Mass Independence of Pendulum Period ```Name: Sam Status: student Age: N/A Location: N/A Country: N/A Date: N/A ``` Question: How come changing the weight of a pendulum does not affect the number of times it swings in 20 seconds? Replies: Sam - It is because a larger pendulum mass requires a greater force to move (or accelerate) it. The amount of force is proportional to the mass. In other words, if the mass is twice a big, it requires twice the force to accelerate it at a given rate... but the force is just that... twice as great. (Gravity pulls harder on a larger mass.) It will always move from the top of the swing to the low point in the same time. Larry Krengel A pendulum is like a falling body. Galileo showed that the speed of heavy objects is the same as light ones (except for light ones that are slowed by air resistance, which is not the case for simple pendulums with reasonably heavy bobs). Since the motion of a pendulum is essentially an object falling toward the center of motion and masses fall at the same rate, the swing time of the pendulum does not depend on mass. Nevertheless there is one very interesting problem associated with a pendulum. In order for the swing time to be independent of mass the ratio of the inertial mass (the measure of resistance that the bob has to acceleration) to the gravitational mass (the measure of the gravitational force) has to be the same for all materials. So far no one has been able to show that there is a change of ratio of inertial to gravitational mass if one changes the bob material. If there is a change it would have to be less than one part in 100 billion. This observation was important in the development of Einstein's general theory of relativity. David S. Kupperman Sam, The driving force for a pendulum is gravity. If the pendulum has twice the mass, gravity pulls twice as hard. Mass is also how hard an object resists the forces it feels. It is much more difficult to start motion for a bowling ball than for a ping-pong ball. Now put these together. A pendulum with twice the mass feels twice the pull, but also has twice the resistance to that pull. These two effects balance out. A pendulum with twice the mass still experiences the same effect. The mass of a pendulum does not affect how it moves. Dr. Ken Mellendorf Physics Instructor Illinois Central College There is a story that Galileo Galilei sat in Church at the Cathedral of Pisa asking himself the same question. According to one story he noticed that the chandelier in the cathedral was swinging, but what attracted his attention was that it was swinging in time to the music that was being played. Since the music was played to a strict rhythm, was the chandelier swinging to a strict rhythm? From about 1602, he began serious study of the properties of pendula (plural of pendulum). His studies showed that the period of the pendulum (that is : the time it takes to swing from left, to right and back to its start position ) was determined by the length of the pendulum, and not by the mass (or weight) of the lump at the end (which is usually called a bob) He also proposed that the period of the pendulum was not affected by the amplitude (size) of the swing, although it has since been shown that this is true for smaller amplitudes, but not for very large ones. In other words, once the length of the string was set, the timing of the pendulum was set no matter how much the bob weighed, or (within reasonable limits) how far the pendulum was set swinging at the start, or how little it moved at the end. This was very useful as a timing device, and has been used in clocks of various sorts since soon after Galileo's discoveries. Curiously, while the mass of the bob has no effect, it turns out that the force of gravity does. On the moon, not only would the bob weigh less, but it would also swing more slowly. Now, to finally answer your question WHY? - the formula for calculating the period of the pendulum is T = 2 pi * Square root of l/g where pi is the constant 3.14159.... G is gravity and l is the length of the pendulum. You will notice that mass is not included in the equation, because it has no effect. The force required to MOVE the mass - (its momentum) and the force provided by gravity (its weight) are both determined my the mass of the object, so they cancel each other out and that is why there is no effect. Nigel Skelton It is because the force accelerating the pendulum comes from its weight. The more massive the pendulum is, the greater its weight, in strict proportion. The inertia, that is, the resistance of the pendulum to acceleration by a force, is also in strict proportion to its mass. Since both the force and inertia vary in strict proportion to the mass, the two effects cancel out and the acceleration is independent of mass. In other words: The more massive something is, the more force is required to make it accelerate at a given rate. The more massive something is, the more force gravity exerts on it. The net result is that acceleration from the force of gravity dies not change as an object's mass changes! Richard Barrans, Ph.D., M.Ed. Department of Physics and Astronomy University of Wyoming Click here to return to the Physics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs