About the Project
NIST

pendulum

AdvancedHelp

(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 θ ( 0 , ± α ) and θ ( ± α , 0 ) have the same “quarter periods” K = K ( sin ( 1 2 α ) ) . …
See accompanying text
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 ( 2 ω t ) . …