Answer :
Final answer:
To find out what time a clock will show after altering the pendulum length by 1.000%, we use the approximation that the period changes by approximately 0.500%. This results in the clock being 7.2 minutes slow if the pendulum is lengthened, and 7.2 minutes fast if shortened after 24 hours.
Explanation:
The student's question involves calculating time changes of a clock with a pendulum that has been altered in length. When the pendulum length of a clock increases, the clock will run slower because the period of the pendulum's oscillation increases. Conversely, if the pendulum length is decreased, the clock runs faster. If the length of the clock's pendulum is changed by 1.000%, we can calculate two possible outcomes depending on if the length was increased or decreased.
To calculate the time, we must understand that the timekeeping of a pendulum clock is proportional to the square root of its length. Since the change is small, we can use the approximation that the period of the pendulum is directly proportional to the square root of its length. The percentage change in the period will therefore be approximately half the percentage change in length, because when x is small, the square root of (1 + x) is approximately 1 + x/2. Thus, a 1.000% increase in length will result in about a 0.500% increase in the period (the clock will gain 0.500% of time), and a 1.000% decrease in length will result in about a 0.500% decrease in the period (the clock will lose 0.500% of time).
Since there are 1440 minutes in a day, a 0.500% change amounts to a 7.2-minute difference over 24 hours (1440 minutes x 0.005). Therefore,
- If the pendulum is lengthened by 1.000%, the clock will be 7.2 minutes slow after 24 hours (it will read 11:52.8 PM).
- If the pendulum is shortened by 1.000%, the clock will be 7.2 minutes fast after 24 hours (it will read 12:07.2 AM).
