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Homework Statement
A 5kg sphere is connected to a thin massless but rigid rod of length L=1.3 m to form a simple pendulum. The rod is connected to a nearby vertical wall by a spring with a spring constant k= 75N/m, connected to it at a distance h=1.1 m below the pivot point of the pendulum. What is the angular frequency (in rad/s) of the system for small amplitude oscillations.
Homework Equations
[tex]\alpha[/tex] = [tex]\omega[/tex]^{2}*[tex]\theta[/tex]
I=m*r^{2}
Torque=f*d*sin[tex]\theta[/tex]
or small oscillations, sin[tex]\theta[/tex]=[tex]\theta[/tex] and cos[tex]\theta[/tex]=1
The Attempt at a Solution
sum the torque and set them equal to [tex]\alpha[/tex]I
gravitational torque= Lmg[tex]\theta[/tex]
if theta is the angle between the rod and its equilibrium position, the angle of the spring torque is (theta + [tex]\pi[/tex]/2), so the spring torque is
f*d*sin(theta + [tex]\pi[/tex]/2). The force is k*[tex]\Delta[/tex]x, or h*sin(theta + [tex]\pi[/tex]/2) and the distance is h, giving
torque from spring =k*h^{2}*sin[tex]\theta[/tex]*sin(theta + [tex]\pi[/tex]/2)
or k*h^{2}*[tex]\theta[/tex]*(theta + [tex]\pi[/tex]/2)
My final equation is
mL^{2}[tex]\omega[/tex]^{2}tex]\theta[/tex] = Lmg[tex]\theta[/tex] + k*h^{2}*[tex]\theta[/tex]*(theta + [tex]\pi[/tex]/2)
The problem is that this gives me a [tex]\omega[/tex] that is dependant on [tex]\theta[/tex], which isn't possible. It should be constant. Can someone please tell me where I'm going wrong?
Homework Statement
Homework Equations
The Attempt at a Solution
Homework Statement
Homework Equations
The Attempt at a Solution
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