A uniform meter stick of mass M is pivoted on a hinge at one end and held horizontal by a spring with spring constant k attached at the other end. If the stick oscillates up and down slightly, what is its frequency? [Hint: Write a torque equation about the hinge.] The length of the beam is 1.25 m.
τ = Iα
Fspring = -kx
The Attempt at a Solution
I have found the question online and have linked it to this comment in case someone needs to see a picture of the scenario.
I started by writing a torque equation, with the beam angled θ above the horizontal and x being the displacement of the spring tip above the horizontal.
The torques about the pivot on the wall on the stick, using the small angle approximation, are:
1) The spring exerts a force, -kx = -kLθ , a distance, L, away from the pivot. Thus, the torque exerted by the spring is, for small angles, -kxL2θ.
2) The center of mass exerts a torque, given by -mgL/2.
The torque equation is. therefore, ∑τ = Iθ'' = -kL2θ - mgL/2.
This is where I was stuck. I don't know how to solve differential equations beyond separable ones as of yet. The book I am using shows that the solution to a simple harmonic oscillator's equation of motion is just a sinusoid; however, this does not seem like a simple oscillator as there is gravity in this system, which is odd as this is the simple oscillatory section of my book.
I tried substituting a function of cosine; however it does not seem to work. If anyone could shed light as to what I did incorrectly or what I am overlooking, it would be greatly appreciated.
Thank you in advance.