Chapter 17 PLANE MOTION OF RIGID BODIES: ENERGY AND MOMENTUM METHODS The principle of work and energy for a rigid body is expressed in the form T1 + U1 2 = T2 where T1 and T2 represent the initial and final values of the kinetic energy of the rigid body and U1 2 the work of the external forces acting on the rigid body. The work of a force F applied at a point A is s2 ò U1 2 = (F cos a) ds s1 where F is the magnitude of the force, a the angle it forms with the direction of motion of A, and s the variable of integration measuring the distance traveled by A along its path. Sharif University-Aerospace Dep. Fall 2004

The work of a couple of moment M applied to a rigid body during a rotation in q of the rigid body is ò U1 2 = M ds q1 The kinetic energy of a rigid body in plane motion is 1 2 1 2 T = mv 2 + Iw2 G w v where v is the velocity of the mass center G of the body, w the angular velocity of the body, and I its moment of inertia about an axis through G perpendicular to the plane of reference.

1 2 1 2 T = mv 2 + Iw2 G The kinetic energy of a rigid body in plane motion may be separated into two parts: (1) the kinetic energy mv 2 associated with the motion of the mass center G of the w v 1 2 1 2 body, and (2) the kinetic energy Iw2 associated with the rotation of the body about G. For a rigid body rotating about a fixed axis through O with an angular velocity w, O 1 2 T = IOw2 w where IO is the moment of inertia of the body about the fixed axis.

T1 + V1 = T2 + V2 dU dt M dq dt Power = = = Mw When a rigid body, or a system of rigid bodies, moves under the action of conservative forces, the principle of work and energy may be expressed in the form which is referred to as the principle of conservation of energy. This principle may be used to solve problems involving conservative forces such as the force of gravity or the force exerted by a spring. T1 + V1 = T2 + V2 The concept of power is extended to a rotating body subjected to a couple dU dt M dq dt Power = = = Mw where M is the magnitude of the couple and w is the angular velocity of the body.

Syst Momenta1 + Syst Ext Imp1 2 = Syst Momenta2 The principle of impulse and momentum derived for a system of particles can be applied to the motion of a rigid body. Syst Momenta1 + Syst Ext Imp1 2 = Syst Momenta2 For a rigid slab or a rigid body symmetrical with respect to the reference plane, the system of the momenta of the particles forming the body is equivalent to a vector mv attached to the mass center G of the body and a couple Iw. The vector mv is associated with translation of the body with G and represents the linear momentum of the body, while the couple Iw corresponds to the rotation of the body about G and represents the angular momentum of the body about an axis through G. (Dm)v mv P Iw

The principle of impulse and momentum can be expressed graphically by drawing three diagrams representing respectively the system of initial momenta of the body, the impulses of the external forces acting on it, and the system of the final momenta of the body. Summing and equating respectively the x components, the y components, and the moments about any given point of the vectors shown in the figure, we obtain three equations of motion which may be solved for the desired unknowns. ò Fdt mv2 y mv1 y y G G Iw1 Iw2 O x O x O x

ò Fdt mv2 y mv1 y y G G Iw1 Iw2 O x O x O x In problems dealing with several connected rigid bodies each body may be considered separately or, if no more than three unknowns are involved, the principles of impulse and momentum may be applied to the entire system, considering the impulses of the external forces only. When the lines of action of all the external forces acting on a system of rigid bodies pass through a given point O, the angular momentum of the system about O is conserved.

(v’B)n - (v’A)n = e[(vA)n - (vB)n] The eccentric impact of two rigid bodies is defined as an impact in which the mass centers of the colliding bodies are not located on the line of impact. In such a situation a relation for the impact involving the coefficient of restitution e holds, and the velocities of points A and B where contact occurs during the impact should be used. B A vB n vA (a) Before impact n B A v’B (v’B)n - (v’A)n = e[(vA)n - (vB)n] n v’A (b) After impact

(v’B)n - (v’A)n = e[(vA)n - (vB)n] (a) Before impact (b) After impact (v’B)n - (v’A)n = e[(vA)n - (vB)n] where (vA)n and (vB)n are the components along the line of impact of the velocities of A and B before impact, and (v’A)n and (v’B)n their components after impact. This equation is applicable not only when the colliding bodies move freely after impact but also when the bodies are partially constrained in their motion.