Time: Simple Pendulum and Ticker-Tape Timer

The Simple Pendulum

A simple pendulum is a small heavy bob hung from a fixed point by a light, inextensible string. When the bob is pulled aside a small angle and released, it swings to and fro in periodic, oscillatory motion under gravity.

The image below shows a simple pendulum.

The following terms (features) are associated with a simple pendulum.

• Oscillation (one complete swing): The bob moves from one extreme, to the other extreme, and back to the start (to and fro motion).

• Period (T): This is the time for the pendulum to complete one full back-and-forth oscillation (one complete oscillation), in seconds.

• Frequency (f): This is the number of complete oscillations the pendulum makes per second (measured in hertz (hz).

• Amplitude: This is the maximum displacement from the rest position.

T = 1/f    and    f = 1/T

T = 2π√(L/g)

Where L is the length of the pendulum (from the point of suspension to the centre of the bob) and g is the acceleration due to gravity (about 9.8 m/s², or 10 m/s² in many exam questions).

For small angles, the period depends on length (L) and on acceleration due to gravity (g). It does not depend on the mass of the bob or on the amplitude. A longer pendulum has a longer period.

Please read the introduction to this topic (concept of time and ways of measuring time) here.

Measuring the Period Accurately in a Simple Pendulum

The period of a simple pendulum can be measured accurately in the laboratory by carryiny out the procedures below:

1. Set up the pendulum and give a small displacement (less than about 10°).

2. Start the stop watch as the bob passes the rest position (start counting at zero).

3. Time 20 or more complete oscillations.

4. Divide the total time by the number of oscillations to get the period -> T = total time / number of oscillations.

5. Repeat and take the average.

Note: Timing many oscillations reduces the effect of your reaction time in starting and stopping the watch.

Please read on fundamental, derived and supplementary quantities in physics here.

Ticker-Tape Timer

A Ticker-tape timer is a classic laboratory device used in physics to measure short time intervals, speed, and acceleration of moving objects

Below is the components and mechanism of action of a ticker-tape timer.

• Vibrating Striker: An internal steel strip or needle vibrates rapidly up and down (usually at 50 Hz or 50 times per second) driven by an alternating current.

• Carbon Paper: A paper tape passes under a carbon disc or paper as an object pulls it.

• Marking Dots: Each vibration strikes the paper and leaves a small dot.

• Time Interval: At a standard 50 Hz frequency, the time between any two consecutive dots is 1/50 s or 0.02 s

A ticker-tape timer has a vibrating arm driven by the alternating current (a.c.) mains, which makes a dot on a paper tape being pulled through it. In many countries the mains frequency is 50 Hz, so the timer makes 50 dots every second.

• One tick (the space between two successive dots) = 0.02 s.

• A ten-tick strip (ten spaces) = 10 × 0.02 = 0.2 s.

• If dots are close together, the tape moved slowly. If they are far apart, the tape moved fast.

Please read on position, distance and displacement here.

Accuracy and Sources of Error in Timing

Accuracy means how close a measurement is to the true value. Precision means how close repeated measurements are to each other.

Below are steps towards minimizing errors in timing.

• Reaction Time: This is a delay of about 0.2 s in starting or stopping a stop watch by hand. This delay can be reduce it by timing a longer interval or many oscillations.

• Zero Error: The timer does not start from zero. It can be corrected (or accounted for) by checking and correcting the timer before use.

• Miscounting Oscillations: Always count carefully from a fixed reference mark.

• Large Amplitude: The pendulum formula only works well for small angles.

• Faulty Instrument: It can be caused by a clock that gains or loses time. It can be corrected by comparing against a reliable standard.

You can read on the basic and fundamental physics concepts here.

Worked Examples

Example 1

Example 2

Example 3

Example 4

Please read the next topic (motion in nature, definition, types causes, examples and effects) here.

Practice Questions

Section A: Objective Questions

1. The SI unit of time is the _____. A. minute B. second C. hour D. hertz

2. Time is best described as _____. A. the distance between two points B. the speed of an event C. the duration between two events D. the direction of an event

3. Time is classified as a _____. A. derived vector quantity B. fundamental scalar quantity C. derived scalar quantity D. fundamental vector quantity

4. Which of the following is an old method of measuring time? A. Quartz watch B. Atomic clock C. Hour glass D. Digital stop watch

5. Which of the following is a modern method of measuring time? A. Sundial B. Water clock C. Candle clock D. Quartz clock

6. A sundial cannot be used to tell the time _____. A. in the morning A. at noon B. at night C. on a sunny day

7. Another name for the water clock is the _____. A. gnomon B. clepsydra C. chronometer D. oscillator

8. An atomic clock uses the vibrations of atoms of _____. A. hydrogen B. oxygen C. caesium-133 D. carbon

9. The time between two successive dots made by a ticker timer of frequency 50 Hz is _____. A. 0.2 s B. 0.05 s C. 0.02 s D. 0.5 s

10. The period of a simple pendulum does not depend on the _____. A. length of the string B. mass of the bob C. acceleration due to gravity D. place where it is used

11. To reduce the error in finding the period of a pendulum, a student should _____. A. use a very large amplitude B. time one oscillation only C. time many oscillations and divide by the number D. use a heavier bob only

12. A pendulum makes 50 oscillations in 100 s. Its frequency is _____. A. 0.5 Hz B. 2 Hz C. 5 Hz D. 50 Hz

13. How many seconds are there in 2 hours? A. 120 s B. 3,600 s C. 7,200 s D. 72,000 s

14. 14. The unit “day” is based on the _____. A. revolution of the Moon around the Earth B. rotation of the Earth on its axis  C. revolution of the Earth around the Sun D. swing of a pendulum

15. Which of these is the most accurate time-measuring device? A. Atomic clock B. Quartz watch C. Hour glass D. Mechanical watch

Section B: Theory Questions

1. (a) Define time. (b) State the SI unit of time and give its standard definition.

2. Explain the difference between an instant (time of day) and a time interval, with an example.

3. List four characteristics that a good timing device should have.

4. Name four periodic events that can be used to measure time.

5. Describe how each of the following works, and state one disadvantage of each: (a) sundial, (b) water clock, (c) hour glass.

6. Describe how a quartz watch works.

7. Why is the atomic clock regarded as the most accurate method of measuring time? State two uses of atomic clocks.

8. Distinguish between old and modern methods of measuring time using at least five points.

9. Describe an experiment to determine the period of a simple pendulum accurately. State two precautions.

10. Why is it better to time 20 oscillations than a single oscillation?

11. List three sources of error when timing an event with a stop watch.

12. What is a ticker-tape timer, and how can it be used to measure time intervals?

Section C: Calculation Questions

Take g = 10 m/s² and π² ≈ 10 unless told otherwise.

1. Convert 2.5 hours to seconds.

2. A pendulum makes 40 complete oscillations in 80 s. Calculate its (a) period, (b) frequency.

3. Calculate the length of a simple pendulum whose period is 2 s.

4. A pendulum bob has frequency 4 Hz. What is the time for 60 oscillations?

5. A student times 20 oscillations of a pendulum and records 36.4 s. Calculate the period.

6. A tape with 30 spaces is pulled through a ticker timer of frequency 50 Hz. How long did the pull take?

7. A quartz crystal vibrates 32,768 times each second. How long does it take to make 98,304 vibrations?

8. A caesium atom completes 4,596,315,885 periods of its radiation. How much time has passed? (1 s = 9,192,631,770 periods)

9. Express 1 ordinary year (365 days) in seconds, in standard form.

10. Two pendulums have lengths 1 m and 4 m. Find the ratio of their periods (T₁ : T₂).