# Pendulem in simple harmonic motion

## Pendulem in simple harmonic motion

In this system, it is called the spring system. Intrigued, Galileo decided to measure how much time it took for each swing, using the only approximately periodic event to which he had ready access: the beating of his own pulse. But do systems execute purely simple harmonic motion in practice? We can learn a lot about the motion just by looking at this case. One more thing you should notice, amplitude does not affect the period of a mass on a spring, and the amplitude, this theta maximum will not affect the period of a pendulum either, as long as your amplitudes are small. So, it's a measure of how sluggish this mass is gonna be to changes in its angular velocity. So, it turns out, it's kinda weird, changing the mass on here does not affect the period at which this swings back and forth. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in [link].

This leaves a net restoring force back toward the equilibrium position at. So, why is this the formula? Gravity's gonna be pulling down and if it pulls down with a greater force, you might think this mass is gonna swing with a greater speed and if it's got a greater speed, it'll complete this cycle in less time because it's moving faster, and since it takes less time, you might think that the period goes down, but these two effects exactly cancel.

In fact, it gets, what physicists call chaotic, which is kind of cool.

## Simple pendulum derivation

So, that's what I wanna talk to you about in this video. Note that the angular amplitude does not appear in the expression for the period. The far more useful and common example of using a variable to describe a pendulum is the angle that the pendulum is at. A stiff spring will have larger k and a soft spring will have small k. But really complicated to describe mathematically. Gravity's gonna be pulling down and if it pulls down with a greater force, you might think this mass is gonna swing with a greater speed and if it's got a greater speed, it'll complete this cycle in less time because it's moving faster, and since it takes less time, you might think that the period goes down, but these two effects exactly cancel. For small displacements, a pendulum is a simple harmonic oscillator. Technically speaking, the simple pendulum is not a perfect simple harmonic oscillator, it's only extremely close to being a simple harmonic oscillator. These systems include- a spring-block system and a simple pendulum. What can we say? Right, this period here, what could we change that would change this period here?

How could you view this sinusoidal motion in a pendulum? When the mass of something goes up, it's more sluggish to accelerations, it's more difficult to move around and change its direction.

So, what does affect the period?

So, bigger g means a bigger force of gravity, pulling downward on this mass, that gives me a larger restoring force. So, how would I apply this equation to this case of a pendulum? So, bigger L gives us bigger moment of inertia.

So, you might say, look, if the length increases, so would the amount of torque. Well, the two pi is just a constant, you get a square root. Show you why it should make sense and hopefully give you a little intuition about why the variables are in here that they are.

Therefore, the mass continues past the equilibrium position, compressing the spring. Using the definition of the angular frequency and the reciprocal relationship between period of time and frequency So the period of the pendulum is proportional to the square root of the length of the pendulum, and inversely proportional to the square root of the acceleration due to gravity.

### Pendulum physics

The linear displacement from equilibrium is , the length of the arc. For any simple mechanical harmonic oscillator: When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium. Using this page, draw a circle on the bowling ball and adjust the threshhold value. Note if the real space and phase space diagram are not co-linear, the phase space motion becomes elliptical. So, this is gonna swing forward and then backward, and then forward and backward. You could have more complicated examples. We've got enough things to study by just studying simple pendulums. Increasing the g, increases the force on the system which gives you a larger acceleration, greater speeds takes less time to go through a period. How could you view this sinusoidal motion in a pendulum? Therefore, a spring system executes simple harmonic motion.

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