Measurement of Time and Motion · Lesson 2 of 6
The Pendulum and Accurate Timekeeping
“Investigate a pendulum’s repeated motion and connect it to units and precision in clocks.”
• Identify the parts and positions of a simple pendulum and trace one full oscillation. • Calculate a pendulum’s time period from several oscillations. • Explain how length and bob mass affect a pendulum’s period in the chapter’s investigations. • Use seconds, minutes, and hours correctly and compare the precision of timekeeping instruments. • Describe how repeated vibrations allow modern clocks to mark time.
A Repeated Swing Can Become a Clock
A swinging lamp gave Galileo a question: does each complete swing take nearly the same time? Christiaan Huygens later used pendulum motion in a clock. The useful property is not that the bob moves quickly, but that a pendulum of a given length at a given place repeats its motion in nearly equal intervals.
A simple pendulum has a small heavy bob hanging from a rigid support by a thread. When it hangs still, the bob is at its mean position. Pull it slightly to one side and release it without a push: the bob swings through the mean position toward the other extreme and returns.
Motion back and forth about a mean position, as shown by a gently swinging pendulum.
Motion that repeats after a regular interval of time.
A complete repetition of the pendulum’s movement: for example, from one extreme to the other and back to the starting extreme.
The time taken to complete one oscillation.
Moving from one extreme to the opposite extreme is only half of the complete extreme-to-extreme-and-back journey. Fix a starting position and count only when the bob returns to it moving in the same direction.
Measure the Period Fairly
Tie a bob to a thread fixed to a support and use a length of about 100 cm. With the string taut, release the bob gently from a small sideways displacement. Time ten complete oscillations with a stopwatch, repeat three or four times, and record each reading. Timing many swings reduces the effect of starting and stopping the watch a little early or late.
Divide each measured time by ten when timing ten swings. The readings should be close, though small timing errors can make them differ. The chapter’s observation table leaves spaces for three trials; comparing the results is more informative than trusting a single reading.
| Trial | Time for 10 oscillations | Time period |
|---|---|---|
| 1 | 20.2 s | 2.02 s |
| 2 | 19.9 s | 1.99 s |
| 3 | 20.1 s | 2.01 s |
Problem
A pendulum completes 10 full oscillations in 20 s. What is its time period?
- 1.The total measured time is 20 s and the number of oscillations is 10.
- 2.Divide the total time by the number of complete oscillations: 20 s ÷ 10.
- 3.The time period is 2 s for one oscillation.
Problem
Three trials for 10 oscillations take 19.8 s, 20.2 s, and 20.0 s. What do these readings suggest?
- 1.Divide each total by 10 to obtain 1.98 s, 2.02 s, and 2.00 s.
- 2.The three period estimates are close to 2 s.
- 3.The small differences may come from measurement variation; repeat and control the release and counting.
What Changes the Period?
To investigate length, keep the same bob and location while changing the thread length. A longer pendulum takes more time per oscillation. To investigate bob mass, keep the length and location fixed while changing the bob; the time period remains approximately the same. Changing one thing at a time allows a fair comparison. Pendulums of the same length have the same period at a given location under comparable conditions.
A playground swing offers a larger-scale investigation. Time ten swings, divide by ten, and repeat. Compare swings of different lengths, then compare riders of different masses on the same swing. Small differences between trials can result from the way a swing is released or timed; they should be discussed, not hidden.
Clocks, Units, and Precision
An old pendulum clock, a quartz clock, and an atomic clock all use a process that repeats. Quartz crystals vibrate very rapidly; atomic clocks use regular behaviour of particular atoms. Their processes differ, but each provides intervals that a clock can count. The increasing precision matters whenever two events occur very close together.
The SI unit of time, written with the symbol s.
A wall clock with a second hand can usually show intervals as small as one second. A stopwatch may show hundredths of a second for a close race; a millisecond is one-thousandth of a second. Heart monitors can measure millisecond changes between beats, while computers can work with microsecond intervals, one-millionth of a second. The smallest division an instrument displays affects how finely it can distinguish times.
Problem
A swing has a period of 2 s. How many full oscillations would it complete in 1 min at this steady period?
- 1.Convert 1 min to 60 s.
- 2.Each complete oscillation takes 2 s, so divide 60 s by 2 s per oscillation.
- 3.It completes 30 oscillations in 1 min.
Write 15 s, 2 min, and 1 h, with a space before the unit. The symbols stay lowercase and do not become plural: write 10 s rather than 10 secs.
Measure a friend’s pulse beats over one minute, then ask whether the beats are steady enough to serve as a rough clock. Compare repeated trials and explain why a pulse is less reliable than a calibrated stopwatch.
Quiz
Which journey is one complete pendulum oscillation when starting at A?
Ten full oscillations take 18 s. What is the estimated period?
In the chapter’s fair comparison, which change affects a simple pendulum’s period?
What is the SI unit of time?
Which expression equals 2 minutes?
Why time ten swings instead of only one?
Practice Problems
- Draw or describe a bob moving through its mean and two extreme positions, and mark one complete oscillation.
- A pendulum makes 10 oscillations in 15 s. Calculate its period and show the division.
- Three timings for ten swings are 20.1 s, 19.9 s, and 20.0 s. Compare their period estimates.
- Describe a fair investigation of how pendulum length affects time period.
- Explain why changing both length and bob mass at once would make the result hard to interpret.
- Convert 3 min 20 s into seconds and state the SI unit.
- A stopwatch reads to 0.01 s while a wall clock reads to 1 s. Which is more useful for a close race, and why?
Key Takeaways
• A complete oscillation returns a pendulum to its starting state after a full back-and-forth movement. • Time period equals total time divided by the number of complete oscillations. • At a given location, pendulum length affects its period, while changing the bob’s mass does not appreciably affect it. • Clocks count repeated processes; the SI unit of time is the second (s). • Use an instrument whose precision suits the interval being measured.