Problem - Consider the crankshaft shown in Figure 1. The system is positioned on a vertical plane
(g = 9.81 m s²) and is formed by two massless rods of length L and 2L, respectively. Three pivots,
about which the two rods can rotate, are placed at points O, 1 and 2 in figure. The two rods are
connected by the pivot corresponding to point 1. Two point-masses, M and 2M, are placed at the
lower ends of the two rods (i.e. at points 1 and 2, respectively). Point O is fixed. Point / has free
in-plane movement. Point 2 is constrained to move in a vertical rail without friction, passing by
the origin. The system has only one degree of freedom.
1
M
L1
lo
/
y
Law of cosines
Y² = W² + V² - 2WV cos?
W² = Y² + V² - 2YV cosp
2L
2
g
2M
q = {y}
Figure 1
W
?
?
Y
Figure 2
a) Using q = {y} as the generalized coordinate for the system, derive a system's Lagrangian
as a function of y and y which only contains any of L, M, and g (1.5 pt.).
L=T(y, y; L, M)-U(y;L,M,g) =
