# What is Pendulum: Definition + 1000 Threads

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.
From the first scientific investigations of the pendulum around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s. The pendulum clock invented by Christiaan Huygens in 1658 became the world's standard timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geo-physical surveys, and even as a standard of length. The word "pendulum" is new Latin, from the Latin pendulus, meaning 'hanging'.

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1. ### To what extent does this system behave as a pendulum?

In the formula above I have that the mechanical momentum of the horizontal force with respect to ##C## is always ##0## because the point of application coincides with the pole. Also, the mechanical momentum of the costraint reactions is ##0## because the costraint is smooth so the reaction is...
2. ### Calculating torque on a pendulum

Let ##m_{r}=1## kg be the mass of the rod and ##m_{s}=0.5## kg be the mass of the sphere. ## \tau = -rFsin\theta ## ## = -r([m_{r}+m_{s}]g)sin\theta ## ## =-1.3(1.5)(9.8)sin30 ## ## \tau = -9.6 ## My book's answer key disagrees and my initial thoughts are that maybe the mass in my...
3. ### How to parametrize motion of a pendulum in terms of Cartesian coordinates?

Let the origin be where the pendulum string is attached to the ceiling. $$\sin{\theta(t)}=\frac{x(t)}{L}\tag{1}$$ $$\cos{\theta(t)}=\frac{y(t)}{L}\tag{2}$$ $$\theta(t)=\sin^{-1}{\frac{x(t)}{L}}\tag{3}$$ $$\dot{\theta}(t)=\frac{\dot{x}(t)}{\sqrt{L^2-x^2(t)}}\tag{4}$$...
4. ### Is the line integral of tangential force in pendulum equal to the negative of change in potential energy?

Consider a simple pendulum as depicted below Consider the integral $$\int \vec{F_g}\cdot d\vec{r}$$ My question is if we can equate this to the negative of a change in a potential energy function, ie ##-\Delta U##? Since ##F_g## is conservative, by the 2nd fundamental theorem of calculus...
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### Double pendulum Lagrangian using small angle approximation formula

For this part (b) of this problem, From (a), we know that ##\mathcal{L}\left(\phi_{1}, \phi_{2}, \dot{\phi}_{1}, \dot{\phi}_{2}\right)=\frac{1}{2} m \ell^{2}\left[2 \dot{\phi}_{1}^{2}+\dot{\phi}_{2}^{2}+2 \cos \left(\phi_{1}-\phi_{2}\right) \dot{\phi}_{1} \dot{\phi}_{2}\right]+m g \ell\left(2...
6. ### A Modeling the damping of a physical rigid pendulum

I'm engaged in a research project focused on pendulums. I'm trying to model a rigid pendulum's motion with a second order differential equation (where time is the independent variable) describing the relationship between θ (theta) , ω (omega) and α (alpha) where: - θ is the angle of the...
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### Pendulum on moving train

The problem and solution are, However, I don't understand why the answer is correct. I think that time should be dilated since ##\Delta t = γ \Delta t_0 = 2γ## where ##γ \geq 1## for ##v \geq 0##. Does anybody please know what I'm doing wrong here? Thanks!
8. ### B Sign conventions in torque and non-uniform circular motion

Sorry for the overly general title but my problem is regarding a specific problem: find the net force on the bob of a pendulum as a function of ##\theta##, the angle it makes with the vertical (assuming the observer is stationary with respect to point from which the string is hung and the...
9. ### Max Impulse on a pendulum

TL;DR Summary: An impulse is given to the pendulum so that it moves in 3 dimensions. What equations apply throughout its motion? A particle of mass ##m## is suspended from a string of length ##\ell##. The string is then deflected at an angle ## \theta ##, where the particle and string are in...
10. M

### Planar pendulum with rotating pivot

For this problem, My working for finding the coordinates of the mass is, ##x = x_p + x_m = R\cos(\omega t) + l\sin(\phi)## ##y = -y_p - y_m = -R\sin(\omega t) -l\cos\phi## However, I am told that correct coordinates of the mass is ##x = x_p + x_m = R\cos(\omega t) + l\sin(\phi)## ##y = y_p -...
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### Spherical pendulum confusion [Issue resolved]

For this problem, I am confused my what they mean by ##\phi##. I have looked at the figure, but it is confusing. Makes it look like the x-axis and y-axis are not perpendicular, even thought I'm assuming they are since this is a right handed coordinate system. Does someone please know what...
12. M

### Simple pendulum with moving support

For this problem, The correct coordinates are, However, I am confused how they got them. So here is my initial diagram. I assume that the point on the vertical circle is rotating counterclockwise, that is, it is rotating from the x-axis to the y-axis. Thus ## \omega t > 0## for the point...
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### Pendulum attached to a rotating vertical disk

For this problem, I correctly got the same coordinates for the pendulum mass using another coordinate system. The coordinate system I used was the other coordinate system rotated counterclockwise by 90 degrees. Why is the pendulum mass coordinates invariant in my cartesian coordinate system...

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29. ### B Why don't two double pendulum apparatus follow the same paths? / Chaos

All double pendulum have same inital position, start from same height at same time, this softwear must have some equations from which calculate their path, so if both use same eqution and same inital position, why they have different paths(this is mathematicaly impossible)? One question about...
30. ### Pendulum SHO but with extra downward acceleration of the pivot

Hey guys, Can someone help me understand how to understand this problem intuitively please? How I understand is that I need to look the acceleration relative to the lift as if it were f.e. on another planet with a different acceleration. this gives me a = g - 5. But then again if I didn't look...
31. ### I Effect of Nearby Mountain on an Ideal Pendulum

Suppose there is a very large mountain adjacent to a pendulum such that there is a horizontal component gravitational force of ##10^{-5}g## acting on the otherwise ideal pendulum. How would one use a perturbation to add that effect to first order? My initial thought would be to figure an angle...
32. ### I A pendulum with viscous friction

Consider the standard pendulum with a weightless rod of length b and a mass point m and mg is applied. In the hinge there is a torque of viscous friction which is proportional ##\omega^2##. Now release the pendulum from the horizontal position. What biggest height does the point m attain after...
33. ### Physics IA: Distance B/w Pendulum & Alum. Block, Finding Damping Coefficient

TL;DR Summary: I am doing an experiment for my Physics IA and don't know the theory behind it I am working on a Physics experiment for my school where I vary the distance between a simple pendulum and an aluminium block, and get the damping coefficient for each distance. Below are the images...
34. ### I Relevant/irrelevant clocks for experimental tests of relativity

Recently, in this forum, highly respected members referred to clocks like pendulum and hourglass as if they are relevant for relativity. Are they really? Besides the lack of accuracy, they depend on acceleration/gravity, so they would not work at all in inertial frames and they could not...
35. ### Why Won't a Disk Rotate on a Frictionless Bearing?

I was able to solve first part I.e. time period of the system when bearing has friction I am unable to figure it out why disk will not rotate when it is mounted to frictionless bearing ? I know that due to absence of friction disk cannot rotate but then Mg is also there which can rotate the...
36. ### Angular Velocity of a Large Pendulum on Earth as seen from the stars

I don't understand the question. how am I supposed to find the magnitudes and directions of the velocity from the figure?
37. ### I On whether the motion of a Foucault pendulum bob is comparable to ballistics

A recurring question is: while the motion of a polar Foucault pendulum is fairly straightforward, the case of a non-polar Foucault pendulum is quite difficult to visualize. In 2020, on physics stackexchange someone submitted that question and I contributed an answer. In a comment to another...
38. ### Help with Inverted Pendulum on Cart EoM

Hi All, My goal is to relearn some control theory and implement a working inverted pendulum on a cart with an industrial linear motor. See video: Working through an example of an inverted pendulum on a cart posted here...
39. ### Torque calculation for a compound pendulum

I need to write an equation for Newton's second law for the above system, where k1=k2 (both springs are the same). The red line represents a bar with m=2kg, l=2m. I know that I*α = M1 + M2 + M3 If we displace the bar by x, we get the angle of displacement theta. M1=M2=-k*x I know that...
40. ### Fictitious forces on a rigid body

I was confused by how to work this problem in a rotating frame. The solution read that the centrifugal force on the mass should be of magnitude 𝑚𝑤𝑅^2. However, I thought it would be 𝑚𝑤𝐿^2 where L is the distance between the mass and the center of the circle (L = l + R). What am I missing here?
41. ### Period of a simple pendulum with a magnet under it

Hi ... I have answered this question and I think that F/mg equals 3. But I've asked it from someone and he told me that F/mg is 4. Can someone help me find out which one is correct ??? My answer :
42. ### How Does a Moving Support Affect Pendulum Motion?

In problem 1 I assumed that the suspension moves up and down in an oscillating manner, so ##y_0(t)=Acos(\omega t)## I am not quite sure about task 2, but I would say you can remove the motion of the suspension and the motion of the system would not change noticeably, as I assume that the...
43. ### Pendulum Question (Matlab Coursework)

Apologies for my lack of knowledge on the equations front, I have burnt by brain out on this and haven't the capacity to learn LaTex right now! So here's a screengrab of it: So, This is a Matlab coursework and I am struggling to work out how best to approach solving it. What I have so far is...
44. ### Physics project confusion (effects of length on a pendulum)

We are seeking to design a project where we use a simple pendulum and a motion sensor (that will give us velocity) in order to study centripetal acceleration by essentially changing the length of the pendulum for each trial. This felt simple enough, however our professor insists that we would...
45. ### I Alternative Approach to Solving Foucault's Pendulum Behavior

In many standard texts the behavior of Foucault’s pendulum is solved by adding the Coriolis force term to equation of motion and deriving two coupled differential equations. Here’s an alternative approach: 4 assumptions are made: Mathematical pendulum (point mass attached to massless rigid...
46. ### B Calculating g with a Conical Pendulum

In analysing the conical pendulum, it can be shown that the period is given by T=2pi.sqrt(L.cos(phi)/g) and that therefore, g = 4.pi^2.L.(cos(phi)/T^2). L = pendulum length, phi is measured at the top of the pendulum (at the point of suspension). Graphing cos(phi) vs T^2 should produce...

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48. ### For a Pendulum: Knowing Acceleration Find Maximum Angle

PS: By the way today I had exams in Physics and this problem was the first one I had to solve :p (unlucky) The question was to find the maximum angle θ that the pendulum can reach if we know that the magnitude of the acceleration is the same when the mass is located in the highest and the lowest...
49. ### I Pendulum Tension Force -- How to calculate the full vector?

Hello! I'm trying to understand how this pendulum works. I found this video that explains how to calculate the T force from the rope. He uses the preservation of kinetic and potential energy in order to find the magnitude of the velocity and then using Newton's second law, he calculates the T...
50. ### Is the Classical Pendulum Formula Still Accurate for Large Theta Values?

Summary: Hi, I'm trying to solve this problem, if it's not right then please help me with a hint without solving it. This formula is just an approximation for small values of theta, but if Vo was greater than the denominator this will lead to large values of theta and then this solution is not...