Solving Rotation Condition for Equation of Motion

[KFC]

If I know the equation of motion of the following form

$$\ddot{\theta} + k^2\sin\theta = 0$$

(for pendulum for example). What's the condition (minimum angular velocity) to keep it rotate instead of just oscillation?

 

[adriank]

Writing $$\omega = \dot \theta$$, that equation becomes (using the chain rule)

$$\omega \frac{d\omega}{d\theta} + k^2 \sin \theta = 0$$

and solving that differential equation gives

$$\frac12 \omega^2 = k^2 \cos \theta + C.$$

Once you have that, what condition on C do you need for $$\theta = \pi$$ to be possible? (Note that each side of this equation corresponds nicely to energy.)

 

[astrokreng]

To solve for the rotation condition in this equation of motion, we need to consider the forces acting on the pendulum. The first term, \ddot{\theta}, represents the angular acceleration and is caused by the restoring force of gravity pulling the pendulum back towards its equilibrium position. The second term, k^2\sin\theta, represents the centripetal force, which is responsible for keeping the pendulum in circular motion.

In order for the pendulum to rotate instead of just oscillating, the centripetal force must be greater than the restoring force. This can be achieved by increasing the angular velocity, which is represented by \dot{\theta}. Therefore, the minimum angular velocity required for the pendulum to rotate is when the centripetal force is equal to the restoring force, or when \dot{\theta} = \sqrt{\frac{g}{l}}, where g is the acceleration due to gravity and l is the length of the pendulum.

It is important to note that this is the minimum angular velocity required for the pendulum to rotate, but higher angular velocities will still allow for rotation. Additionally, other factors such as air resistance and friction may also play a role in determining the minimum angular velocity for rotation. Further analysis and experimentation may be needed to fully understand and accurately determine this condition.