- 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

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.
*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**)
The position is derived by a fairly simple application of trigonometry. The velocity and acceleration are then 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*T* 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 **i**= unit vector in horizontal direction**j**= unit vector in vertical direction

acceleration =

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

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''* = − ^{g}⁄_{R} *θ*
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
where *θ*_{0} is the initial angle and *t* is time. The period is the time it takes for *θ*(*t*) to repeat, so
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;

© Erik Neumann, 2004 | Home | Top | Next |