Why is pendulum period independent of mass

On rounding to the nearest half-integer, this result indicates that 2. A ha'penny is kept by the clock. The position of the hands cannot be read to satisfactory precision which is why the time of the first stroke of the hour is used as a reference.

Moreover, the error in the time of the first stroke is slightly different at different hours of the day which is why a particular hour 8 a. The pendulum incorporates a temperature compensation mechanism but it suffers from time lag and requires human intervention to help it on its way. It is worth noting that the time is almost never adjusted, but only the rate.

The clock is stopped for occasional maintenance and when the clocks go back an hour in October each year. It may take a week or two for the timing to recover after such incidents which typically occur once or twice a year. The correction may be made "on the fly" without stopping the swing of the pendulum.

Grandfather clocks are adjusted by a screw which moves the bob up and down the pendulum shaft. This requires stopping the clock during adjustment. Another pendulum problem: Suspend a flat square metal plate from strings at its four corners, so that it lies in a horizontal plane and the strings are parallel and vertical.

Let the string lengths be, say 1. Pull the plate aside and it swings like a pendulum, remaining horizontal at all times.

Find its period of oscillation. Now cut two strings which support one edge. The plate drops and now hangs from two parallel strings in a vertical plane. Pull the plate aside and release it so it swings in its own plane, its upper and lower edge remaining horizontal at all times, and its other two edges remaining vertical. Clearly the average distance of the plate's mass from the suspension point is significantly greater than before, but its mass hasn't changed.

Is its period now a very much greater than before, b slightly greater than before, c the same as before, d slightly smaller than before, e very much smaller than before? The outcome is easily tested by timing, say, 20 swings with a stopwatch. Or your pulse, as Galileo might have done. This is easy to demonstrate. Find or construct a rigid support to suspend strings from. A flat metal plate, the larger the better for class use.

For smaller groups I've used a large flat Meccano or Erector metal plate, about 5 x 7 inches. Flat plate pendulum. Note that the direction of swing is such that in the second case B the plate moves in its own plane. Compare the two periods in the two cases by timing about 20 swings. I've done this for audiences of physics teachers, many of whom were surprised at the result. In my demos for teachers lecture I remark "If we teachers can't understand simple everyday things like this well enough to predict them, and explain them to students, we kid ourselves if we suppose we can teach subtle things like quantum mechanics.

Then I slide into a consideration of the mechanism of the platform rocker chair. Don't teachers claim they like physics applications to everyday things? Analysis of the flat plate pendulums.

Figure 5. A horizontal rod pendulum. We need to compare a swinging plate with what we have learned from the previous analysis.

Start with a small coin as before Fig. Pendulums are in common usage. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 1.

Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. We begin by defining the displacement to be the arc length s. Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. For the simple pendulum:. This result is interesting because of its simplicity.

The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. Even simple pendulum clocks can be finely adjusted and accurate. Note the dependence of T on g. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity.

Consider Example 1. What is the acceleration due to gravity in a region where a simple pendulum having a length We are asked to find g given the period T and the length L of a pendulum. This method for determining g can be very accurate. This is why length and period are given to five digits in this example.

It is due to the effect of gravity. Because acceleration remains the same, so does the time over which the acceleration occurs. In a clock's pendulum the bob is often a hanging chunk of lead. The rest of the pendulum is as light as possible so that the wire the bob hangs from can be ignored. This is the approximation I indicated in the "Answer" above.

That assumption needs to be valid in the construction of the pendulum. If you increase the mass of the bob, that is no problem since it makes the above assumption more valid mass is even more concentrated at the end. 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. 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.

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. Ask students:. 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": the path is always the same, it can be reproduced time after time, and variations in the set-up can be used to test their effect on the falling behavior.

Note: Make sure that the groups understand that by changing the value of only one variable at a time mass, starting angle, or length , they 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. 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. 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.