Try using the graph and changing parameters like mass, length, gravity to answer these questions about the spring simulation:

- What is the relationship between angular acceleration and angle?
- How do mass, length, or gravity affect the relationship between angular acceleration and angle?
- For small oscillations, how do length or gravity affect the period or frequency of the oscillation?

Note: Leave damping and drive frequency set to zero here (they complicate things). You'll find the answers below.

Hint: Try starting the pendulum from an almost vertically "up" position.

The pendulum is modeled as a point mass at the end of a massless rod. We define the following variables:

θ = angle of pendulum (0= vertical)R = length of rodm = mass of pendulumg = gravitational constant

We will derive the equation of motion for the pendulum using the rotational analog of Newton's second law for motion about a fixed axis, which is

τ = net torqueI = rotational inertiaα = θ''= angular acceleration

The rotational inertia about the pivot is ^{2}

-R m g sin θ = m R^{2} α

which simplifies to

This is the equation of motion for the pendulum.

Most students are less familiar with rotational inertia and torque than with the simple mass and acceleration found in Newton's second law, **F** = m **a**

To show that there is nothing new in the rotational version of Newton's second law, we derive the equation of motion here without the rotational dynamics. As you will see, this method involves more algebra.

We'll need the standard unit vectors, **i**, **j**

unit vector in horizontal direction**i**= unit vector in vertical direction**j**=

The kinematics of the pendulum are then as follows

**i** - R cos θ **j**

velocity = R θ' cos θ **i** + R θ' sin θ **j**

acceleration = R(θ'' cos θ **i** - θ'^{2} sin θ **i** **j** ^{2} cos θ **j**)

The position is derived by a fairly simple application of trigonometry. The velocity and acceleration are then just the first and second derivatives of the position.

Next we draw the free body diagram for the pendulum. The forces on the pendulum are the tension in the rod and gravity. So we can write the net force as:

**F** = T cos θ **j** - T sin θ **i** - m g **j**

Using Newton's law **F** = m **a**

**j** - T sin θ **i** - m g **j** = **i** ^{2} sin θ **i**
**j** ^{2} cos θ **j**)

We can write the vector components of the above equation as separate equations. This gives us two simultaneous equations.

^{2} sin θ)

T cos θ - m g = m R(θ'' sin θ + θ'^{2} cos θ)

Next we do some algebraic manipulations to eliminate the unknown

^{2}θ - θ'^{2} sin θ cos θ)

T cos θ sin θ - m g sin θ = m R(θ'' sin^{2}θ ^{2} sin θ cos θ)

A few more rearrangements:

^{2}θ - θ'^{2} sin θ cos θ)

T cos θ sin θ = m R(θ'' sin^{2}θ ^{2} sin θ cos θ)

Now we can equate the two right hand sides above and divide by

^{2}θ + θ'^{2} sin θ cos θ = θ'' sin^{2}θ ^{2} sin θ cos θ

Using the trig identity ^{2}θ + sin^{2}θ = 1

There is yet a third way to derive the equations of motion for the pendulum. This is to use the "indirect" energy based method associated with the terms "Lagrangian", "Euler-Lagrange equations", "Hamiltonian", and others.

While this method isn't shown here, you can see an example of it for the Pendulum+Cart simulation.

To solve the equations of motion numerically, so that we can drive the simulation, we use the Runge-Kutta method for solving sets of ordinary differential equations. First we define a variable for the angular velocity

θ' = ω

ω' = -(g/R) sin θ

This is the form needed for using the Runge-Kutta method.

Question: What is the relationship between angular acceleration and angle?

Answer: It is a sine wave relationship as given by equation (1)

Question: How do mass, length, or gravity affect the relationship between angular acceleration and angle?

Answer: From equation (1) we see that:

- Mass doesn't affect the motion at all.
- The amplitude of the sine relationship is proportional to gravity.
- The amplitude of the sine relationship is inversely proportional to length of the pendulum.

Question: For small oscillations, how do length or gravity affect the period or frequency of the oscillation?

Answer: For small oscillations we can use the approximation that

This is a linear relationship. You can see that the graph of acceleration versus angle is a straight line for small oscillations. This is the same form of equation as for the single spring simulation. The analytic solution is

θ(t) = θ_{0} cos(sqrt(g/R) t)

where _{0}

frequency = (1/2π) sqrt(g/R)

So we predict that

- increasing length by 4 times doubles the period and halves the frequency;
- increasing gravity by 4 times halves the period and doubles the frequency;