How Did The Change In The Weight Of The Bob Affect The Resulting Period And Frequency?

Pendulum Motility

A simple pendulum consists of a relatively massive object hung by a string from a stock-still support. It typically hangs vertically in its equilibrium position. The massive object is affectionately referred to equally the pendulum bob. When the bob is displaced from equilibrium and and so released, it begins its back and forth vibration about its stock-still equilibrium position. The movement is regular and repeating, an example of periodic motion. Pendulum move was introduced earlier in this lesson as we made an effort to understand the nature of vibrating objects. Pendulum motion was discussed again every bit we looked at the mathematical properties of objects that are in periodic motion. Here nosotros volition investigate pendulum motion in even greater detail every bit nosotros focus upon how a variety of quantities change over the course of fourth dimension. Such quantities volition include forces, position, velocity and energy - both kinetic and potential energy.

Force Analysis of a Pendulum

Before in this lesson we learned that an object that is vibrating is acted upon by a restoring force. The restoring force causes the vibrating object to wearisome down equally it moves away from the equilibrium position and to speed up equally it approaches the equilibrium position. It is this restoring force that is responsible for the vibration. So what forces act upon a pendulum bob? 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 World's mass attracting the mass of the bob. And there is a tension force acting upwards 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 ever opposes the motion of the bob as it swings to and fro. The air resistance force is relatively weak compared to the ii ascendant forces.

The gravity strength is highly anticipated; information technology is always in the same direction (down) and ever of the same magnitude - mass*nine.8 N/kg. The tension strength is considerably less predictable. Both its management and its magnitude change as the bob swings to and fro. The management of the tension force is always towards the pivot point. So equally the bob swings to the left of its equilibrium position, the tension force is at an angle - directed upwardly and to the correct. And as the bob swings to the correct of its equilibrium position, the tension is directed upwardly and to the left. The diagram below depicts the direction of these two forces at 5 different positions over the form of the pendulum's path.

In physical situations in which the forces acting on an object are not in the same, contrary or perpendicular directions, information technology is customary to resolve one or more of the forces into components. This was the exercise used in the analysis of sign hanging problems and inclined aeroplane problems. Typically one or more of the forces are resolved into perpendicular components that lie forth coordinate axes that are directed in the management of the dispatch or perpendicular to information technology. Then in the example of a pendulum, it is the gravity force which gets resolved since the tension forcefulness 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 centrality organisation is sketched on the diagram and the force of gravity is resolved into ii 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; information technology is labeled Fgrav-perp. You volition notice that the perpendicular component of gravity is in the contrary management of the tension forcefulness. You might besides 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 exist a internet strength which is perpendicular to the arc of the bob's movement. This must be the case since we await that objects that move along round paths will experience an inward or centripetal strength. The tangential component of gravity (Fgrav-tangent) is unbalanced by whatsoever other force. So there is a cyberspace force directed along the other coordinate axes. It is this tangential component of gravity which acts every bit 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 in a higher place 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. Withal the process of resolving gravity into ii components along axes that are perpendicular and tangent to the arc remains the same. The diagram below shows the results of the forcefulness analysis for several other positions.

There are a couple comments to exist made. First, observe the diagram for when the bob is displaced to its maximum displacement to the right of the equilibrium position. This is the position in which the pendulum bob momentarily has a velocity of 0 yard/s and is irresolute its direction. The tension force (Ftens) and the perpendicular component of gravity (Fgrav-perp) rest each other. At this instant in time, there is no net strength directed forth the axis that is perpendicular to the motion. Since the motion of the object is momentarily paused, at that place is no need for a centripetal force.

2nd, observe the diagram for when the bob is at the equilibrium position (the string is completely vertical). When at this position, in that location is no component of force along the tangent direction. When moving through the equilibrium position, the restoring forcefulness is momentarily absent. Having been restored to the equilibrium position, there is no restoring forcefulness. The restoring strength is simply needed when the pendulum bob has been displaced abroad from the equilibrium position. Yous 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 forth a circular arc, there must exist a cyberspace centripetal force at this position.

The Sinusoidal Nature of Pendulum Motion

In the previous part of this lesson, we investigated the sinusoidal nature of the motion of a mass on a spring. We will conduct a similar investigation here for the motility of a pendulum bob. Permit's suppose that we could mensurate the corporeality 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 deportation 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. time plot is shown below. Similar to what was observed for the mass on a leap, the position of the pendulum bob (measured along the arc relative to its balance 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 fourth dimension. As the pendulum bob does the dorsum and forth, the velocity is continuously irresolute. In that location volition 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). And of course there will exist moments in fourth dimension at which the velocity is 0 yard/s. If the variations in velocity over the form of time were plotted, the resulting graph would resemble the i shown beneath.

At present let's effort 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 so 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. At present 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 correct is regarded as a positive position. An assay of the plots shows that the velocity is least when the deportation is greatest. And the velocity is greatest when the displacement of the bob is to the lowest degree. 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 tin can be explained by the fact that as the bob moves away from the equilibrium position, at that place is a restoring force that opposes its move. This force slows the bob down. Then as the bob moves leftward from position D to E to F to G, the force and dispatch is directed rightward and the velocity decreases as information technology moves along the arc from D to M. At 1000 - the maximum displacement to the left - the pendulum bob has a velocity of 0 m/s. You might recollect of the bob every bit being momentarily paused and prepare to modify its management. Next the bob moves rightward forth the arc from G to F to E to D. As it does, the restoring strength is directed to the right in the aforementioned direction as the bob is moving. This forcefulness will accelerate the bob, giving information technology a maximum speed at position D - the equilibrium position. As the bob moves past position D, it is moving rightward along the arc towards C, and so B and then A. Every bit information technology does, in that location 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 strength. Once the bob reaches position A - the maximum deportation to the right - it has attained a velocity of 0 m/s. Once more, the bob'due south velocity is least when the displacement is greatest. The bob completes its cycle, moving leftward from A to B to C to D. Forth this arc from A to D, the restoring force is in the direction of the motion, thus speeding the bob up. Then it would be logical to conclude that as the position decreases (forth the arc from A to D), the 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 blitheness at the correct (used with the permission of Wikimedia Eatables; special thank you 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 find that this vector is entirely tangent to the arc when at maximum deportation; this is consistent with the force assay discussed above. And the vector is vertical (towards the center of the arc) when at the equilibrium position. This likewise is consequent with the force analysis discussed higher up.

Energy Analysis

In a previous affiliate of The Physics Classroom Tutorial, the free energy possessed by a pendulum bob was discussed. We will expand on that discussion here as we brand an endeavour to acquaintance the move characteristics described above with the concepts of kinetic energy, potential energy and total mechanical energy.

The kinetic energy possessed past an object is the energy it possesses due to its motion. It is a quantity that depends upon both mass and speed. The equation that relates kinetic energy (KE) to mass (1000) and speed (v) is

KE = ½•m•vⁱⁱ

The faster an object moves, the more kinetic energy that information technology volition possess. We can combine this concept with the discussion to a higher place most how speed changes during the form 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 farther abroad from the equilibrium position.

The potential energy possessed by an object is the stored energy of position. Ii types of potential energy are discussed in The Physics Classroom Tutorial - gravitational potential energy and rubberband potential energy. Rubberband potential free energy is only present when a spring (or other rubberband medium) is compressed or stretched. A simple pendulum does not consist of a spring. The form of potential free energy possessed past a pendulum bob is gravitational potential energy. The amount of gravitational potential free energy is dependent upon the mass (m) of the object and the height (h) of the object. The equation for gravitational potential free energy (PE) is

PE = grand•g•h

where g represents the gravitational field strength (sometimes referred to as the dispatch acquired by gravity) and has the value of 9.8 Northward/kg.

The height of an object is expressed relative to some arbitrarily assigned zero level. In other words, the top must exist measured as a vertical altitude above some reference position. For a pendulum bob, it is customary to telephone call the lowest position the reference position or the zero level. And so when the bob is at the equilibrium position (the lowest position), its superlative is nothing and its potential energy is 0 J. As the pendulum bob does the dorsum and along, there are times during which the bob is moving away from the equilibrium position. As it does, its height is increasing as it moves further and further abroad. Information technology reaches a maximum tiptop as it reaches the position of maximum displacement from the equilibrium position. As the bob moves towards its equilibrium position, it decreases its pinnacle and decreases its potential free energy.

At present permit'southward put these ii concepts of kinetic free energy and potential free energy together equally nosotros consider the motility of a pendulum bob moving along the arc shown in the diagram at the right. Nosotros will utilize an free energy bar nautical chart to stand for the changes in the two forms of free energy. The amount of each form of energy is represented by a bar. The height of the bar is proportional to the amount of that course of energy. In improver to the potential energy (PE) bar and kinetic energy (KE) bar, at that place is a third bar labeled TME. The TME bar represents the total amount of mechanical energy possessed by the pendulum bob. The total mechanical energy is only the sum of the two forms of energy – kinetic plus potential energy. Take some time to inspect the bar charts shown below for positions A, B, D, F and G. What practise you observe?

When you lot inspect the bar charts, information technology is evident that as the bob moves from A to D, the kinetic energy is increasing and the potential free energy is decreasing. Withal, the total amount of these ii forms of energy is remaining constant. Whatever potential energy is lost in going from position A to position D appears every bit kinetic energy. There is a transformation of potential energy into kinetic free energy equally the bob moves from position A to position D. Withal the total mechanical free energy remains constant. We would say that mechanical energy is conserved. As the bob moves past position D towards position G, the opposite is observed. Kinetic energy decreases equally the bob moves rightward and (more importantly) upwardly toward position G. At that place is an increment in potential energy to accompany this decrease in kinetic energy. Energy is beingness transformed from kinetic form into potential form. Withal, as illustrated by the TME bar, the full corporeality of mechanical free energy is conserved. This very principle of energy conservation was explained in the Energy affiliate of The Physics Classroom Tutorial.

The Period of a Pendulum

Our final discussion will pertain to the period of the pendulum. Equally discussed previously in this lesson, the period is the time it takes for a vibrating object to consummate its cycle. In the case of pendulum, it is the time for the pendulum to start at ane extreme, travel to the opposite extreme, then render to the original location. Here nosotros volition be interested in the question What variables bear upon the flow of a pendulum? We will concern ourselves with possible variables. The variables are the mass of the pendulum bob, the length of the cord on which it hangs, and the angular displacement. The angular displacement or arc bending is the bending that the string makes with the vertical when released from residual. These 3 variables and their effect on the period are easily studied and are frequently the focus of a physics lab in an introductory physics grade. The information tabular array below provides representative data for such a study.

Trial	Mass (kg)	Length (m)	Arc Bending (°)	Menstruation (south)
1	0.02-	0.xl	15.0	1.25
2	0.050	0.40	xv.0	1.29
3	0.100	0.40	fifteen.0	i.28
4	0.200	0.twoscore	15.0	1.24
5	0.500	0.xl	15.0	one.26
6	0.200	0.lx	15.0	i.56
7	0.200	0.80	15.0	i.79
8	0.200	1.00	15.0	ii.01
9	0.200	ane.20	xv.0	2.19
ten	0.200	0.forty	10.0	1.27
11	0.200	0.40	20.0	1.29
12	0.200	0.40	25.0	1.25
13	0.200	0.40	30.0	1.26

In trials 1 through 5, the mass of the bob was systematically contradistinct while keeping the other quantities constant. By and so doing, the experimenters were able to investigate the possible effect of the mass upon the period. Equally tin can be seen in these five trials, alterations in mass have piddling effect upon the period of the pendulum.

In trials 4 and 6-9, the mass is held abiding at 0.200 kg and the arc angle is held constant at 15°. Withal, the length of the pendulum is varied. By and then doing, the experimenters were able to investigate the possible event of the length of the string upon the period. As tin can be seen in these 5 trials, alterations in length definitely have an effect upon the period of the pendulum. Equally the cord is lengthened, the flow of the pendulum is increased. There is a direct relationship between the menstruation and the length.

Finally, the experimenters investigated the possible effect of the arc bending upon the menstruation in trials four and x-13. The mass is held constant at 0.200 kg and the string length is held abiding at 0.400 yard. As tin exist seen from these five trials, alterations in the arc bending take little to no effect upon the menses of the pendulum.

So the conclusion from such an experiment is that the i variable that effects the period of the pendulum is the length of the string. Increases in the length lead to increases in the period. But the investigation doesn't accept to stop there. The quantitative equation relating these variables can be determined if the information is plotted and linear regression analysis is performed. The 2 plots below represent such an analysis. In each plot, values of period (the dependent variable) are placed on the vertical axis. In the plot on the left, the length of the pendulum is placed on the horizontal axis. The shape of the curve indicates some sort of power relationship between catamenia and length. In the plot on the right, the square root of the length of the pendulum (length to the ½ ability) is plotted. The results of the regression analysis are shown.


Slope: 1.7536 Y-intercept: 0.2616 COR: 0.9183	Slope: two.0045 Y-intercept: 0.0077 COR: 0.9999

The analysis shows that there is a better fit of the data and the regression line for the graph on the right. As such, the plot on the right is the footing for the equation relating the period and the length. For this information, the equation is

Period = two.0045•Length^0.v + 0.0077

Using T as the symbol for period and L as the symbol for length, the equation tin can be rewritten as

T = 2.0045•50^0.5 + 0.0077

The commonly reported equation based on theoretical development is

T = 2•Π•(L/g)^0.5

where yard is a constant known as the gravitational field forcefulness or the dispatch of gravity (9.8 Due north/kg). The value of 2.0045 from the experimental investigation agrees well with what would be expected from this theoretically reported equation. Substituting the value of k into this equation, yields a proportionality constant of 2Π/g^0.v, which is 2.0071, very similar to the 2.0045 proportionality constant developed in the experiment.

Investigate!

Employ the Investigating a Pendulum widget beneath to investigate the effect of the pendulum length upon the menses of the pendulum. Merely type in a value of length into the input field and click on the Submit button. Experiment with various values of pendulum length.

Check Your Understanding

1. A pendulum bob is pulled dorsum to position A and released from rest. The bob swings through its usual circular arc and is caught at position C. Determine the position (A, B, C or all the same) where the …

a. … forcefulness of gravity is the greatest?
b. … restoring force is the greatest?
c. … speed is the greatest?
d. … potential energy is the greatest?
e. … kinetic free energy is the greatest
f. … total mechanical energy is the greatest?

2. Utilise energy conservation to fill in the blanks in the following diagram.

iii. A pair of trapeze performers at the circus is swinging from ropes attached to a big elevated platform. Suppose that the performers tin be treated as a uncomplicated pendulum with a length of 16 m. Determine the catamenia for one complete back and along cycle.

4. Which would have the highest frequency of vibration?

Pendulum A: A 200-g mass fastened to a i.0-one thousand length string
Pendulum B: A 400-thousand mass attached to a 0.5-m length string

v. Anna Litical wishes to make a unproblematic pendulum that serves as a timing device. She plans to make information technology such that its menstruum is one.00 second. What length must the pendulum have?