# What is the Lagrangian of a simple pendulum?

## What is the Lagrangian of a simple pendulum?

The Lagrangian is L = T −V = m ˙y2/2−mgy, so eq. (6.22) gives ¨y = −g, which is simply the F = ma equation (divided through by m), as expected.

### What are examples of physical pendulum?

A physical pendulum refers to an object which oscillates back and forth, in contrast to the rather idealized simple pendulum where all the mass is concentrated in a single point (usually the mass hanging on the end of the massless rope). One example of a physical pendulum is a baseball bat swinging back and forth.

How do you find the mass of a physical pendulum?

It consists of any rigid body that oscillates about a pivot point. For small amplitudes, the period of a physical pendulum only depends on the moment of inertia of the body around the pivot point and the distance from the pivot to the body’s center of mass. It is calculated as: T=2π√Imgh T = 2 π I mgh .

What is Lagrangian equation of motion?

One of the best known is called Lagrange’s equations. The Lagrangian L is defined as L = T − V, where T is the kinetic energy and V the potential energy of the system in question.

## What is meant by physical pendulum?

: a rigid body so mounted on a horizontal axis through its center of suspension that when the body is displaced it vibrates freely about its position of equilibrium —distinguished from simple pendulum.

### What is the period of a physical pendulum?

The period of a physical pendulum has a period of T = 2π√ImgL. Use the moment of inertia to solve for the length L: T=2π√ImgL=2π√13ML2MgL=2π√L3g;L=3g(T2π)2=3(9.8m/s2)(2s2π)2=2.98m.

What affects the period of a physical pendulum?

The period of a simple pendulum depends on its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass and the maximum displacement.

What forces are acting on a pendulum?

The forces acting on the bob of a pendulum are its weight and the tension of the string. It is useful to analyze the pendulum in the radial/tangential coordinate system. The tension lies completely in the radial direction and the weight must be broken into components.

## Does physical pendulum depend on mass?

The period of a pendulum does not depend on the mass of the ball, but only on the length of the string. With the assumption of small angles, the frequency and period of the pendulum are independent of the initial angular displacement amplitude. A pendulum will have the same period regardless of its initial angle.

### What is period of physical pendulum?

The period of a simple pendulum is T=2π√Lg T = 2 π L g , where L is the length of the string and g is the acceleration due to gravity. The period of a physical pendulum T=2π√ImgL T = 2 π I m g L can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L.

When does SHM occur in a simple pendulum?

SHM in a Pendulum The motion of a simple pendulum is very close to Simple Harmonic Motion (SHM). SHM results whenever a restoring force is proportional to the displacement, a relationship often known as Hooke’s Law when applied to springs.

How is the motion of a simple pendulum related to the displacement?

The motion of a simple pendulum is very close to Simple Harmonic Motion (SHM). SHM results whenever a restoring force is proportional to the displacement, a relationship often known as Hooke’s Law when applied to springs. F = -kx. Where F is the restoring force, k is the spring constant, and x is the displacement.

## How can you see the sinusoidal motion of a pendulum?

You can observe the sinusoidal motion of a pendulum in a Physlet by Andrew Duffy of Boston University. This model for the period of a pendulum only applies for the small angle approximation. As the amplitude becomes greater than 10 degrees, the period deviates from this equation.

### What causes the restoring force on a pendulum?

FORCE ANALYSIS. In this case the tension is always perpendicular to the motion of the pendulum. Thus although the tension changes the direction of the pendulum, it does not change the speed of the pendulum. The restoring force on a pendulum is caused by gravity. Incidentally, because of the pendulum’s own inertia,…