Pendulum

Pendulum Torque  The pendulum is driven by forces of gravity and tension. Constraint force tensionConstraint force tension One generalized force from gravityOne generalized force from gravity  The angle is the natural generalized coordinate. Generalized force torqueGeneralized force torque Tension exerts no torqueTension exerts no torque mg FTFT mg sin   l

Small Angles  A small angle approximation gives a simple harmonic solution. Angular acceleration Angular acceleration  Moment of inertia I = ml 2.Moment of inertia I = ml 2. Small angles sin  = Small angles sin  =   Compare angle and Angle(rad)Sine 1  ( )  ( )  ( )  ( )  ( )  ( )  ( )

One Dimensional Potential  A conservative system has a position-dependent potential.  The velocity can be found in terms of the energy.  The time and position can be found by integration. Not always analyticNot always analytic

Pendulum Potential  Single variable is the angle s = l s = l  q = q =  V(q) = mgl(1 – cosq)V(q) = mgl(1 – cosq)  Use unitless variables. mg v l cos   l

Elliptic Integral  The velocity can be found from the potential.  The velocity can be integrated by substitution.  The integral takes on different forms for different energy ranges. E < 2 E = 2 E > 2

Bound Motion  Energy below threshold will have bound motion. E < 2E < 2  The integral is approximated by a power series. Multiply by 4 for the periodMultiply by 4 for the period

Unbound Motion  For energy above the threshold the motion is unbounded. E > 2E > 2 Similar substitutionSimilar substitution  This is non-uniform circular motion Depends on energyDepends on energy Acceleration of gravityAcceleration of gravity

Critical Energy  At the threshold energy there is no elliptical integral. E = 2E = 2 Non-periodic motionNon-periodic motion Non-circular motionNon-circular motion  The pendulum reaches its peak at infinite time.

Phase Portrait  A phase diagram is a plot of position vs. velocity. Match energies to curvesMatch energies to curves E < 2 E = 2 E > 2 next