# Pendulum period and length relationship

### Exploring Pendulums - Science NetLinks

For small amplitudes, the period of such a pendulum can be approximated by: Show. For pendulum length. L = cm = m the pendulum period is. T. To Investigate The Relationship Between The Period And The Length Of A Simple Pendulum And From This To Calculate Acceleration Due To Gravity (g). The reliability of pendulum clocks is based on the predictable relationship between the length of a pendulum and the time it takes the pendulum to complete one.

For years, from its discovery around until development of the quartz clock in the s, the pendulum was the world's standard for accurate timekeeping. Pendulums require great mechanical stability: Pendulum clock Pendulums in clocks see example at right are usually made of a weight or bob b suspended by a rod of wood or metal a.

In quality clocks the bob is made as heavy as the suspension can support and the movement can drive, since this improves the regulation of the clock see Accuracy below. A common weight for seconds pendulum bobs is 15 pounds 6. This avoids the friction and 'play' caused by a pivot, and the slight bending force of the spring merely adds to the pendulum's restoring force.

A few precision clocks have pivots of 'knife' blades resting on agate plates. The impulses to keep the pendulum swinging are provided by an arm hanging behind the pendulum called the crutch, ewhich ends in a fork, f whose prongs embrace the pendulum rod.

The crutch is pushed back and forth by the clock's escapementg,h. Each time the pendulum swings through its centre position, it releases one tooth of the escape wheel g. The force of the clock's mainspring or a driving weight hanging from a pulley, transmitted through the clock's gear traincauses the wheel to turn, and a tooth presses against one of the pallets hgiving the pendulum a short push.

The clock's wheels, geared to the escape wheel, move forward a fixed amount with each pendulum swing, advancing the clock's hands at a steady rate. The pendulum always has a means of adjusting the period, usually by an adjustment nut c under the bob which moves it up or down on the rod. Some precision clocks have a small auxiliary adjustment weight on a threaded shaft on the bob, to allow finer adjustment.

Some tower clocks and precision clocks use a tray attached near to the midpoint of the pendulum rod, to which small weights can be added or removed. This effectively shifts the centre of oscillation and allows the rate to be adjusted without stopping the clock.

Pendulum clocks should be attached firmly to a sturdy wall. The most common pendulum length in quality clocks, which is always used in grandfather clocksis the seconds pendulumabout 1 metre 39 inches long. Only a few large tower clocks use longer pendulums, the 1. The wood had to be varnished to prevent water vapor from getting in, because changes in humidity also affected the length. Mercury pendulum The first device to compensate for this error was the mercury pendulum, invented by George Graham [56] in In a mercury pendulum, the pendulum's weight bob is a container of mercury.

Factors Affecting Time Period of a Simple Pendulum

With a temperature rise, the pendulum rod gets longer, but the mercury also expands and its surface level rises slightly in the container, moving its centre of mass closer to the pendulum pivot. By using the correct height of mercury in the container these two effects will cancel, leaving the pendulum's centre of mass, and its period, unchanged with temperature.

Explain the features of this demonstration to your students: In this demonstration, you can vary the length of the pendulum and the acceleration of gravity by entering numerical values or by moving the slide bar. Also, you can click on the bob and drag the pendulum to its starting position.

This demonstration allows you to measure the period of oscillation of a pendulum. To participate in this demonstration, students should follow these steps: Press the "Start" button of the stopwatch just at the moment when the pendulum is going through its deepest point.

Count "one" when it goes again through its deepest point coming from the same side. Repeat counting until "ten.

### Pendulum Motion

Dividing the time in the display by ten yields the period of oscillation. Students can also measure the frequency of a pendulum, or the number of back-and-forth swings it makes in a certain length of time. By counting the number of back-and-forth swings that occur in 30 seconds, students can measure the frequency directly. What is meant by the period of oscillation?

It is a way of measuring the back and forth swing of the pendulum. How does changing the length of the bob affect the period of oscillation? The longer the length of the bob, the longer the period of oscillation will be. What is meant by the acceleration of gravity? Is the acceleration of gravity always the same on earth? The acceleration of gravity is the force gravity exerts on an object. The force of gravity will always be the same on earth. The force of gravity on other planets will be different from earth's force of gravity.

How does changing the acceleration of gravity affect the period of oscillation? Increasing the acceleration of gravity increases the period of oscillation. How does changing the starting point or angle affect the period of oscillation? Increasing the angle increases the period of oscillation.

What happens if you start the pendulum in an upside down position of degrees? The pendulum will not move.

At this point, students should understand that gravitational forces cause the pendulum to move. They should also understand that changing the length of the bob or changing the starting point will affect the distance the pendulum falls; and therefore, affect its period and frequency.

Divide students in cooperative groups of two or three to work together to complete this activity. As outlined, students will first make predictions and then construct and test controlled-falling systems, or pendulums, using the materials listed and following the directions on the worksheet.

This controlled-falling system is a weight bob suspended by a string from a fixed point so that it can swing freely under the influence of gravity. If the bob is pushed or pulled sideways, it can't move just horizontally, but has to move on the circle whose radius is the length of the supporting string.

It has to move upward from where it started as well as sideways. If the bob is now let go, it falls because gravity is pulling it back down.

• Pendulum Motion
• Pendulum Period

It can't fall straight down, but has to follow the circular path defined by its support. This is "controlled falling": Make sure that the groups understand that by changing the value of only one variable at a time mass, starting angle, or lengththey can determine the effect that it has on the rate of the pendulum's swing.

Also, students should be sure the measurements with all the variables are reproducible, so they are confident about and convinced by their answer.

After students have completed the experiments, discuss their original predictions on the activity sheet and compare them with their conclusions based on the data and the results of the tests.

### The Simple Pendulum

Students should have been able to arrive at the following conclusions: Heavier and lighter masses fall at the same rate. Increasing the angle, or amplitude, increases the distance that the bob falls; and therefore, the frequency, or number of back and forth swings in a set time frame will be less.

Increasing the length of string to which the bob is attached, increases the radius of the circle on which the bob moves; and therefore, the frequency, or number of back and forth swings in a set time frame, will be less.

Older students should probably learn how the downward force of gravity on the bob is split into a component tangential to the circle on which it moves and a component perpendicular to the tangent coincident with the line made by the supporting string and directed away from the support. The tangential force moves the bob along the arc and the perpendicular force is exactly balanced by the taut string. Now, based on these observations, determine what conclusions students can make about the nature of gravity.

Students should conclude that gravitational force acting upon an object changes its speed or direction of motion, or both. If the force acts toward a single center, the object's path may curve into an orbit around the center.

Read More Assessment Assess the students' understanding by having them explore the Pendulums on the Moon lesson, found on the DiscoverySchool. Students should click the link for "online Moon Pendulum," found under the "Procedure" section of the lesson.

## Exploring Pendulums

This activity simulates the gravitational force on the moon. Students should experiment for approximately minutes, changing the mass, length, and angle to observe the effect it has on the pendulum. Instruct students to change only one variable at a time. Then, ask students these questions: How do you get the quickest swing?

Shorten the length of the string and decrease the angle. How do you get the longest swing? Increase the length of the string and increase the angle. In your own words, describe the relationship between mass, length of string, and angle. Mass does not affect the pendulum's swing. The longer the length of string, the farther the pendulum falls; and therefore, the longer the period, or back and forth swing of the pendulum.

The greater the amplitude, or angle, the farther the pendulum falls; and therefore, the longer the period. How does the force of gravity on the Moon compare with the force of gravity on Earth? What effect do you think the difference in gravitational forces would have on the pendulum? The force of gravity is less on the moon than on the Earth. Since the force of gravity is less on the Moon, the pendulum would swing slower at the same length and angle and its frequency would be less.

Extensions Make Coupled Resonant Pendulums This experiment demonstrates that two pendulums suspended from a common support will swing back and forth in intriguing patterns if the support allows the motion of one pendulum to influence the motion of the other.

The directions for this experiment are on the Exploratorium website. Measuring Falling Time When Galileo was studying medicine at the University of Pisa, he noticed something interesting about the periods of a pendulum.

In church one day, he watched a chandelier swing back and forth in what seemed like a steady pattern of swings. He timed each swing and discovered that each period was the same length same amount of time. In the previous activity, students measured the periods of their pendulums using either digital watches or stopwatches.

Galileo did not have these tools, so he used his pulse. In this activity, students will time the periods of their pendulums using their pulses and compare their results with those obtained with a watch.

Show students how to find their pulse by pressing two fingers on the artery next to their wrist. Make sure that students have been at rest for several minutes before doing this so that they can obtain a steady pulse rate.

## Simple Pendulum

Working in teams, have one student set the pendulum in motion while another measures the pulse beats that occur during five complete swings and then ten complete swings. Students should reproduce the distances they used in the earlier experiment, Testing Falling, for the amplitude and length of string.

Record the number of pulse beats. Repeat this procedure with different students measuring their pulse rates. Then have students measure and record five complete swings and ten complete swings using a stopwatch or digital watch.