1 Class #13 of 30 Lagrange’s equations Worked examples  Pendulum with sliding support You solve it  T7-17 Atwood’s machine with massive pulley  T7-4, 7-16 Masses on ramps :02

2 Pendulum with sliding support-I :10 x1x1 L m1m1 x2x2 z2z2 m2m2 A pendulum with mass m 2 and length L is suspended from a block of mass m 1 resting on a frictionless plane. What is the frequency of small oscillations of the pendulum? What is the center of mass motion?

3 Pendulum with sliding support-II :15

4 Pendulum with sliding support-III :20

5 Pendulum with sliding support-IV :25

6 Pendulum with sliding support-V :40 A pendulum with mass m 2 and length L is suspended from a block of mass m 1 resting on a frictionless plane. What is the frequency of small oscillations of the pendulum? x1x1 L m1m1 x2x2 z2z2 m2m2

7 Pendulum with sliding support-VI :45 A pendulum with mass m 2 and length L is suspended from a block of mass m 1 resting on a frictionless plane. What is the center of mass motion? x1x1 L m1m1 x2x2 z2z2 m2m2

8 Atwood’s Machine Lagrangian recipe :60 m1m1 m2m2 m1m1 m2m2  R

9 T7-17 Atwood’s Machine with massive pulley Lagrangian recipe :70 m1m1 m2m2  R 1) Write down T and U in any convenient coordinate system. 2) Write down constraint equations 3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables

10 T7-4, T7-16 Masses on ramps :65 A block of mass “m” starts from rest and slides down a ramp of height “h” and angle “theta”. a) Calculate acceleration “a” at top of ramp, time “t” to get to bottom of ramp and velocity “v” at bottom of ramp. Use the Lagrangian formalism. b) Do the same for a rolling disk (mass “m”, radius “r”) O y x m h m y x

11 Class #13 Windup New variables can be introduced so long as add additional constraints Generalized coordinates do not need to be of same type (e.g. angle / position). Office hours today 3-5 Wed 4-5:30 :72