دانلود کتاب راهنمای حل مسائل مکانیک کلاسیک و الکترودینامیک

Four mechanical systems are studied. In all cases the number of degrees of freedom are specified, and an appropriate set of generalized coordinates is chosen. a) The first system consists of a pendulum attached to a block which in turn is attached to a spring. We assume all motion takes place in a twodimensional, vertical plane. The block is constrained to move in the horizontal direction, and the pendulum is constrained by the constant length of the rod. Starting from two degrees of freedom for each of the two objects, the two constraints reduce the number of degrees of freedom to two, one for each object. A natural choice of generalized coordinates is the horizontal displacement x of the block and the angle θ of the rod relative to the vertical direction. b) The second system consists of a pendulum attached to a vertical disk, which rotates with a fixed angular frequency. Also here we consider the motion restricted to a two-dimensional, vertical plane. There is no degree of freedom related to the rotating disk, since it has an externally determined angular frequency. The pendulum is again only constrained by the fixed length of the rod, and the number of degrees of freedom of the system is therefore one. A natural choice of generalized coordinate is the angle θ between the pendulum rod and the vertical direction. c) In the third case a rigid rod can tilt without sliding on the top of the cylinder, while the cylinder can roll on a horizontal plane. Assuming again that the motion is restricted to a two-dimensional, vertical plane, the starting point is three degrees of freedom for each object. For the cylinder this corresponds to two coordinates for its center of mass and one for its angle of rotation. For the rod there are two coordinates needed to determine the position of its center of mass, and one coordinate to determine the angle of the rod relative to the horizontal (or vertical) direction.