a compound pendulum is a physical system with a more complex structure than a simple pendulum, incorporating its mass distribution and dimensions into its oscillatory motion around a fixed axis. Understanding its dynamics involves principles of rotational mechanics and the interplay between gravitational potential energy and kinetic energy. Compound pendulums are used in various scientific and engineering applications, such as seismology for measuring earthquakes, in clocks to maintain accurate timekeeping, and in mechanical systems to study oscillatory motion dynamics.
1. The document discusses Newton's laws of motion and gravity and how they can be used to derive Kepler's laws of planetary motion.
2. It shows that assuming a central force directed at a central body, the conservation of angular momentum implies that equal areas are swept out in equal times, in accordance with Kepler's second law.
3. By applying Newton's gravity equation and conservation of angular momentum to the Lagrangian of a central-force problem, the document derives that the differential equation governing planetary orbits is identical in form to that of an ellipse, explaining Kepler's first law that planets follow elliptical orbits.
1. The document describes 5 experiments related to mechanical oscillations and vibrations. Experiment 1 verifies the relation between the period of a simple pendulum and its length. Experiment 2 determines the radius of gyration of a compound pendulum.
2. Experiment 3 uses a bifilar suspension to determine the radius of gyration of a bar. Experiment 4 studies longitudinal vibrations of a helical spring and determines the frequency theoretically and experimentally.
3. Experiment 5 examines vibrations of a system with springs in series. Each experiment involves setting up the apparatus, collecting observations in tables, performing calculations, and drawing conclusions by comparing experimental and theoretical results.
Modulus of rigidity is an elastic constant. that measures the elastic behavior of a material when it is twisted or sheared.
torsion pendulum consists of a weight of given shape hanged to the ceiling through a metallic wire, on twisting this load through an angle and left, it starts to oscillate, back and forth, The number of oscillations per unit time is measured.
In this article the theory and experimental procedure of the measurement of modulus rigidity of the given metallic wire is presented by using the principle of torsional pendulum
Introduction to oscillations and simple harmonic motion
Physics presentation about Simple Harmonic Motion of Hooke's Law springs and pendulums with derivation of formulas and connections to Uniform Circular Motion.
References include links to illustrative youtube clips and other powerpoints that contributed to this peresentation.
Torsional vibrations and buckling of thin WALLED BEAMS
The document discusses the torsional vibrations and buckling of thin-walled beams on elastic foundation using a dynamic stiffness matrix method. It develops analytical equations to model the behavior of clamped-simply supported beams under an axial load and resting on an elastic foundation. Numerical results are presented for natural frequencies and buckling loads for different values of warping and foundation parameters. The dynamic stiffness matrix approach can accurately analyze beams with non-uniform cross-sections and complex boundary conditions.
1) The document presents three theoretical physics problems involving spinning balls, charged particles in loops, and laser cooling of atoms.
2) Problem 1 considers a spinning ball falling and rebounding, calculating the rebound angle, horizontal distance traveled, and minimum spin rate. Problem 2 analyzes relativistic effects on charged particles in a moving loop in an electric field.
3) Problem 3 describes using lasers to cool atoms by resonant absorption. It calculates the laser frequency needed, velocity range absorbed, direction change upon emission, maximum velocity decrease, number of absorption events to slow to zero velocity, and distance traveled during cooling.
Fourier Series for Continuous Time & Discrete Time Signals
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
This document discusses several key concepts related to open and cross belt drives:
1) It defines the variables used to calculate the total length of a belt, including the radii of the pulleys and the distance between their centers.
2) It provides equations to calculate the total belt length based on the arcs subtended by the pulleys.
3) It also discusses how power is transmitted by a belt drive based on the tensions on the tight and slack sides of the belt.
A Route to Chaos for the Physical Double Pendulum by
This document summarizes research examining the route to chaos for the physical double pendulum. The Lagrangian and Hamiltonian are derived for the physical double pendulum system. Poincare sections show that as the parameter epsilon increases, the quasi-periodic tori in phase space collapse into points, indicating periodic motion of the bottom pendulum every n oscillations. Bifurcation diagrams also show splitting corresponding to period n orbits just before the onset of global chaos. The research demonstrates that the physical double pendulum exhibits ordered periodic motion just prior to becoming fully chaotic.
Chapter 9 tutorial exercises with solutions 19 02 2013
1. The document describes a system consisting of two blocks placed at the ends of a horizontal massless board that rests on an axis of rotation. The moment of inertia of the system is given as 12 kg·m2.
2. It then presents a problem involving a wrecking ball supported by a boom. Forces acting on the system include the weight of the wrecking ball, the weight of the boom, and the tension in a support cable.
3. The document solves the problems by applying principles of equilibrium, including analyzing torque equations and using trigonometric relationships between forces and angles. Solutions are obtained for requested values like tension in the support cable and magnitude of the force on the lower end of the boom
Why Does the Atmosphere Rotate? Trajectory of a desorbed molecule
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Kármán Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
This document summarizes a physics lecture on rotational equilibrium and rotational dynamics. It discusses rotational kinetic energy, moment of inertia, torque, and Newton's second law for rotational motion. Key points include:
- Rotational kinetic energy is analogous to linear kinetic energy but depends on the moment of inertia and angular velocity.
- Moment of inertia depends on both the mass and its distribution from the axis of rotation. It can be calculated for point masses, composite particles, and extended objects.
- Torque is a measure of the effectiveness of a force in producing angular acceleration. It depends on the magnitude of force, distance from the rotation axis, and angle of application.
- Newton's second law for rotation states
This document summarizes a physics lecture on rotational equilibrium and rotational dynamics. It discusses rotational kinetic energy, moment of inertia, torque, and Newton's second law for rotational motion. Key points include:
- Rotational kinetic energy is analogous to linear kinetic energy but depends on the moment of inertia and angular velocity.
- Moment of inertia depends on both the mass and its distribution from the axis of rotation. It can be calculated for point masses, composite particles, and extended objects.
- Torque is a measure of the effectiveness of a force in producing angular acceleration. It depends on the magnitude of force, distance from the rotation axis, and angle of application.
- Newton's second law for rotation states
NASA’s new campaign of lunar exploration will see astronauts visiting sites of scientific or strategic
interest across the lunar surface, with a particular focus on the lunar South Pole region.[1] After landing
crew and cargo at these destinations, local mobility around landing sites will be key to movement of
cargo, logistics, science payloads, and more to maximize exploration returns.
NASA’s Moon to Mars Architecture Definition Document (ADD)[2] articulates the work needed to achieve
the agency’s human lunar exploration objectives by decomposing needs into use cases and functions.
Ongoing analysis of lunar exploration needs reveals demands that will drive future concepts and elements.
Recent analysis of integrated surface operations has shown that the transportation of cargo on the
surface from points of delivery to points of use will be particularly important. Exploration systems will
often need to support deployment of cargo in close proximity to other surface infrastructure. This cargo
can range from the crew logistics and consumables described in the 2023 “Lunar Logistics Drivers and
Needs” white paper,[3] to science and technology demonstrations, to large-scale infrastructure that
requires precision relocation.
1) The document is an open letter from Le Van Cuong to scientists and professors suggesting they correct Einstein's theory of special relativity before teaching it to students.
2) Cuong argues that Einstein's postulate that the speed of light is constant is incorrect and has confused science for over a century. He believes light speed depends on the motion of the light source.
3) Cuong provides a proof using equations and diagrams showing that when an object emitting light is in motion, the observed speed of the light (c') is not actually equal to the constant speed of light (c) and is given by c' = (c^2 + v^2)^1/2, where v is the velocity
[1] A simple pendulum consists of a bob suspended by a light, inextensible thread from a fixed point that allows it to oscillate in a vertical plane along an arc. [2] Simple pendulums exhibit simple harmonic motion (SHM) because the acceleration is directly proportional to the displacement from the equilibrium position. [3] The time period of a simple pendulum depends only on the length of the thread and acceleration due to gravity according to the equation T = 2π(l/g)^1/2.
The document describes several theoretical physics problems involving mechanics, thermodynamics, and radioactivity dating.
Problem 1 describes a bungee jumper attached to an elastic rope, deriving expressions for the distance dropped before coming to rest, maximum speed, and time taken.
Problem B involves a heat engine operating between two bodies at different temperatures, deriving an expression for the final temperature if maximum work is extracted, and using this to find the maximum work.
Problem C uses radioactive decay of uranium isotopes to date the age of the Earth, deriving equations relating isotope ratios to time and obtaining an approximate age of 5.38 billion years.
1. The document discusses Newton's laws of motion and gravity and how they can be used to derive Kepler's laws of planetary motion.
2. It shows that assuming a central force directed at a central body, the conservation of angular momentum implies that equal areas are swept out in equal times, in accordance with Kepler's second law.
3. By applying Newton's gravity equation and conservation of angular momentum to the Lagrangian of a central-force problem, the document derives that the differential equation governing planetary orbits is identical in form to that of an ellipse, explaining Kepler's first law that planets follow elliptical orbits.
1. The document describes 5 experiments related to mechanical oscillations and vibrations. Experiment 1 verifies the relation between the period of a simple pendulum and its length. Experiment 2 determines the radius of gyration of a compound pendulum.
2. Experiment 3 uses a bifilar suspension to determine the radius of gyration of a bar. Experiment 4 studies longitudinal vibrations of a helical spring and determines the frequency theoretically and experimentally.
3. Experiment 5 examines vibrations of a system with springs in series. Each experiment involves setting up the apparatus, collecting observations in tables, performing calculations, and drawing conclusions by comparing experimental and theoretical results.
Modulus of rigidity is an elastic constant. that measures the elastic behavior of a material when it is twisted or sheared.
torsion pendulum consists of a weight of given shape hanged to the ceiling through a metallic wire, on twisting this load through an angle and left, it starts to oscillate, back and forth, The number of oscillations per unit time is measured.
In this article the theory and experimental procedure of the measurement of modulus rigidity of the given metallic wire is presented by using the principle of torsional pendulum
Introduction to oscillations and simple harmonic motionMichael Marty
Physics presentation about Simple Harmonic Motion of Hooke's Law springs and pendulums with derivation of formulas and connections to Uniform Circular Motion.
References include links to illustrative youtube clips and other powerpoints that contributed to this peresentation.
Torsional vibrations and buckling of thin WALLED BEAMSSRINIVASULU N V
The document discusses the torsional vibrations and buckling of thin-walled beams on elastic foundation using a dynamic stiffness matrix method. It develops analytical equations to model the behavior of clamped-simply supported beams under an axial load and resting on an elastic foundation. Numerical results are presented for natural frequencies and buckling loads for different values of warping and foundation parameters. The dynamic stiffness matrix approach can accurately analyze beams with non-uniform cross-sections and complex boundary conditions.
1) The document presents three theoretical physics problems involving spinning balls, charged particles in loops, and laser cooling of atoms.
2) Problem 1 considers a spinning ball falling and rebounding, calculating the rebound angle, horizontal distance traveled, and minimum spin rate. Problem 2 analyzes relativistic effects on charged particles in a moving loop in an electric field.
3) Problem 3 describes using lasers to cool atoms by resonant absorption. It calculates the laser frequency needed, velocity range absorbed, direction change upon emission, maximum velocity decrease, number of absorption events to slow to zero velocity, and distance traveled during cooling.
Fourier Series for Continuous Time & Discrete Time SignalsJayanshu Gundaniya
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
This document discusses several key concepts related to open and cross belt drives:
1) It defines the variables used to calculate the total length of a belt, including the radii of the pulleys and the distance between their centers.
2) It provides equations to calculate the total belt length based on the arcs subtended by the pulleys.
3) It also discusses how power is transmitted by a belt drive based on the tensions on the tight and slack sides of the belt.
A Route to Chaos for the Physical Double Pendulum by Daniel Berkowitz
This document summarizes research examining the route to chaos for the physical double pendulum. The Lagrangian and Hamiltonian are derived for the physical double pendulum system. Poincare sections show that as the parameter epsilon increases, the quasi-periodic tori in phase space collapse into points, indicating periodic motion of the bottom pendulum every n oscillations. Bifurcation diagrams also show splitting corresponding to period n orbits just before the onset of global chaos. The research demonstrates that the physical double pendulum exhibits ordered periodic motion just prior to becoming fully chaotic.
Chapter 9 tutorial exercises with solutions 19 02 2013TRL4EVER
1. The document describes a system consisting of two blocks placed at the ends of a horizontal massless board that rests on an axis of rotation. The moment of inertia of the system is given as 12 kg·m2.
2. It then presents a problem involving a wrecking ball supported by a boom. Forces acting on the system include the weight of the wrecking ball, the weight of the boom, and the tension in a support cable.
3. The document solves the problems by applying principles of equilibrium, including analyzing torque equations and using trigonometric relationships between forces and angles. Solutions are obtained for requested values like tension in the support cable and magnitude of the force on the lower end of the boom
Why Does the Atmosphere Rotate? Trajectory of a desorbed moleculeJames Smith
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the Kármán Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
This document summarizes a physics lecture on rotational equilibrium and rotational dynamics. It discusses rotational kinetic energy, moment of inertia, torque, and Newton's second law for rotational motion. Key points include:
- Rotational kinetic energy is analogous to linear kinetic energy but depends on the moment of inertia and angular velocity.
- Moment of inertia depends on both the mass and its distribution from the axis of rotation. It can be calculated for point masses, composite particles, and extended objects.
- Torque is a measure of the effectiveness of a force in producing angular acceleration. It depends on the magnitude of force, distance from the rotation axis, and angle of application.
- Newton's second law for rotation states
This document summarizes a physics lecture on rotational equilibrium and rotational dynamics. It discusses rotational kinetic energy, moment of inertia, torque, and Newton's second law for rotational motion. Key points include:
- Rotational kinetic energy is analogous to linear kinetic energy but depends on the moment of inertia and angular velocity.
- Moment of inertia depends on both the mass and its distribution from the axis of rotation. It can be calculated for point masses, composite particles, and extended objects.
- Torque is a measure of the effectiveness of a force in producing angular acceleration. It depends on the magnitude of force, distance from the rotation axis, and angle of application.
- Newton's second law for rotation states
Similar to Mechanics:- Simple and Compound Pendulum (19)
Lunar Mobility Drivers and Needs - ArtemisSérgio Sacani
NASA’s new campaign of lunar exploration will see astronauts visiting sites of scientific or strategic
interest across the lunar surface, with a particular focus on the lunar South Pole region.[1] After landing
crew and cargo at these destinations, local mobility around landing sites will be key to movement of
cargo, logistics, science payloads, and more to maximize exploration returns.
NASA’s Moon to Mars Architecture Definition Document (ADD)[2] articulates the work needed to achieve
the agency’s human lunar exploration objectives by decomposing needs into use cases and functions.
Ongoing analysis of lunar exploration needs reveals demands that will drive future concepts and elements.
Recent analysis of integrated surface operations has shown that the transportation of cargo on the
surface from points of delivery to points of use will be particularly important. Exploration systems will
often need to support deployment of cargo in close proximity to other surface infrastructure. This cargo
can range from the crew logistics and consumables described in the 2023 “Lunar Logistics Drivers and
Needs” white paper,[3] to science and technology demonstrations, to large-scale infrastructure that
requires precision relocation.
TOPIC: INTRODUCTION TO FORENSIC SCIENCE.pptximansiipandeyy
This presentation, "Introduction to Forensic Science," offers a basic understanding of forensic science, including its history, why it's needed, and its main goals. It covers how forensic science helps solve crimes and its importance in the justice system. By the end, you'll have a clear idea of what forensic science is and why it's essential.
ScieNCE grade 08 Lesson 1 and 2 NLC.pptxJoanaBanasen1
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A slightly oblate dark matter halo revealed by a retrograde precessing Galact...Sérgio Sacani
The shape of the dark matter (DM) halo is key to understanding the
hierarchical formation of the Galaxy. Despite extensive eforts in recent
decades, however, its shape remains a matter of debate, with suggestions
ranging from strongly oblate to prolate. Here, we present a new constraint
on its present shape by directly measuring the evolution of the Galactic
disk warp with time, as traced by accurate distance estimates and precise
age determinations for about 2,600 classical Cepheids. We show that the
Galactic warp is mildly precessing in a retrograde direction at a rate of
ω = −2.1 ± 0.5 (statistical) ± 0.6 (systematic) km s−1 kpc−1 for the outer disk
over the Galactocentric radius [7.5, 25] kpc, decreasing with radius. This
constrains the shape of the DM halo to be slightly oblate with a fattening
(minor axis to major axis ratio) in the range 0.84 ≤ qΦ ≤ 0.96. Given the
young nature of the disk warp traced by Cepheids (less than 200 Myr), our
approach directly measures the shape of the present-day DM halo. This
measurement, combined with other measurements from older tracers,
could provide vital constraints on the evolution of the DM halo and the
assembly history of the Galaxy.
Possible Anthropogenic Contributions to the LAMP-observed Surficial Icy Regol...Sérgio Sacani
This work assesses the potential of midsized and large human landing systems to deliver water from their exhaust
plumes to cold traps within lunar polar craters. It has been estimated that a total of between 2 and 60 T of surficial
water was sensed by the Lunar Reconnaissance Orbiter Lyman Alpha Mapping Project on the floors of the larger
permanently shadowed south polar craters. This intrinsic surficial water sensed in the far-ultraviolet is thought to be
in the form of a 0.3%–2% icy regolith in the top few hundred nanometers of the surface. We find that the six past
Apollo Lunar Module midlatitude landings could contribute no more than 0.36 T of water mass to this existing,
intrinsic surficial water in permanently shadowed regions (PSRs). However, we find that the Starship landing
plume has the potential, in some cases, to deliver over 10 T of water to the PSRs, which is a substantial fraction
(possibly >20%) of the existing intrinsic surficial water mass. This anthropogenic contribution could possibly
overlay and mix with the naturally occurring icy regolith at the uppermost surface. A possible consequence is that
the origin of the intrinsic surficial icy regolith, which is still undetermined, could be lost as it mixes with the
extrinsic anthropogenic contribution. We suggest that existing and future orbital and landed assets be used to
examine the effect of polar landers on the cold traps within PSRs
Collaborative Team Recommendation for Skilled Users: Objectives, Techniques, ...Hossein Fani
Collaborative team recommendation involves selecting users with certain skills to form a team who will, more likely than not, accomplish a complex task successfully. To automate the traditionally tedious and error-prone manual process of team formation, researchers from several scientific spheres have proposed methods to tackle the problem. In this tutorial, while providing a taxonomy of team recommendation works based on their algorithmic approaches to model skilled users in collaborative teams, we perform a comprehensive and hands-on study of the graph-based approaches that comprise the mainstream in this field, then cover the neural team recommenders as the cutting-edge class of approaches. Further, we provide unifying definitions, formulations, and evaluation schema. Last, we introduce details of training strategies, benchmarking datasets, and open-source tools, along with directions for future works.
Deploying DAPHNE Computational Intelligence on EuroHPC Vega for Benchmarking ...University of Maribor
Slides from talk:
Aleš Zamuda, Mark Dokter:
Deploying DAPHNE Computational Intelligence on EuroHPC Vega for Benchmarking Randomised Optimisation Algorithms.
2024 International Conference on Broadband Communications for Next Generation Networks and Multimedia Applications (CoBCom), 9--11 July 2024, Graz, Austria
https://www.cobcom.tugraz.at/
Dalghren, Thorne and Stebbins System of Classification of AngiospermsGurjant Singh
The Dahlgren, Thorne, and Stebbins system of classification is a modern method for categorizing angiosperms (flowering plants) based on phylogenetic relationships. Developed by botanists Rolf Dahlgren, Robert Thorne, and G. Ledyard Stebbins, this system emphasizes evolutionary relationships and incorporates extensive morphological and molecular data. It aims to provide a more accurate reflection of the genetic and evolutionary connections among angiosperm families and orders, facilitating a better understanding of plant diversity and evolution. This classification system is a valuable tool for botanists, researchers, and horticulturists in studying and organizing the vast diversity of flowering plants.
El Nuevo Cohete Ariane de la Agencia Espacial Europea-6_Media-Kit_english.pdfChamps Elysee Roldan
Europe must have autonomous access to space to realise its ambitions on the world stage and
promote knowledge and prosperity.
Space is a natural extension of our home planet and forms an integral part of the infrastructure
that is vital to daily life on Earth. Europe must assert its rightful place in space to ensure its
citizens thrive.
As the world’s second-largest economy, Europe must ensure it has secure and autonomous access to
space, so it does not depend on the capabilities and priorities of other nations.
Europe’s longstanding expertise in launching spacecraft and satellites has been a driving force behind
its 60 years of successful space cooperation.
In a world where everyday life – from connectivity to navigation, climate and weather – relies on
space, the ability to launch independently is more important than ever before. With the launch of
Ariane 6, Europe is not just sending a rocket into the sky, we are asserting our place among the
world’s spacefaring nations.
ESA’s Ariane 6 rocket succeeds Ariane 5, the most dependable and competitive launcher for decades.
The first Ariane rocket was launched in 1979 from Europe’s Spaceport in French Guiana and Ariane 6 will continue the adventure.
Putting Europe at the forefront of space transportation for nearly 45 years, Ariane is a triumph of engineering and the prize of great European industrial and political
cooperation. Ariane 1 gave way to more powerful versions 2, 3 and 4. Ariane 5 served as one of the world’s premier heavy-lift rockets, putting single or multiple
payloads into orbit – the cargo and instruments being launched – and sent a series of iconic scientific missions to deep space.
The decision to start developing Ariane 6 was taken in 2014 to respond to the continued need to have independent access to space, while offering efficient
commercial launch services in a fast-changing market.
ESA, with its Member States and industrial partners led by ArianeGroup, is developing new technologies for new markets with Ariane 6. The versatility of Ariane 6
adds a whole new dimension to its very successful predecessors
SCIENTIFIC INVESTIGATIONS – THE IMPORTANCE OF FAIR TESTING.pptxJoanaBanasen1
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1. Compound Pendulum
# Topics
a) Time period of compound pendulum
b) Maximum and Minimum Time periods of a compound pendulum
2. Simple Pendulum
A pendulum is suspended body oscillating under the
influence of the gravitational force acting on it. This is the
most accurate method for determining of acceleration
due to gravity.
The simple pendulum whose period of oscillation is,
g
l
T
2
where, g = acceleration due to gravity
l = is the length of pendulum
3. Compound Pendulum
The real (or rigid body) pendulum which can be of any shape object or a rigid body
capable of oscillating in a vertical plane freely about a horizontal axis passing
through any point of it is called physical pendulum or compound pendulum.
The moment (or torque) of a force about a
turning point is the force multiplied by the
perpendicular distance to the force from the
turning point.
Moments are measured in newton metres
(Nm).
Moment = F d
F is the force in Newton
d is the perpendicular distance
Example: A 10N force acts at a perpendicular distance of 0.50m from the turning point.
What is the moment of the force?
Moment = F . d
= 10 x 0.50
= 5.0 Nm
4. A couple is two equal forces which act in opposite directs
on an object but not through the same point so they
produce a turning effect.
The moment (or torque) of a couple is calculated by
multiplying the size of one of the force (F) by the
perpendicular distance between the two forces (s).
E.g. a steering wheel in a car;
Moment of Couple = F . s
6. Equate equation (1) and (2), we get
This is the differential equation of angular SHM showing that the motion of the compound
pendulum is simple harmonic. The time period of oscillation is given by,
mgl
I
T
T
2
2
0
---------------------(3)
---------------------(4)
7. Let Ig be the moment of inertia of the rigid body about an axis parallel to the axis of
suspension but passing through its centre of gravity G. Then from the parallel axis
theorem, we can write,
I = Ig + ml2
If k be the radius of gyration of the rigid body about an axis passing through the centre of
gravity G, then Ig = mk2 . Thus we have,
I = mk2 + ml2 = m(k2 + l2 )
Hence, the time period becomes,
mgl
l
k
m
T
)
(
2
2
2
gl
l
k
T
)
(
2
2
2
---------------------(5)
The moment of inertia about any axis parallel to that axis through the
center of mass
the moment of inertia about a parallel axis is the center of mass
moment plus the moment of inertia of the entire object treated as a
point mass at the center of mass.
8. Length of Equivalent Simple Pendulum
mgl
l
k
m
T
)
(
2
2
2
The time period of compound pendulum is,
g
l
l
k
gl
l
k
T
2
2
2
2
)
(
2
Comparing this time period of a compound pendulum with the time period of an
ideal simple pendulum of length L
g
L
T
e
i
2
.
.
l
l
k
L
2
if we use above equation, then two time periods would be the same.
This length L of a compound pendulum is known as the length of equivalent simple pendulum.
If the whole mass of the compound pendulum is concentrated at a point C
at a distance from the point of suspension, as shown in Fig.
l
l
k
L
2
This pendulum behave like a simple pendulum of same period.
The point C is then the centre of oscillation.
9. Maximum and Minimum Time Periods of a Compound Pendulum
gl
l
k
T
2
2
2
The time period of compound pendulum is,
Squaring on both sides, we get,
l
l
k
g
T
2
2
2
2 4
l
k
l
g
T
2
2
2 4
Differentiating the above eqn. with respect to l,
we get,
2
2
2
1
4
2
l
k
g
dl
dT
T
(a) If l = 0, the R.H.S. of eqn. (2) becomes
infinite. Hence,
------------(1)
------------(2)
The time period of the pendulum
gradually increases as the point of
suspension is shifted slowly towards the
centre of mass.
The time period of the pendulum is
maximum when the axis of suspension
passes through the centre of gravity of
the pendulum.
Hence the time period of the pendulum
is minimum, when the distance between
the centre of suspension and centre of
gravity (c.g.) becomes equal the radius of
gyration of the pendulum about the
horizontal axis passing through c.g., we
then have,
(b) If l = k, the R.H.S. of eqn. (2) is zero.
gl
l
k
T
2
2
min 2
gl
l
l
T
2
2
2
g
l
gl
l
T
2
2
2
2
2
10. Centre of suspension & centre of oscillation are mutually interchangeable
The point O through which the horizontal axis of suspension passes is known as the centre
of suspension. The length of equivalent simple pendulum is,
l
l
k
L
2
If the line OG is extended up to a point C, then the point C is called the
centre of oscillation. Thus, OC = OG + GC
l
l
k
L
2
Let the distance of the c.g. from the centre of oscillation C = l′. Then we have,
l
k
l
2
'
2
' k
ll
If the pendulum is
suspended about the point
O, then the time period
becomes,
gl
l
k
T
2
2
2
gl
l
ll
T
2
'
2
g
l
l
T
'
2
Again, when the pendulum
is inverted and suspended
about the point C, then the
time period T′ is,
𝑇′ = 2𝜋
𝑘2 + 𝑙′2
𝑔𝑙′
𝑇′ = 2𝜋
𝑙𝑙′ + 𝑙′2
𝑔𝑙′
g
l
l
T
'
2
'
T
T
'
-------(1)
-------(2)
Eqn. (1) & (2)
11. Thus, the centre of suspension and the centre of oscillation are
interchangeable. If the distance between the centre of suspension and the
centre of oscillation is L and knowing the time period about either of them,
we have the value of g as follows,
g
L
T
2
2
2
4
T
L
g
------------------------------------------(3)
12. Reversible Compound Pendulum
A compound pendulum with two knife edges (i.e., the point of suspension and point of
oscillation) so placed that the periods of oscillation when suspended from either is the
same is called a reversible compound pendulum.
The time period of the compound
pendulum from one side is,
gl
l
k
T
2
2
2
squaring on both sides,
gl
l
k
T
)
(
4
2
2
2
2
)
(
4 2
2
2
2
l
k
g
l
T
and the time period of the compound
pendulum from other side is,
'
'
2
'
2
2
gl
l
k
T
------------(1)
------------(2)
squaring on both sides,
'
)
'
(
4
'
2
2
2
2
gl
l
k
T
)
'
(
4
'
' 2
2
2
2
l
k
g
l
T
------------(3)
------------(4)
Let us put T′ = T, and subtract eqn. (4) from eqn. (2), we get,
)
'
(
4
)
'
( 2
2
2
2
l
l
g
l
l
T
13. Continued…… If l ≠ l′, then
)
'
(
4
)
'
(
)
'
)(
'
(
4 2
2
2
l
l
g
l
l
l
l
l
l
g
T
g
l
l
T
'
2
2
2
)
'
(
4
T
l
l
g
or ------------(5)
The distance between any two points on opposite sides of G and at unequal distances
from G, the time periods about them being exactly equal, then the distance (l + l’) is
equal to the length of the equivalent simple pendulum and is given by
L = l + l′
Experimentally, we can measure the values of l, l′ and the time period T = T′. Hence, the
acceleration due to gravity can be found using eqn. (5).
)
'
(
4
)
'
( 2
2
2
2
l
l
g
l
l
T
14. In actual practice, it is extremely difficult to find the positions of the axes for the time
period T and T′ to be exactly equal.
Continued……
According to Friedrich Wilhelm Bessel, it is found that, if the two periods are nearly equal is
also sufficient.
Then the time periods of the compound pendulum of both sides (after squaring) using eqns.
(1) and (3) can be written as,
gl
l
k
T
)
(
4
2
2
2
2
------------(1)
'
)
'
(
4
2
2
2
2
gl
l
k
T
------------(3)
2
2
2
2
4
l
k
glT
2
2
2
2
'
4
'
l
k
glT
On subtracting, above equations, we get,
2
2
2
2
2
'
]
'
'
[
4
l
l
T
l
lT
g
2
2
2
2
2
'
'
'
4
l
l
T
l
lT
g
------------(6) ------------(7)
)
'
)(
'
(
'
'
4 2
2
2
l
l
l
l
T
l
lT
g
------------(8)
15. Continued…… It can be solved by using partial fractions as follow,
)
'
(
)
'
(
4 2
l
l
B
l
l
A
g
where A and B are undetermined constants.
)
'
)(
'
(
)
'
(
)
'
(
4 2
l
l
l
l
l
l
B
l
l
A
g
)
'
)(
'
(
)
(
'
)
(
4 2
l
l
l
l
B
A
l
B
A
l
g
Comparing the coefficient of l and l′ of eqns. (8) and (10), we get,
------------(9)
------------(10)
)
'
)(
'
(
'
'
4 2
2
2
l
l
l
l
T
l
lT
g
------(8)
A + B = T2 and A – B = T′2
2
'
&
2
' 2
2
2
2
T
T
B
T
T
A
Substituting the values of A and B in eqn. (9), we get,
)
'
(
2
'
)
'
(
2
'
4 2
2
2
2
2
l
l
T
T
l
l
T
T
g
)
'
(
2
'
)
'
(
2
'
4
2
2
2
2
2
l
l
T
T
l
l
T
T
g
------------(11)
16. Continued……
Equation (11) is called Bessel’s formula which is used to
determine the acceleration due to gravity ‘g’ using reversible
compound pendulum.
)
'
(
2
'
)
'
(
2
'
4
2
2
2
2
2
l
l
T
T
l
l
T
T
g
------------(11)
As T = T′, (T2 – T′2) → 0 but (l – l′) is appreciable. As the second term in the
denominator of eqn. (11) is very small as compared to the first term, hence ignoring
the second term.
It is enough to determine the position of centre of gravity by balancing the pendulum
on a horizontal knife edge and measuring l and l′. Thus,
2
2
2
'
)
'
(
8
T
T
l
l
g
------------(12)
17. Kater’s Reversible Pendulum is the compound pendulum,
and this is used to give an accurate value of g.
The principle is based on reversibility of centre of
suspension and centre of oscillation. It is therefore called
Kater’s reversible pendulum.
It consists of a metal rod of uniform cross section, a heavy
cylindrical weight W1 can be
Fixed near its one end so that the centre of mass of the rod
is shifted towards this end.
Two weights W1 and W2, one heavy of metal and others
light of wood can be made to slide along the length of the
bar and clamped at any position.
Two knife edges k1 and k2 facing each other may be adjusted
near the two edges as shown in Fig.
Kater’s Reversible Pendulum
Kater’s Pendulum
18. Kater’s Reversible Pendulum
Continued……
The pendulum is suspended with a knife edge k1 on
the horizontal rigid support and time period T is
measured.
The pendulum is reversed and allowed to oscillate
about the knife edge k2. The time period T′ are again
measured.
The mass W1 is adjusted up or downwards to make
the time periods from the two knife edges nearly equal.
Fix up this mass W1, the mass W2 is moved by means
of a screw attached with it to make the final adjustment
to equality of the time periods.
The rod is then balanced on a sharp knife edge
horizontally and its centre of mass is marked.
Distance of two knife edges from centre of mass give l
and l′.
Knowing l, l′, T and T′ we can calculate the value of g
using Bessel’s formula.
2
2
2
'
)
'
(
8
T
T
l
l
g
19. Advantages of a Compound Pendulum Over a Simple Pendulum
(i) In compound pendulum the length being the distance
between the knife edges can be accurately measured.
In the simple pendulum since the point of suspension
and the centre of mass of the bob are both indefinite,
thus the effective length of the pendulum cannot be
measured acurately.
(i) In compound pendulum, being of large mass, can
oscillate for a very long time before coming to rest and
hence the period of oscillation can be determined with
great accuracy of measuring the time for 100 to 200
oscillations. In a simple pendulum the oscillation die
out very soon on account of small mass of the bob and
the accuracy is limited.
20. Moment of inertia ( I ) is defined as the
sum of the products of the mass of each
particle of the body and square of its
perpendicular distance from the axis. It is
also known as rotational inertia.
The moment of inertia reflects the
mass distribution of a body or a system
of rotating particles, with respect to an
axis of rotation.
The moment of inertia only depends
on the geometry of the body and the
position of the axis of rotation, but it
does not depend on the forces involved
in the movement.
1. A thin uniform bar, one meter long is allowed to oscillate under the influence of
gravity about a horizontal axis passing through its one end. Calculate: (a) the length
of equivalent simple pendulum, (b) the period of oscillation of the bar and (c) its
angular velocity.
Solution:
21. M.I. of uniform bar about the axis passing through its one end and perpendicular
to its length is given by,
12
12
2
2
L
k
Mk
ML
I
(a) The length of equivalent simple pendulum is,
l
L
l
l
k
l
L
12
'
2
2
12
L
k
In this case, the distance between centre of suspension and centre of gravity
m
L 6667
.
0
5
.
0
12
)
1
(
5
.
0
'
2
m
L
l 5
.
0
2
(b) The period of oscillation of the bar,
.
sec
634
.
1
8
.
9
6667
.
0
14
.
3
2
'
2
2
2
g
L
g
l
k
l
T
s
rad
T
/
845
.
3
634
.
1
14
.
3
2
2
(c) The angular velocity of the bar is,
22. 2. A uniform circular disc of radius R oscillates in a vertical plane about a horizontal
axis. Find the distance of the axis of rotation from the centre for which the period is
minimum. Also evaluate the value of this period.
Assume circular disc acts as a compound pendulum, the time period is given by,
Solution:
g
l
k
l
T
2
2
where, l is the distance between the centre of suspension and the centre of mass of the disc, and
k is the radius of gyration. The period of a compound pendulum is minimum when l = k
g
k
T
2
2
min
For the disc, the momentum of inertia about an axis parallel to the axis of suspension and
passing through its centre of mass, i.e., about its own axis is given by,
2
2
2
1
Mk
MR
I
For example, consider a solid disk with radius = RR and total mass = mm.
2
R
k
24. Q.1. Define compound pendulum. Show that a compound pendulum executives SHM.
Find its periodic time.
Q.1.Explain the concept of length of equivalent simple pendulum.
Q.2.Derive the condition for maximum and minimum time period of the compound
pendulum.
Q.3. Define centre of suspension and centre of oscillation. Show that in compound
pendulum they are interchangeable.
Q.2.Show that in reversible compound pendulum the Bessel’s formula to calculate the
acceleration due to gravity g is given by,
Q.4. Explain Kater’s reversible pendulum to measure the accurate value of acceleration
due to gravity g.
Long answer questions
Short answer questions