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the head. When external vibrations match 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 as the force required to compress or extend the spring by a unit distance. In this scenario, it represents the effective stiffness of the musculature that holds the eyeball in its socket. A higher spring constant indicates a stiffer system, which would resist displacement more than a system with a lower spring constant.

Resonance

Resonance occurs when a system is driven at its natural frequency, resulting in increased amplitude of oscillation. 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 significantly. This phenomenon can lead to visual disturbances, as the eye's position changes rapidly, affecting its ability to focus.

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