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timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s.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'.
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...
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...
Let the origin be where the pendulum string is attached to the ceiling.
$$\sin{\theta(t)}=\frac{x(t)}{L}\tag{1}$$
$$\cos{\theta(t)}=\frac{y(t)}{L}\tag{2}$$
$$\theta(t)=\sin^{-1}{\frac{x(t)}{L}}\tag{3}$$
$$\dot{\theta}(t)=\frac{\dot{x}(t)}{\sqrt{L^2-x^2(t)}}\tag{4}$$...
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...
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...
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...
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!
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...
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...
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 -...
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...
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...
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...
Hello. This is the figure of the problem:
First, we should determine the Lagrangian of the system. I have already completed this part without any issues. To respect everyone’s time, I won’t go into the details of how I accomplished it.
$$L=\dfrac {M+m}{2}\dot x^2+ml\dot x \dot \theta \cos...
Pretty much what the title says. Everything that I can find online addresses the pendulum problem with the assumption that the rope is under tension to start. What if the rope is slack to start? Will there be an increase in force applied to the pivot point, say if a mass is dropped off a...
I'm the technical director for a theatre group and am looking at the best way to get an actor to use a rope to swing on stage. As we rent the theatre we use I need to run everything by their staff, who doesn't have a solid grasp of mechanics and physics. I proposed that we lower a pipe from...
I tried using the formulas x=xi+vit and y=yi+voyt-1/2g(t^2)
I assumed voy would be 0 and I almost arrive to the answer but idk how to get rid of the negative
Hi all,
This is a question that has been bothering me and applies to kinetic sculptures.
Let's say you have a wheel. The wheel is situated in the vertical plane (like a clock hanging on the wall). For the sake of easy math, let's say the wheel has a radius of 1m, and has the entirety of it's...
I have a problem with the method that they solved. This is what I mean ##\delta t= \frac{\pi L\alpha \delta T}{\sqrt{gL}}##. You can derive this equation by using errors and approximations here delta t is a tiny(not infinitesimal) change in time period, delta T is a tiny change in temperature...
I put the answer as (IV) but that happens to be wrong (or maybe it was only one of the multiple correct answers). Here is my reasoning:
I. the force is dependent on mass, but isn't always constant.
II. It's not always in the same direction, it points towards the rest point. Consider a point at...
Started by analyzing the change in energy from the initial position to the final position which gives us mgh=1/2mv^2
Since we are trying to find speed, we rearrange the equation to solve for v, which gives us √2gL.
My question is, do we need to take a component of L for √2gL because it is at...
While not exactly correct, we will continue to use Newtonian gravitational force and tension force in the lab frame. We will not concern ourselves with GR, besides the approximation is reasonable for low velocity and small mass.
In the lab frame, the forces acting on the pendulum is weight and...
Consider a pendulum swinging at the North Pole in an inertial frame of reference xyz with the x and y axes in the plane tangent to the pole and the suspension placed at elevation z = l. There is no induction \mathbf{B} and in the chosen system the plane of oscillation xz of the pendulum is...
Hi,
I'm working on a simple benchmark problem for FEA. It's a pendulum initially positioned at an angle of ##45^{\circ}## and then subjected to gravity. I'm interested in the maximum velocity (when the pendulum is in the vertical position). So far, I've been using this formula: $$v=\omega \cdot...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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?
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 :
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...
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...
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...
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...
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...
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...
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...
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...