# Double pendulum velocity derivation

• TheFerruccio
In summary, the conversation discusses the process of deriving the velocity of the second mass in a double pendulum example in terms of the angles of the two rods. The first mass velocity is straightforward, while the second mass velocity is expressed as a combination of terms involving the first mass velocity and the cosine of the difference between the two angles. The information is being referenced from a webpage, but the person eventually figures it out on their own.
TheFerruccio
I'm doing a personal project and I'm refreshing my knowledge on Lagrangian mechanics. Right now, I am looking at the double pendulum example. How is the velocity of the 2nd mass derived, in terms of the angles of the two rods? 2nd mass as in, the mass that hangs lowest.

I have not been able to figure out the procedure to doing a derivation. The first mass velocity is relatively straightforward.

I should probably give examples:

I got $$v_1^2$$ rather easily:

$$v_1^2 = l_1^2\dot\theta_1^2$$

But, v2 is expressed as:

$$v_2^2 = l_1^2\dot\theta_1^2+l_2^2\dot\theta_2^2+2l_1l_2\dot\theta_1\dot\theta_2\cos{(\theta_1-\theta_2)}$$

http://scienceworld.wolfram.com/physics/DoublePendulum.html

Last edited:
I figured it out! Never mind!

## 1. What is a double pendulum?

A double pendulum is a physical system that consists of two pendulums attached to each other, with the second pendulum attached to the end of the first one. This creates a more complex motion compared to a single pendulum, with the two pendulums swinging and rotating in different directions.

## 2. How is the velocity of a double pendulum derived?

The velocity of a double pendulum is derived using the principles of classical mechanics, specifically the equations of motion and conservation of energy. The equations are solved using mathematical methods such as Lagrange's equations or Hamiltonian mechanics to determine the velocity of each pendulum at any given time.

## 3. What factors affect the velocity of a double pendulum?

The velocity of a double pendulum is affected by several factors, including the length and mass of each pendulum, the initial conditions (such as the angle and speed of the pendulums), and external forces acting on the system (such as friction and air resistance).

## 4. Why is the velocity of a double pendulum important?

The velocity of a double pendulum is important because it determines the motion and behavior of the system. It can help predict how the pendulums will move and how they will respond to different initial conditions and external forces. It is also a fundamental concept in understanding more complex systems and their dynamics.

## 5. Can the velocity of a double pendulum be calculated using computer simulations?

Yes, the velocity of a double pendulum can be calculated using computer simulations. By inputting the relevant parameters and initial conditions into a simulation program, the velocity of each pendulum can be calculated at any given time during the motion of the double pendulum. This method is often used to study the behavior of complex systems and to verify the results of analytical calculations.

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