Math Model

To develop a mathematical model you must first decide what the variables are and then choose a coordinate frame.

If the independent variable is time, t, which of the following is the dependent variable?

To fix a coordinate frame you must choose

What do you think would be good choices? Compare your answer to the correct one.

The bob moves on a circular arc centered at the pivot point where the rod is attached to the support. Therefore, it is natural to choose a polar coordinate system with its origin at the pivot point. The polar axis is straight down and the positive angular direction is counterclockwise. The radial coordinate is measured in centimeters and the angle in degrees.

This diagram shows these choices.

polar coordinate system

As you proceed, keep in mind how the choice of positive direction might affect the derivation.

A good starting point for the study of this system is Newton's Second Law

F = ma

which states that the product of the vector acceleration a and the mass m is equal to the net force F. To apply Newton's Second Law, you should identify the forces acting on the mass and show them in a free-body diagram.

What are the forces acting on the mass?

In the diagram, identify the forces by clicking on the arrows.

Newton's Second Law is a vector equation. For the pendulum system it is convenient to choose one component in the positive direction (tangent to the arc described by the bob's motion), and one in the r direction (perpendicular to the arc described by the bob's motion). Since the bob doesn't move in the radial direction (its r coordinate is the constant l), we can concentrate on the tangential equation

where and are the theta components of force and acceleration respectively.

To proceed further, you need to rewrite and in terms of t, (t), l, m, the tension T in the rod, and any necessary universal constants, such as the gravitational acceleration g.

To rewrite you need to find the theta components of the forces. Draw pictures and resolve the forces into their and r components. When you are ready, compare your answers to the correct ones.

Since is the sum of the components of gravity and tension,

What about the components in the radial direction? To investigate further go to collaborate.

Now you need to rewrite .

If the pendulum moves through an angle , what is the distance traveled by the bob? That is, what is the length of its arc?

Given the displacement of the bob, what is its acceleration?

Now put your expressions for and into , so

Thus,

Notice that we found this equation of motion without determining the tension in the rod. This is an advantage of the coordinate system we chose.

Would this equation be different if we had chosen theta to be positive in the clockwise direction? What steps along the way would be different?

This equation for the angular motion of the pendulum bob is a nonlinear differential equation that cannot be solved in terms of elementary functions.

To find a linear approximation for this nonlinear equation we can use the Taylor series for about ,

For small we can neglect the higher order terms and approximate by . If we do so, how big will the error be? If you are not sure, go to practice.

This approximation leads to the linear differential equation

To describe the problem completely, it is also necessary to specify the initial state of the system.

How many initial conditions do you need? (What is the order of the differential equation?)

What are good choices for initial conditions?

The differential equation along with the initial conditions is a mathematical model for this pendulum system.
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