Physics-based simulation of a simple pendulum.

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For small oscillations the simple pendulum has *linear* behavior meaning that it's equation of motion can be characterized by a linear equation (no squared terms or sine or cosine terms), but for larger oscillations the it becomes very non-linear with a sine term in the equation of motion.

- 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?

pendulum variables

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

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 *θ*= angle of pendulum (0=vertical)*R*= length of rod*T*= tension in rod*m*= mass of pendulum*g*= gravitational constant

*τ*= net torque*I*= rotational inertia*α*=*θ''*= angular acceleration

−*R* *m* *g* sin *θ* = *m* *R*^{2} *α*

θ'' = − ^{g}⁄_{R} sin θ
| (1) |

pendulum forces

We'll need the standard unit vectors, **i**, **j**. We use bold and overline to indicate a vector.

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

Using Newton's law F = **i**= unit vector in horizontal direction**j**= unit vector in vertical direction

position = *R* sin *θ* **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** +
*θ''* sin *θ* **j** +
*θ'* ^{2} cos *θ* **j**)

Next we draw the free body diagram for the pendulum. The forces on the pendulum are the tension in the rod

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

*T* cos *θ* **j** − *T* sin *θ* **i** − *m* *g* **j** =
*m* *R*(*θ''* cos *θ* **i** −
*θ'* ^{2} sin *θ* **i** +
*θ''* sin *θ* **j** +
*θ'* ^{2} cos *θ* **j**)

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

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

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

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

−*θ''* cos^{2}*θ* + *θ'* ^{2} sin *θ* cos *θ* = *θ''* sin^{2}*θ* +
*θ'* ^{2} sin *θ* cos *θ* +
^{g}⁄_{R} sin *θ*

*θ''* = − ^{g}⁄_{R} sin *θ*

*θ'* = *ω*

*ω*' = − ^{g}⁄_{R} sin *θ*

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

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

*θ''* = − ^{g}⁄_{R} sin *θ*

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 sin *θ* = *θ*. Then the equation of motion becomes
*θ*_{0} is the initial angle and *t* is time. The period is the time it takes for *θ*(*t*) to repeat, so

*θ''* = − ^{g}⁄_{R} *θ*

where

period =

The frequency of oscillation is the inverse of the period:frequency =

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;

TO DO:

- Give derivation for damping here? or on a separate page?
- Give better explanations for equation of motion derivation.
- No overline for i, j vectors?

This web page was first published April 2001.