Pendulum clocks are very important for the history of exploration. Accurate time

Question

Pendulum clocks are very important for the history of exploration.   Accurate time was the best way to determine one's longitude, since the time of sunrise or sunset depends sensitively on your longitude. In this regard, the great enemy of accuracy was thermal expansion of the pendulum used for the chronometer.

(A) If a brass rod lengthens by 0.002% for a 1°C temperature rise, by how much would the period of a 1 m long brass-rod pendulum change if the temperature was changed 10°C?

(B) How much error would that lead to in a day?

(C) Given that 24 hours corresponds to 360 degrees of longitude, how many degrees of longitude error does your answer to B imply?

(D) Using the radius of the earth from your book, and your answer above, how far away from your expected location would you be assuming you are at the equator?   (recall that this is the error in 1 day; the total will accumulate.)

Explanation / Answer

Let L be the initial length of the brass rod. L = 1 m

For 10 C temperature rise, the change is 0.002 %

for 100 C temperature rise, the change will be 10 x 0.002% = 0.02 %

So the new length will be L' = Actual length + change in length = L + 0.02L = 1.02L

The formula for time period of pendulum is T = 2 ( L/g )1/2 = 2 ( 1/ 9.8)1/2 = 2.00 seconds

New Time period T' will be T' = 2 ( L' /g )1/2 = 2 ( 1.02 L/g )1/2 = 2 ( L/g )1/2 x 1.021/2 = 1.021/2 x T

T' =  1.021/2 x T = 1.009950494T ~ 1.01 T = 1.01 x 2 = 2.02 seconds

So the time period increases by .02 seconds

b.) In a day there are 24 hours = 24 x 60 minutes = 24 x 60 x 60 seconds = 86400 seconds

Actual Time period should be T = 2seconds

So with in a day, that corresponds to 86400 /T = 86400 /2 = 43200 cycles.

For every cycles there is an error of 0.02 seconds

So in a day, for 43200 cycles the error would be 43200 x 0.02 = 864 seconds = 14 minutes 24 seconds

c.) If 24 hours, i.e 86400 seconds correspond to 360 degrees, 864 seconds correspond to (864 / 86400 ) x 360

= 3.6 degrees

d.) Radius of the earth is 6371 km. So, the perimeter around will be = 2 xR = 2 x 6371 = 40030.17359 km

This corresponds to complete angle around the earth, i.e 360 degrees

So the corresponding length of arc subtended by an angle of 3.6 degrees will be ( 3.6 / 360 ) x 40030.17359 km

= 400.3 km

This is just the error of one day. Imagine an ancient sailor who was on a voyage of say, 2 or 3 months!

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.