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- Homework Statement
- For this problem you can consider an ideal pendulum,

for which you will calculate the effect of changing its environment.

(d) Most substances expand with increasing temperature, and so a

metal rod will expand in length by a fraction α δT if the temperature is changed by δT, where α is called the coefficient of linear thermal expansion. Consider the effect of this expansion on a pendulum clock with a pendulum made from brass, with α = 19 × 10^-6 K^-1 .

What temperature change can this clock tolerate if it is to remain accurate to 1 second in 24 hours?

- Relevant Equations
- P=2π √(L/g)

1) I do not quite understand how the phrase remain accurate to 1 second in 24 hours? , means ΔP = 1 second,

2) I also don't understand how pendulum period P should be 24 hours

What is the reasoning for both?

The solution is as such

P = 2π √(L/g) P' = 2π √((L+L α δT)/g)

ΔP = P'- P = 2π √((L(1+ α δT)/g) - 2π √(L/g)

where 2π √(L/g) factored out from above to give :

ΔP = 2π √(L/g) ( (√(1+ α δT) - 1)

ΔP = P ( (√(1+ α δT) - 1)

ΔP = 1 sec (which I don't understand this part)

also P = 24 hours = 24*3600 secs ( which I don't understand how one pendulum period should be 24 hours?, I don't understand the reasoning behind this.)

ΔP/P = 1/24*3600

ΔP/P = P ( (√(1+ α δT) - 1) /P

√(1+ α δT) - 1 =1/24*3600

then the value for α can be plugged in and rearranged to find δT.

2) I also don't understand how pendulum period P should be 24 hours

What is the reasoning for both?

The solution is as such

P = 2π √(L/g) P' = 2π √((L+L α δT)/g)

ΔP = P'- P = 2π √((L(1+ α δT)/g) - 2π √(L/g)

where 2π √(L/g) factored out from above to give :

ΔP = 2π √(L/g) ( (√(1+ α δT) - 1)

ΔP = P ( (√(1+ α δT) - 1)

ΔP = 1 sec (which I don't understand this part)

also P = 24 hours = 24*3600 secs ( which I don't understand how one pendulum period should be 24 hours?, I don't understand the reasoning behind this.)

ΔP/P = 1/24*3600

ΔP/P = P ( (√(1+ α δT) - 1) /P

√(1+ α δT) - 1 =1/24*3600

then the value for α can be plugged in and rearranged to find δT.

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