Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A clockmaker in Kyoto builds a pendulum clock. The time of one swing is , where is the length of the pendulum and . She measures the length as , and the measurement could be wrong by .
Use a differential to find the percentage error in the period that this causes. Then find the error in the period itself, in seconds. Find also the time that the clock would gain or lose in one day of seconds, if it ran at the wrong rate all day. Say also whether a pendulum that is a little longer than planned makes the clock gain time or lose time, and why.
and . The relative error is , the percentage error is , and for the relative error in is times that in .
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The relative error in . .
Step 2. The relative error in . Since , we have . The percentage error is .
Step 3. The period. . Then .
Step 4. One day. . The clock would be out by about every day.
Gain or loss. A longer pendulum has a longer period, so each swing takes more time than the clock expects. The clock loses time.
Check. A direct derivative gives , and .
Answer: , and in a day.
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