# pendulum

(0.001 seconds)

## 2 matching pages

##### 1: 22.19 Physical Applications
###### §22.19(i) Classical Dynamics: The Pendulum
With appropriate scalings, Newton’s equation of motion for a pendulum with a mass in a gravitational field constrained to move in a vertical plane at a fixed distance from a fulcrum is …The periodicity and symmetry of the pendulum imply that the motion in each four intervals $\theta\in(0,\pm\alpha)$ and $\theta\in(\pm\alpha,0)$ have the same “quarter periods” $K=K\left(\sin\left(\frac{1}{2}\alpha\right)\right)$. … Figure 22.19.1: Jacobi’s amplitude function am ⁡ ( x , k ) for 0 ≤ x ≤ 10 ⁢ π and k = 0.5 , 0.9999 , 1.0001 , 2 . When k < 1 , am ⁡ ( x , k ) increases monotonically indicating that the motion of the pendulum is unbounded in θ , corresponding to free rotation about the fulcrum; compare Figure 22.16.1. …This corresponds to the pendulum being “upside down” at a point of unstable equilibrium. … Magnify
##### 2: 28.33 Physical Applications
###### §28.33(iii) Stability and Initial-Value Problems
If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as $\cos\left(2\omega t\right)$. …