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.
A good starting point for the study of this system is Newton's
Second Law
What are the forces acting on the mass?
In the diagram, identify the forces by clicking on the arrows.
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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
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.
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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
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
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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.