text: # Relationship between mass and length of pendulum
  
  ### Pendulum Motion
  
  For small oscillations the simple pendulum has linear behavior meaning that its How do mass, length, or gravity affect the relationship between angular. A simple pendulum consists of a mass m hanging from a string of length L and fixed at a With the assumption of small angles, the frequency and period of the . A simple pendulum consists of a point mass 'm', suspended from a fixed point the relationship between the period and the length of a simple pendulum to.
  
  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. 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. How can you get the shortest period? Decrease the length, and decrease the angle.
  
  How can you get the longest period? Increase the length, and increase the angle. Explain why the pendulum continues to move without stopping or slowing down once it is set in motion. According to the law of inertia, a body in motion will continue in motion, unless acted upon by a force. 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.
  
  ### Exploring Pendulums - Science NetLinks
  
  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.
  
  Time Period of a Pendulum - How Does it Depend on its Length?
  
  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. 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. 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.
  
  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. 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.
  
  ## Pendulum Motion
  
  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?
  
  And what is the restoring force for a pendulum? There are two dominant forces acting upon a pendulum bob at all times during the course of its motion. There is the force of gravity that acts downward upon the bob. It results from the Earth's mass attracting the mass of the bob.
  
  And there is a tension force acting upward and towards the pivot point of the pendulum. The tension force results from the string pulling upon the bob of the pendulum. In our discussion, we will ignore the influence of air resistance - a third force that always opposes the motion of the bob as it swings to and fro.
  
  The air resistance force is relatively weak compared to the two dominant forces. The tension force is considerably less predictable. Both its direction and its magnitude change as the bob swings to and fro.
  
  The direction of the tension force is always towards the pivot point. So as the bob swings to the left of its equilibrium position, the tension force is at an angle - directed upwards and to the right.
  
  ## Exploring Pendulums
  
  And as the bob swings to the right of its equilibrium position, the tension is directed upwards and to the left. The diagram below depicts the direction of these two forces at five different positions over the course of the pendulum's path. In physical situations in which the forces acting on an object are not in the same, opposite or perpendicular directions, it is customary to resolve one or more of the forces into components. This was the practice used in the analysis of sign hanging problems and inclined plane problems.
  
  Typically one or more of the forces are resolved into perpendicular components that lie along coordinate axes that are directed in the direction of the acceleration or perpendicular to it.
  
  So in the case of a pendulum, it is the gravity force which gets resolved since the tension force is already directed perpendicular to the motion.
  
  The diagram at the right shows the pendulum bob at a position to the right of its equilibrium position and midway to the point of maximum displacement. A coordinate axis system is sketched on the diagram and the force of gravity is resolved into two components that lie along these axes. One of the components is directed tangent to the circular arc along which the pendulum bob moves; this component is labeled Fgrav-tangent. The other component is directed perpendicular to the arc; it is labeled Fgrav-perp.
  
  You will notice that the perpendicular component of gravity is in the opposite direction of the tension force. You might also notice that the tension force is slightly larger than this component of gravity. The fact that the tension force Ftens is greater than the perpendicular component of gravity Fgrav-perp means there will be a net force which is perpendicular to the arc of the bob's motion.
  
  This must be the case since we expect that objects that move along circular paths will experience an inward or centripetal force. The tangential component of gravity Fgrav-tangent is unbalanced by any other force. So there is a net force directed along the other coordinate axes. It is this tangential component of gravity which acts as the restoring force. As the pendulum bob moves to the right of the equilibrium position, this force component is directed opposite its motion back towards the equilibrium position.
  
  The above analysis applies for a single location along the pendulum's arc. At the other locations along the arc, the strength of the tension force will vary. Yet the process of resolving gravity into two components along axes that are perpendicular and tangent to the arc remains the same.
  
  The diagram below shows the results of the force analysis for several other positions.
  
  There are a couple comments to be made. First, observe the diagram for when the bob is displaced to its maximum displacement to the right of the equilibrium position.
  
  The tension force Ftens and the perpendicular component of gravity Fgrav-perp balance each other. At this instant in time, there is no net force directed along the axis that is perpendicular to the motion. Since the motion of the object is momentarily paused, there is no need for a centripetal force.
  
  ### Pendulum Period | Science Primer
  
  Second, observe the diagram for when the bob is at the equilibrium position the string is completely vertical. When at this position, there is no component of force along the tangent direction.
  
  When moving through the equilibrium position, the restoring force is momentarily absent. Having been restored to the equilibrium position, there is no restoring force.
  
  The restoring force is only needed when the pendulum bob has been displaced away from the equilibrium position. You might also notice that the tension force Ftens is greater than the perpendicular component of gravity Fgrav-perp when the bob moves through this equilibrium position. Since the bob is in motion along a circular arc, there must be a net centripetal force at this position. The Sinusoidal Nature of Pendulum Motion In the previous part of this lessonwe investigated the sinusoidal nature of the motion of a mass on a spring.
  
  We will conduct a similar investigation here for the motion of a pendulum bob. Let's suppose that we could measure the amount that the pendulum bob is displaced to the left or to the right of its equilibrium or rest position over the course of time. A displacement to the right of the equilibrium position would be regarded as a positive displacement; and a displacement to the left would be regarded as a negative displacement. Using this reference frame, the equilibrium position would be regarded as the zero position.
  
  And suppose that we constructed a plot showing the variation in position with respect to time. The resulting position vs. Similar to what was observed for the mass on a spring, the position of the pendulum bob measured along the arc relative to its rest position is a function of the sine of the time. Now suppose that we use our motion detector to investigate the how the velocity of the pendulum changes with respect to the time.
  
  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.
  
  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.