Dynamics

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By A. Frank D'Souza

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Determine (a) the velocity and acceleration of point Pat the base of the cone shown in Fig. 3, and (b) the velocity and acceleration of C, the mass center of the cone. The coordinate system 0 X YZ is inertial with the Z axis being vertical. The cone rolls on the horizontal XY plane. In Fig. 3, xyz is a coordinate system with origin at the fixed point 0 and rotating at constant angular velocity roo about the vertical Z axis. , cos t'Xk. (a) In order to determine the angular velocity Q of the body, we first determine the spin of the cone about the x axis.

12) Hence, it follows that if the origin 0 coincides with center of mass C, then ; c = 0. 16) where is the position of mass particle dm from the center of mass. 16) is valid even though the moving center of mass has a velocity ~c· The motivation for choosing the origin of the coordinate system either at its center of mass or at a fixed point of the body, if such a point exists, becomes obvious. The rotational equations of motion are uncoupled from the translational. Henceforth, we shall assume, unless mentioned otherwise, that the origin of the coordinate system to describe the motion of a rigid body has been selected judiciously in this manner.

4 A particle of mass m 1 is free to slide on a horizontal bar with Coulomb friction under the action of a force P. Mass m 2 is pivoted from m 1 by a massless rigid link of length b. Obtain the equations of motion for this system of particles shown in Fig. 9. 22). 9 System of two particles. 27) =e. m1x3 a - = Xl .. -t P In order to express these equations as a set of first-order equations, we choose the displacements and velocities as the state variables. 27) become This problem is solved by drawing the free-body diagram for each particle as shown in Fig.

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