A seconds pendulum clock has steel wire. The clock

Question

A seconds pendulum clock has steel wire. The clock Shows correct time at 25 degree centigrade. How much time does the clock loose or gain in one week when the temp. is increased to 35 degree centigrade?

Arun · Accepted Answer

The period of a pendulum is:

T = 2*π*sqrt( L/g )

With thermal expansion, the period will be longer. Let L = L*(1 + Δ) be the pendulum length at 35C. The longer period is:

T2 = 2*π*sqrt( L*(1 + Δ) /g )

The correct number of periods counted per day is:

N = 60*60*24 / ( 2*π*sqrt( L/g ) ) = 86400 / ( 2*π*sqrt( L/g ) )

The clock would count the same number of periods at 35C, but the time taken would be:

t2 = N*2*π*sqrt( L*(1 + Δ) /g )

t2 = 86400 *2*π*sqrt( L*(1 + Δ) /g ) / ( 2*π*sqrt( L/g ) )

A lot of stuff cancels, and:

t2 = 86400 * sqrt( 1 + Δ )

At the end of counting N periods at 35C, the clock has taken an amount of time greater than one day by:

t2 - 86400 = 86400* ( sqrt(1 + Δ ) - 1)

For brass, the thermal coefficient of expansion is α = 19*10^-6 C^-1. The length of a pendulum at 35C would be:

ΔL = L1*α*ΔT = L * 19*10^-6 * 10 = L * 19*10-5

L2 = L + ΔL = L * ( 1 + α*ΔT )

In the above derivation, this was expressed as:

L = L*( 1 + Δ )

Δ = 19*10^-5

So, the time error after N periods (one day) would be:

Δt = 86400* ( sqrt(1 + Δ ) - 1) = 86400*( sqrt( 1 + 19*10^-5 ) - 1 ) = 8.2 sec

Hence in one week = 7 * 8.2 = 57.4 sec

General Physics> A seconds pendulum clock has steel wire. ...

A seconds pendulum clock has steel wire. The clock Shows correct time at 25 degree centigrade. How much time does the clock loose or gain in one week when the temp. is increased to 35 degree centigrade?

Suvendu Mohapatra , 8 Years ago

Arun

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General Physics> A seconds pendulum clock has steel wire. ...

A seconds pendulum clock has steel wire. The clock Shows correct time at 25 degree centigrade. How much time does the clock loose or gain in one week when the temp. is increased to 35 degree centigrade?

Suvendu Mohapatra , 8 Years ago

Arun

LIVE ONLINE CLASSES

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