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**Algebra - "Quartz Tuning Fork" Watch**

## Homework Statement

Many modern watches operate based on a small quartz tuning fork which transduces a mechanical oscillation into an electrical signal. The frequency of the tuning fork is

inversely proportional to sqrt(

*l*), with

*l*being the length of the fork. If the watch keeps perfect time at 20

*°C*, what is the fractional gain or less in time for a quartz tuning fork that is 6mm long at:

(a) 0

*°C*

(b) 30

*°C*

*Hint: try working this out algebraically. The changes are small and prone to rounding issues.*

*α*= 0.59 × 10

_{Quartz}^{-6}/

*°C*

## Homework Equations

Not sure if the following equations are useful, but my topic of study right now is based on thermal expansion, as well as calculations based on heat energy (

*Q = mcΔT*).

*ΔL = αL*(linear thermal expansion)

_{0}ΔTIf a body has length L

_{0}at temperature

*T*, then its length

_{0}*L*at a temperature

*T = T*

L = L

_{0}+ ΔT is:L = L

_{0}+ ΔL = L_{0}+ αL_{0}ΔT = L_{0}(1 + αΔT)*Q = mcΔT*

## The Attempt at a Solution

This question appears to me on a topic that I haven't touched so far in my Physics class; however, I know that the topics I am studying is interconnected with this question somehow. From what I read from the question, I concluded that

**frequency = 1/√length**but I don't know how to incorporate that into an equation in which I can find a "ratio." I am stuck here and do not know how to further approach this problem.