massless stiff rod with two masses m1 and m2, rotating in the plane
around a point P under the influence of gravity (and without
friction). The masses. are a distance r1 and r2, respectively, from
P, and we assume r1m1 > r2m2 for the proper functioning of the
metronome. At the attachment point the metronome has a torsion
spring which acts just like a linear spring in storing potential
energy, but as a function of the angle instead of the
position, V = 1/2(k θ^2) , where k > 0 is the spring
constant and θ is the angle of deflection away from the
undeformed state, which for the metronome rod is the vertical
position.
- We Will Be Focusing On The Motion Of Metronomes Consisting Of A Massless Stiff Rod With Two Masses M1 And M2 Rotating 1 (84.91 KiB) Viewed 70 times
space of the system, including also large angles. Interpret the
motion shown. Reasonable parameters for the metronome could for
example be m1 = 0.16 kg, m2 = 0.01 kg, r1 = 0.02 m, r2 = 0.16 m.
Consider different values of the spring constant k. [3 points]
a) Show that the Lagrangian for this system is 1 L(0,0) = 10 + M Rg cos 8 – 5 mo? = Ꮎ (1) 2 where 0 is the angle to the vertical for the rod, I the moment of inertia for the two masses for the rotation around P, M = mi+ m2 the total mass, and R the centre-of-mass distance from P. Justify the choices you make, such as for the coordinates. [2.5 points] b) Find the equation of motion for the system and solve it for the small angle approximation 0 « 1. [2 points] c) Identify any equilibrium solutions and interpret these physically. [2 points]