# How does a large-angle pendulum oscillate?

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• Peter Jones
In summary, the conversation discusses the behavior of a pendulum with no air resistance, specifically when the angle of oscillation is not small. According to energy conservation, the pendulum will still oscillate, but it is no longer considered simple harmonic motion. Instead, it exhibits non-linear behavior and its period is dependent on the amplitude of oscillation. This type of oscillation is known as the cycloidal pendulum and can be solved using elliptic functions.
Peter Jones
So in high school i studied the small oscillation of a simple pendulum with no air resistence. It reaches harmonic oscillation when the angle is small enough, so it is an approximation right? But what happens if the angle isn't small, will it still oscillate and how? According to energy conservation, it is supposed to. I came across this problem in a Mechanics book from Landau, and the result of cycle is a function of the amplitude of the oscillations. My question is that if this is still considered as a form of oscillation, what type is it?
Here's the picture of the problem from the book.

Delta2
It is still considered an oscillaion as Landau states in his Problem statement. It is no longer considered simple harmonic oscillation. It is non-linear in nature. As the problem solution indicates the period takes longer than for simple harmonic motion.

Hamiltonian
For small oscillations the period does not depend on the amplitude;
for non small oscillations it does
for the cycloidal pendulum the period does not depend on the amplitude even for non small oscillations

Haborix and vanhees71
wrobel said:
for the cycloidal pendulum the period does not depend on the amplitude even for non small oscillations
What @wrobel says is true regarding the cycloidal pendulum, but it requires a pair of shoulders to constrain the motion.

There is an interesting problem. The total energy of the pendulum is
$$H=\frac{1}{2}\dot\varphi^2-k\cos\varphi,\quad k>0.$$
Let ##\tau(h)## be a period of the trajectory on the energy level ##H=h##.
Find an asymptotic formula
$$\tau(h)\sim \,?\quad\mbox{as}\quad 1)\,h\to k+,\quad 2) \,h\to k-$$

Last edited:
Peter Jones said:
But what happens if the angle isn't small, will it still oscillate and how?
If the deflection angle is π it might get stuck, but otherwise it will oscillate.

Or it turns around. The solution are of course elliptic functions. There's a plethora of all kinds of approximations for such problems in the literature. Nowadays, I think it's no problem to just use elliptic functions, because they are as easily available via numerics as are the socalled "elementary functions".

## 1. What is a large-angle pendulum?

A large-angle pendulum is a type of pendulum that swings in a wide arc, typically greater than 10 degrees. It is different from a small-angle pendulum, which swings in a much smaller arc and follows the laws of simple harmonic motion.

## 2. How does a large-angle pendulum differ from a small-angle pendulum?

A large-angle pendulum experiences non-linear motion, meaning its period and amplitude are not constant. It also has a more complex equation of motion compared to a small-angle pendulum, which follows a sinusoidal pattern.

## 3. What factors affect the oscillation of a large-angle pendulum?

The oscillation of a large-angle pendulum is affected by its length, mass, and initial angle of release. The force of gravity and air resistance also play a role in its motion.

## 4. How does the amplitude of a large-angle pendulum change over time?

The amplitude of a large-angle pendulum decreases over time due to the effects of air resistance and friction. This is known as damping and can be seen in the gradual decrease in the pendulum's swing.

## 5. What is the significance of studying large-angle pendulum oscillation?

Studying large-angle pendulum oscillation allows scientists to understand the principles of non-linear motion and how it differs from simple harmonic motion. It also has practical applications in fields such as engineering and physics, where pendulums are used in various devices and experiments.

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