Pendulum Motion: Difference between revisions

Latest revision as of 18:20, 4 December 2015

created by: Mary Briatta

The pendulum is a mass hanging from a string of negligible mass that is fixed to a point. The equilibrium position of the pendulum is the position when the string and mass hang vertically downward. When pulled back away from this equilibrium state, the string and mass will swing back and forth. If there is no friction or air resistance applied then the pendulum will swing forever.

The Main Idea

Properties of Pendulum Motion

Animation of a pendulum showing forces acting on the mass: the tension T in the rod and the gravitational force mg. Link

A pendulum is just a mass hanging from a spring moving back and forth. In order to describe a pendulum, you need to understand its properties and parameters. There is a string with length L, a mass m hanging from the string, and it is pulled away from its equilibrium, there is an angle θmeasured off the vertical. The two forces acting on the pendulum when it is pulled away from its equilibrium are the string tension, Ft and the gravity, F = mg.

As long as the initial angle or amplitude of the swing is small, the major factor involved in the equations for calculating the frequency of a pendulum is the length of the rod or wire. Once you know the length of the pendulum, you can determine its frequency and speed.

Period of Oscillation

The period of oscillation of the pendulum, T, is defined in terms of the acceleration due to gravity, g, and the length of the pendulum, L, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ₀, called the amplitude

[math]\displaystyle{ T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 \, }[/math]

This is a pretty scary-looking equation, but there’s really only one thing you need to gather from it: the longer the pendulum rope, the longer it will take for the pendulum to oscillate back and forth. You should also note that the mass of the pendulum bob and the angle of displacement play no role in determining the period of oscillation.

Energy

As long as there is no friction applied, mechanical energy in a pendulum is conserved. Total mechanical energy is a combination of gravitational potential energy kinetic energy. As the pendulum swings back and forth, there is a constant exchange between kinetic energy and gravitational potential energy. The potential energy of a pendulum, mgh, increases with the height of the weight as it swings. The image below shows that the potential energy is minimized at the equilibrium and is maximized at θ = ±θ_max The kinetic energy and velocity of the pendulum are maximized at the equilibrium point and minimized when θ = ±θ_max

In the figure above, v is velocity, F_r is the restoring force, F_t is the tension in the pendulum string, U_s is potential energy, and KE is kinetic energy.

Velocity

The velocity can be approximated from the gravity equation for a weight dropping from a height.

In order to calculate the the velocity of the pendulum mass at the equilibrium position, you have arrange the coordinate system so that the height of the mass at the equilibrium position is equal to zero. From this we can use the total mechanical energy which is equal to the kinetic energy at the equilibrium point where U = 0. This total mechanical energy also equals the total potential energy at where KE = 0. By putting these equations together we get:

But what is height and how do we find it? The height is determined by the angle from the vertical that is the starting point of the pendulum's swing.

From the figure, we see that If we plug that value into the equation above, we can solve for v:

Examples

Simple

Middling

Difficult

History

The first to study the properties of the pendulum was Italian scientist Galileo Galilei around 1602 which was was recorded in an extant report of his research in a letter to Guido Ubaldo dal Monte from Padua. It was said by his biographer and student, Vincenzo Viviani, that the swinging motion of a chandelier in the Pisa cathedral was what initiated his interest in the topic around 1582. The isochronism states that is a period of the pendulum is approximately independent of the amplitude or width of the swing. Galileo discovered this crucial property which helped him figure out that makes pendulums useful timekeepers. From his research he determined that the period is independent of the mass and proportional to the square root of the length of the pendulum. In 1641 Galileo created a design for a pendulum clock and left it for his son Vincenzo to construct. The pendulum was the first harmonic oscillator used by man.

Connectedness

As you see in the pendulum history and usage in time telling, the pendulum motion can be seen in our everyday life. As an architecture major, I have always been interested art and design. Different sculptures, installations, and art pieces can even be achieved throughout the application of a pendulum motion. Take a look here and here and here

@@ Line 1: / Line 1: @@
---[[User:Mbriatta3|mbriatta3]] ([[User talk:Mbriatta3|talk]]) 14:17, 3 December 2015 (EST)
+created by: Mary Briatta
 The pendulum is a mass hanging from a string of negligible mass that is fixed to a point. The equilibrium position of the pendulum is the position when the string and mass hang vertically downward. When pulled back away from this equilibrium state, the string and mass will swing back and forth. If there is no friction or air resistance applied then the pendulum will swing forever.
@@ Line 13: / Line 13: @@
   gravitational force mg. [https://en.wikipedia.org/wiki/Pendulum Link]]]
-A pendulum is just a mass hanging from a spring moving back and forth. In order to describe a pendulum, you need to understand its properties and parameters. There is a string with length '''L''', a mass '''''m''''' hanging from the string, and it is pulled away from its equilibrium, there is an angle measured off the vertical. The two forces acting on the pendulum when it is pulled away from its equilibrium are the string tension, '''Ft''' and the gravity, '''F = mg'''.
+A pendulum is just a mass hanging from a spring moving back and forth. In order to describe a pendulum, you need to understand its properties and parameters. There is a string with length '''L''', a mass '''''m''''' hanging from the string, and it is pulled away from its equilibrium, there is an angle '''θ'''measured off the vertical. The two forces acting on the pendulum when it is pulled away from its equilibrium are the string tension, '''Ft''' and the gravity, '''F = mg'''.
 [[File:Mary2.gif|center|]]
+As long as the initial angle or amplitude of the swing is small, the major factor involved in the equations for calculating the frequency of a pendulum is the length of the rod or wire. Once you know the length of the pendulum, you can determine its frequency and speed.
 ===Period of Oscillation===
@@ Line 27: / Line 29: @@
 ===Energy===
-The mechanical energy of the pendulum is a conserved quantity. The potential energy of the pendulum, mgh, increases with the height of the bob; therefore the potential energy is minimized at the equilibrium point and is maximized at . Conversely, the kinetic energy and velocity of the pendulum are maximized at the equilibrium point and minimized when .
-The figure below summarizes this information in a qualitative manner, which is the manner in which you are most likely to find it on SAT II Physics. In this figure, v signifies velocity,  signifies the restoring force,  signifies the tension in the pendulum string, U signifies potential energy, and KE signifies kinetic energy.
+As long as there is no friction applied, mechanical energy in a pendulum is conserved. Total mechanical energy is a combination of gravitational potential energy kinetic energy.  As the pendulum swings back and forth, there is a constant exchange between kinetic energy and gravitational potential energy. The potential energy of a pendulum, '''mgh''', increases with the height of the weight as it swings. The image below shows that the potential energy is minimized at the equilibrium and is maximized at ''θ = &plusmn;θ<sub>max</sub>'' The kinetic energy and velocity of the pendulum are maximized at the equilibrium point and minimized when ''θ = &plusmn;θ<sub>max</sub>''
 [[Image:mary1.png|center|]]
+In the figure above,  '''v''' is velocity,  '''F<sub>r</sub>''' is the restoring force, '''F<sub>t</sub>''' is the tension in the pendulum string, '''U<sub>s</sub>''' is potential energy, and '''KE''' is kinetic energy.
+[[File:mary12.gif|frame|center|[http://www.physicsclassroom.com/mmedia/energy/pe.cfm Energy Transformation for a Pendulum]]]
 ===Velocity===
@@ Line 37: / Line 43: @@
   velocity and acceleration vectors. [https://en.wikipedia.org/wiki/Pendulum Link]]]
-Calculating the velocity of the pendulum bob at the equilibrium position requires that we arrange our coordinate system so that the height of the bob at the equilibrium position is zero. Then the total mechanical energy is equal to the kinetic energy at the equilibrium point where U = 0. The total mechanical energy is also equal to the total potential energy at  where KE = 0. Putting these equalities together, we get
+The velocity can be approximated from the gravity equation for a weight dropping from a height.
+In order to calculate the the velocity of the pendulum mass at the equilibrium position, you have arrange the coordinate system so that the height of the mass at the equilibrium position is equal to zero. From this we can use the total mechanical energy which is equal to the kinetic energy at the equilibrium point where U = 0.
+This total mechanical energy also equals the total potential energy at where KE = 0. By putting these equations together we get:
 [[File:mary6.gif|center|]]
-But what is h?
+But what is height and how do we find it? The height is determined by the angle from the vertical that is the starting point of the pendulum's swing.
 [[File:mary5.gif|center|]]
 From the figure, we see that [[File:mary4.gif]]
 If we plug that value into the equation above, we can solve for v: [[File:mary3.gif]]
@@ Line 47: / Line 59: @@
 ===Simple===
+[[File:Mary8.jpg|1000px]]
 ===Middling===
-===Difficult===
-==Connectedness==
+[[File:Mary9.jpg|1000px]]
-In 1656 the Dutch scientist Christiaan Huygens built the first pendulum clock.[34] This was a great improvement over existing mechanical clocks; their best accuracy was increased from around 15 minutes deviation a day to around 15 seconds a day.[35] Pendulums spread over Europe as existing clocks were retrofitted with them.[36]
+===Difficult===
-The English scientist Robert Hooke studied the conical pendulum around 1666, consisting of a pendulum that is free to swing in two dimensions, with the bob rotating in a circle or ellipse.[37] He used the motions of this device as a model to analyze the orbital motions of the planets.[38] Hooke suggested to Isaac Newton in 1679 that the components of orbital motion consisted of inertial motion along a tangent direction plus an attractive motion in the radial direction. This played a part in Newton's formulation of the law of universal gravitation.[39][40] Robert Hooke was also responsible for suggesting as early as 1666 that the pendulum could be used to measure the force of gravity.[37]
+[[File:Mary10.jpg|1000px]]
-During his expedition to Cayenne, French Guiana in 1671, Jean Richer found that a pendulum clock was 2 1⁄2 minutes per day slower at Cayenne than at Paris. From this he deduced that the force of gravity was lower at Cayenne.[41][42] In 1687, Isaac Newton in Principia Mathematica showed that this was because the Earth was not a true sphere but slightly oblate (flattened at the poles) from the effect of centrifugal force due to its rotation, causing gravity to increase with latitude.[43] Portable pendulums began to be taken on voyages to distant lands, as precision gravimeters to measure the acceleration of gravity at different points on Earth, eventually resulting in accurate models of the shape of the Earth.[44]
+==History==
-As you see, the pendulum motion can be seen in our everyday life. As an architecture major, I have always been interested art and design. Different sculptures, installations, and art pieces can even be achieved throughout the application of a pendulum motion. Take a look [https://www.youtube.com/watch?v=3rigdgiHmnQ here] and [https://www.youtube.com/watch?v=9Ba-KNFEdfY here] and [https://www.youtube.com/watch?v=fk0OLIPZ3GE here]
+[[File: download.jpeg|thumb|]]
+The first to study the properties of the pendulum was Italian scientist Galileo Galilei around 1602 which was was recorded in an extant report of his research in  a letter to Guido Ubaldo dal Monte from Padua. It was said by his biographer and student, Vincenzo Viviani, that the swinging motion of a chandelier in the Pisa cathedral was what initiated his interest in the topic around 1582. The isochronism  states that is a period of the pendulum is approximately independent of the amplitude or width of the swing. Galileo discovered this crucial property which helped him figure out that makes pendulums useful timekeepers. From his research he determined that the period is independent of the mass and proportional to the square root of the length of the pendulum. In 1641 Galileo created a design for a pendulum clock and left it for his son Vincenzo to construct. The pendulum was the first harmonic oscillator used by man.
-==History==
+==Connectedness==
+[[File:Pend1.png|thumb| The first pendulum clock [http://www.torgoen.com/blog/wp-content/uploads/2014/11/pend1.png Link]]]
-Italian scientist Galileo Galilei was the first to study the properties of pendulums, beginning around 1602.[28] The earliest extant report of his research is contained in a letter to Guido Ubaldo dal Monte, from Padua, dated November 29, 1602.[29] His biographer and student, Vincenzo Viviani, claimed his interest had been sparked around 1582 by the swinging motion of a chandelier in the Pisa cathedral.[30][31] Galileo discovered the crucial property that makes pendulums useful as timekeepers, called isochronism; the period of the pendulum is approximately independent of the amplitude or width of the swing.[32] He also found that the period is independent of the mass of the bob, and proportional to the square root of the length of the pendulum. He first employed freeswinging pendulums in simple timing applications. His physician friend, Santorio Santorii, invented a device which measured a patient's pulse by the length of a pendulum; the pulsilogium.[28] In 1641 Galileo conceived and dictated to his son Vincenzo a design for a pendulum clock;[32] Vincenzo began construction, but had not completed it when he died in 1649.[33] The pendulum was the first harmonic oscillator used by man.[32]
+As you see in the pendulum history and usage in time telling, the pendulum motion can be seen in our everyday life. As an architecture major, I have always been interested art and design. Different sculptures, installations, and art pieces can even be achieved throughout the application of a pendulum motion. Take a look [https://www.youtube.com/watch?v=3rigdgiHmnQ here] and [https://www.youtube.com/watch?v=9Ba-KNFEdfY here] and [https://www.youtube.com/watch?v=fk0OLIPZ3GE here]
 == See also ==
@@ Line 80: / Line 96: @@
 *Michael R. Matthews, Colin Gauld and Arthur Stinner (2005) The Pendulum: Its Place in Science, Culture and Pedagogy. Science & Education, 13, 261-277.
 *Schlomo Silbermann,(2014) "Pendulum Fundamental; The Path Of Nowhere" (Book)
-*Matthys, Robert J. (2004). Accurate Pendulum Clocks. UK: Oxford Univ. Press. ISBN 0-19-852971-6.
 *Nelson, Robert; M. G. Olsson (February 1986). "The pendulum – Rich physics from a simple system". American Journal of Physics 54 (2): 112–121. Bibcode:1986AmJPh..54..112N. doi:10.1119/1.14703.
 *L. P. Pook (2011). Understanding Pendulums: A Brief Introduction (Springer).
-===External links===
-[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
 ==References==
@@ Line 92: / Line 104: @@
 *"Pendulum Motion." Pendulum Motion. Physics Classroom, 1996. Web. 03 Dec. 2015.
 *"Pendulum." Wikipedia. Wikimedia Foundation, n.d. Web. 03 Dec. 2015.
+*"Pendulums." SparkNotes. SparkNotes, 2011. Web. 03 Dec. 2015.
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