Question

5. A particle of mass m slides without friction and starting from rest down a block of mass M in which the surface is shaped into a quadrant of a circle of radius R. The block also slides without friction on a table top. Choose two suitable generalized coordinates a) Using the Lagrange-multiplier formalism, find the equations of motion of the mass and the block and the reaction force of the block on the mass (ie. the constraint force b) Identify one of the coordinates as being ignorable ed momentum? What is the corresponding conserv Use this to eliminate one of the four integrations. c) Use conservation of energy to eliminate another integration and, in the end, express the constraint force solely in terms of the angle that gives the location of the mass relative to the center of the circle 0-2sin where M+m (1-asin') and , is the initial position of the mass before it is released from rest (zero initial velocity

Answer 1

Equation of motion, Let at t' = t position of m e.r.to mass m = (R, R) initially position of m e.r.to mass M = (R sin theta, R cos theta) finally Now position of M e.r.to ground at t' = t = x_1 I rightarrow d vector m rightarrow M = d vector m rightarrow g rightarrow d vector_m rightarrow m + d_M rightarrow g displacement of m w.r.to ground (d vector_m rightarrow g)_x = (R - R sin theta) (-i) + x_1 I - R cos theta j rightarrow d vector m rightarrow g = -[R - R sin theta + x_1] I + (- R cos theta) j = (- R + R sin theta - x_1) I - R cos theta j Hence T (kinetic energy of system)