We know θ and we know v (^) b , so we can solve for L .
Now that we know L , we can find the frequency.
The Sinusoidal Nature of SHM and the Second-Order Differential Equation
Consider the force acting on an object in simple harmonic motion: F (^) net = −kx . Well, F (^) net = ma , and
acceleration is the second derivative of position. So the equation for the motion of the pendulum becomes
This type of equation is called a differential equation, where a derivative of a function is proportional to
the function itself. Specifically, since the second derivative is involved, this is called a “second-order”
differential equation.
You don’t necessarily need to be able to solve this equation from scratch. However, you should be
able to verify that the solution x = A cos(wt ) satisfies the equation, where
(How do you verify this? Take the first derivative to get dx /dt = −Aω sin(ωt ); then take the second
derivative to get −Aω 2 cos(ωt ). This second derivative is, in fact, equal to the original function
multiplied by −k /m .)
What does this mean? Well, for one thing, the position–time graph of an object in simple harmonic
motion is a cosine graph, as you might have been shown in your physics class. But more interesting is the
period of that cosine function. The cosine function repeats every 2p radians. So, at time t = 0 and at time t
= 2p /ω , the position is the same. Therefore, the time 2p /ω is the period of the simple harmonic motion.
And plugging in the ω value shown above, you see that—voila! —