# A pendulum system and ODEs

• mathgirl
In summary, a pendulum system is a physical system that follows the principles of simple harmonic motion and is used to model the motion of a pendulum. This is related to ordinary differential equations (ODEs), which are used to describe the motion of the pendulum and determine its position, velocity, and acceleration. The pendulum equation is derived using Newton's second law of motion and factors such as length, mass, and amplitude can affect its motion. Pendulum systems have various real-world applications, including timekeeping devices, seismology, engineering, and physics experiments.

#### mathgirl

Can anyone help me figure out how to model this pendulum system using ODE's? It is a two-mass system in which the two masses are placed at opposite ends of a massless rod, with a fulcrum somewhere in the middle. The smaller mass is length k away from the fulcrum and the larger mass is length L away from the fulcrum.

Could u attach a drawing,please and explain what sort of oscillations are u looking for...?

The physics involved would be only a wise application of Newton's second law for nonlinear movement (extended bodies).

Daniel.

Sure, I can try to help you with modeling this pendulum system using ODEs. First, let's define some variables:

- Let m1 be the mass of the smaller mass and m2 be the mass of the larger mass.
- Let k be the distance of the smaller mass from the fulcrum and L be the distance of the larger mass from the fulcrum.
- Let x1 and x2 be the horizontal positions of the smaller and larger masses respectively, with x1 = -k and x2 = L.
- Let θ be the angle made by the rod with the vertical direction.

Now, we can use Newton's second law to write down the equations of motion for the two masses:

m1x1'' = T1sinθ - m1gcosθ

m2x2'' = T2sinθ - m2gcosθ

where T1 and T2 are the tensions in the rod acting on the smaller and larger masses respectively.

Next, we need to relate the horizontal positions x1 and x2 to the angle θ. Since the rod is massless, the distance between the two masses remains constant, and we can use the Pythagorean theorem to write:

(x2 - x1)^2 + (L - k)^2 = (L + k)^2

Expanding and simplifying, we get:

x2^2 - 2x1x2 + x1^2 + L^2 - 2Lk + k^2 = L^2 + 2Lk + k^2

Simplifying further, we get:

x2^2 - 2x1x2 + x1^2 = 4Lk

Now, we can use the small angle approximation sinθ ≈ θ and cosθ ≈ 1 to rewrite the equations of motion as:

m1x1'' = T1θ - m1g

m2x2'' = T2θ - m2g

We also need to express the tensions T1 and T2 in terms of θ. From the diagram, we can see that:

T1sinθ = m1x1'' + m1gcosθ

T2sinθ = m2x2'' + m2gcosθ

Substituting these into the equations of motion, we get:

## 1. What is a pendulum system and how does it work?

A pendulum system is a physical system consisting of a mass attached to a fixed point by a string or rod, which allows it to swing back and forth under the influence of gravity. It follows the principles of simple harmonic motion, where the pendulum's period (time for one complete swing) is independent of its amplitude (distance from the equilibrium position).

## 2. What is an ordinary differential equation (ODE) and how is it related to pendulum systems?

An ordinary differential equation (ODE) is a mathematical equation that describes how a variable changes in relation to its independent variable(s). In pendulum systems, ODEs are used to model the motion of the pendulum and determine its position, velocity, and acceleration at any given time.

## 3. How is the pendulum equation derived?

The pendulum equation is derived using Newton's second law of motion, which states that the net force acting on an object is equal to its mass times its acceleration. By applying this law to the forces acting on a pendulum (gravity and tension), a second-order linear ODE can be obtained to describe the motion of the pendulum.

## 4. What factors affect the motion of a pendulum?

The motion of a pendulum is affected by several factors, including the length of the pendulum, the mass of the pendulum bob, and the amplitude of its swings. In addition, the force of gravity and air resistance can also impact the motion of a pendulum.

## 5. How are pendulum systems used in real-world applications?

Pendulum systems have many real-world applications, including timekeeping devices such as grandfather clocks and metronomes. They are also used in seismology to measure earthquakes and in engineering to study the effects of vibrations on structures. In addition, pendulums are used in physics experiments to demonstrate principles of simple harmonic motion and energy conservation.