pendulum formula

The pendulum formulae can be found Here. It depends on gravity and the length of the string. Mass doesn't factor in. (Unless you're trying to calculate the air resistance effecting it. Then it would, but that would add another level of complexity to it)

strictly speaking, the closed form solution of this system is not known.

linearized around the rest position however, it is quite solvable, given you define the problem a little better.

google clearly is your friend if you want to know the whole story.

So, basically, it looks like a good approximation is this:

theta = theta0 * cos(sqrt(gravity/length) * time)

So, when you say "linearized", are you talking about using the small angle approximation to make "sin(theta)" equal to "theta"?


mike
http://www.coolgroups.com/

Yep, that's what he's talking about.

Quote:

Original post by mike74 Does anyone know what the formula is for the angle of a standard pendulum in terms of time? Basically, I'm looking for something like theta = f(time)

Let θ be the angle of the pendulum made with the vertical, L the length of the connecting rod, T the force of tension in the rod, m the mass of the pendulum, and g the gravitational constant.

Using Newton's second law for rotational forces (τ = Iα), observe that the pendulum's rotational inertia around the pivot point is I = mR². It works out that the torque due to gravity is τ = -Lmg · sin(θ). This gives us -Lmg · sin(θ) = mR²α, which in turn simplifies to θ" = -g/L · sin(θ).

Now we want θ as a function of an external variable, time (t), or θ(t). For small oscillations, we can use the approximation θ = sin(θ) (the "linearization" mentioned before), so we get θ(t) = θ₀ · cos(sqrt[g/L] · t) as the final answer.