# Pendulum relationship between period and angle

### Pendulum Motion

There is a direct relationship between the period and the length. five trials, alterations in the arc angle have little to no effect upon the period of the pendulum. To understand the relationship between gravitational forces and the mass of objects, the . How does changing its starting point or angle affect the period?. The period of swing of a simple gravity pendulum depends on the difference between the true period and the small angle.

In Henry Kater used this idea to produce a type of reversible pendulum, now known as a Kater pendulumfor improved measurements of the acceleration due to gravity. History Replica of Zhang Heng's seismometer. The pendulum is contained inside. One of the earliest known uses of a pendulum was a 1st-century seismometer device of Han Dynasty Chinese scientist Zhang Heng. Galileo's research Italian scientist Galileo Galilei was the first to study the properties of pendulums, beginning around He first employed freeswinging pendulums in simple timing applications.

His physician friend, Santorio Santoriiinvented a device which measured a patient's pulse by the length of a pendulum; the pulsilogium. The pendulum clock The first pendulum clock In the Dutch scientist Christiaan Huygens built the first pendulum clock. This played a part in Newton's formulation of the law of universal gravitation. From this he deduced that the force of gravity was lower at Cayenne.

Huygens' Horologium Oscillatorium In17 years after he invented the pendulum clock, Christiaan Huygens published his theory of the pendulum, Horologium Oscillatorium sive de motu pendulorum.

By a complicated method that was an early use of calculushe showed this curve was a cycloidrather than the circular arc of a pendulum, [48] confirming that the pendulum was not isochronous and Galileo's observation of isochronism was accurate only for small swings.

Temperature compensated pendulums The Foucault pendulum in was the first demonstration of the Earth's rotation that did not involve celestial observations, and it created a "pendulum mania". In this animation the rate of precession is greatly exaggerated.

During the 18th and 19th century, the pendulum clock 's role as the most accurate timekeeper motivated much practical research into improving pendulums. It was found that a major source of error was that the pendulum rod expanded and contracted with changes in ambient temperature, changing the period of swing. As the pendulum bob does the back and forth, the velocity is continuously changing.

There will be times at which the velocity is a negative value for moving leftward and other times at which it will be a positive value for moving rightward. If the variations in velocity over the course of time were plotted, the resulting graph would resemble the one shown below.

Now let's try to understand the relationship between the position of the bob along the arc of its motion and the velocity with which it moves. Suppose we identify several locations along the arc and then relate these positions to the velocity of the pendulum bob. The graphic below shows an effort to make such a connection between position and velocity. As is often said, a picture is worth a thousand words. Now here come the words.

The plot above is based upon the equilibrium position D being designated as the zero position. A displacement to the left of the equilibrium position is regarded as a negative position. A displacement to the right is regarded as a positive position.

An analysis of the plots shows that the velocity is least when the displacement is greatest. And the velocity is greatest when the displacement of the bob is least. The further the bob has moved away from the equilibrium position, the slower it moves; and the closer the bob is to the equilibrium position, the faster it moves. This can be explained by the fact that as the bob moves away from the equilibrium position, there is a restoring force that opposes its motion.

This force slows the bob down. So as the bob moves leftward from position D to E to F to G, the force and acceleration is directed rightward and the velocity decreases as it moves along the arc from D to G.

## Exploring Pendulums

You might think of the bob as being momentarily paused and ready to change its direction. Next the bob moves rightward along the arc from G to F to E to D. As it does, the restoring force is directed to the right in the same direction as the bob is moving. This force will accelerate the bob, giving it a maximum speed at position D - the equilibrium position. As the bob moves past position D, it is moving rightward alo ng the arc towards C, then B and then A.

As it does, there is a leftward restoring force opposing its motion and causing it to slow down. So as the displacement increases from D to A, the speed decreases due to the opposing force.

Once again, the bob's velocity is least when the displacement is greatest. The bob completes its cycle, moving leftward from A to B to C to D. Along this arc from A to D, the restoring force is in the direction of the motion, thus speeding the bob up.

So it would be logical to conclude that as the position decreases along the arc from A to Dthe velocity increases. Once at position D, the bob will have a zero displacement and a maximum velocity. The velocity is greatest when the displacement is least.

The animation at the right used with the permission of Wikimedia Commons ; special thanks to Hubert Christiaen provides a visual depiction of these principles. The acceleration vector that is shown combines both the perpendicular and the tangential accelerations into a single vector.

You will notice that this vector is entirely tangent to the arc when at maximum displacement; this is consistent with the force analysis discussed above. And the vector is vertical towards the center of the arc when at the equilibrium position. This also is consistent with the force analysis discussed above. Energy Analysis In a previous chapter of The Physics Classroom Tutorial, the energy possessed by a pendulum bob was discussed.

We will expand on that discussion here as we make an effort to associate the motion characteristics described above with the concepts of kinetic energypotential energy and total mechanical energy. The kinetic energy possessed by an object is the energy it possesses due to its motion. It is a quantity that depends upon both mass and speed. We can combine this concept with the discussion above about how speed changes during the course of motion. This blending of concepts would lead us to conclude that the kinetic energy of the pendulum bob increases as the bob approaches the equilibrium position.

And the kinetic energy decreases as the bob moves further away from the equilibrium position. The potential energy possessed by an object is the stored energy of position. Two types of potential energy are discussed in The Physics Classroom Tutorial - gravitational potential energy and elastic potential energy.

Elastic potential energy is only present when a spring or other elastic medium is compressed or stretched.

A simple pendulum does not consist of a spring. The form of potential energy possessed by a pendulum bob is gravitational potential energy. The amount of gravitational potential energy is dependent upon the mass m of the object and the height h of the object. The height of an object is expressed relative to some arbitrarily assigned zero level. In other words, the height must be measured as a vertical distance above some reference position.

In easily verifiable experiments or demonstrations it can be shown that the period swing of a pendulum is independent of the pendulum's mass. It depends instead on the length of the pendulum. This would suggest that objects fall at a rate independent of mass. The greater the amount of the unbalanced force, the more rapidly a given object's speed or direction of motion changes; the more massive an object is, the less rapidly its speed or direction changes in response to any given force.

In this lesson, students will explore websites with simulations of pendulums, where they'll be able to change the length and angle of the bob and observe its effects. They will then construct and test their own controlled-falling systems, or pendulums, to further observe and verify these theories. Read More Motivation Ask students the following questions in order to get a feel for their current knowledge and perceptions of pendulums.

Answers to these questions are provided for you, but don't expect or lead students to these answers yet. At this point, simply gather and keep a good record of students' current ideas; students will have a chance to refine these after the website exploration that follows.

How would you define a pendulum?

## Pendulum Motion

A pendulum is loosely defined as something hanging from a fixed point which, when pulled back and released, is free to swing down by gravity and then out and up because of its inertia, or tendency to stay in motion. How does a pendulum work? What are the parts of a pendulum? A simple pendulum consists of a mass called the bob attached to the end of a thin cord, which is attached to a fixed point.

When the mass is drawn upwards and let go, the force of gravity accelerates it back to the original position. The momentum built up by the acceleration of gravity causes the mass to then swing in the opposite direction to a height equal to the original position.

This force is known as inertia. What is the period of a pendulum? A period is one swing of the pendulum over and back. What is the frequency of a pendulum?

The frequency is the number of back and forth swings in a certain length of time. What variables affect the rate of a pendulum's swing? Students may come up with a variety of answers, but the four that they will be testing in this lesson are: Length of the pendulum-Changing the length of a pendulum while keeping other factors constant changes the length of the period of the pendulum.

Longer pendulums swing with a lower frequency than shorter pendulums, and thus have a longer period. Starting angle of the pendulum-Changing the starting angle of the pendulum how far you pull it back to get it started has only a very slight effect on the frequency.

Mass of the bob at the end of the pendulum-Changing the mass of the pendulum bob does not affect the frequency of the pendulum.

Force of gravity-This accelerates the pendulum down. The momentum built up by the acceleration of gravity causes the mass to swing in the opposite direction to a height equal to the original position.

Many students believe that changing any of the variables string length, mass, or where we release the pendulum will change the frequency of the pendulum. Give them a chance to debate and discuss their answers before continuing. Where do you see pendulums in everyday life?

### Exploring Pendulums - Science NetLinks

How are they useful? Pendulums can be found in swing sets, grandfather clocks, swinging a baseball bat, and the circus trapeze. Pendulums are useful in timekeeping because varying the length of the pendulum can change the frequency. After your discussion, have students explore these websites: What is a Pendulum? After students have explored these sites, review with them their list of answers to the initial questions about pendulums, revising it with the current information based on the students' exploration of the websites.

As you review their answers to the question, "What variables affect the rate of a pendulum's swing? Read More Development Begin this part of the lesson by telling students that they will explore websites to learn more about how pendulums help us learn about gravitational forces. In the second part of the lesson, students will work in groups to construct their own pendulums and test what they have observed on the websites. Have students run the demonstration called the Pendulum Lab. With this lab, students can play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing.

Make sure they understand how to run the experiment by telling them the following: With this demonstration, you can observe how one or two pendulums suspended on rigid strings behave. You can click on the bob the object at the end of the string and drag the pendulum to its starting position.

Also, you can adjust the length and mass of the pendulum by adjusting the the controls in the green box on the right side of the page. The pendulum can be brought to its new starting position by clicking on the "Reset" button.

You also can measure the period by choosing the "photogate timer" option in the green box. Point out that the program measures the period, or one swing of the pendulum over and back.

### Pendulum - Wikipedia

How does changing the length of the bob affect the period? The shorter the length of the bob, the shorter the period will be. How does changing its starting point or angle affect the period? The smaller the angle, the shorter the period will be.