chapt11 - Plane Motion of Rigid Bodies: Energy and Momentum...

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Plane Motion of Rigid Bodies: Energy and Momentum Methods
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Introduction • Method of work and energy and the method of impulse and momentum will be used to analyze the plane motion of rigid bodies and systems of rigid bodies. • Principle of work and energy is well suited to the solution of problems involving displacements and velocities. 2 2 1 1 T U T = + • Principle of impulse and momentum is appropriate for problems involving velocities and time. () 2 1 2 1 2 1 2 1 O t t O O t t H dt M H L dt F L G G G G G G = + = + • Problems involving eccentric impact are solved by supplementing the principle of impulse and momentum with the application of the coefficient of restitution.
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Principle of Work and Energy for a Rigid Body • Method of work and energy is well adapted to problems involving velocities and displacements. Main advantage is that the work and kinetic energy are scalar quantities. • Assume that the rigid body is made of a large number of particles. 2 2 1 1 T U T = + = 2 1 , T T = 2 1 U initial and final total kinetic energy of particles forming body total work of internal and external forces acting on particles of body. • Internal forces between particles A and B are equal and opposite. • In general, small displacements of the particles A and B are not equal but the components of the displacements along AB are equal. • Therefore, the net work of internal forces is zero.
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Work of Forces Acting on a Rigid Body • Work of a force during a displacement of its point of application, () = = 2 1 2 1 cos 2 1 s s A A ds F r d F U α G G • Consider the net work of two forces forming a couple of moment during a displacement of their points of application. F F G G and M G θ d M d Fr ds F r d F r d F r d F dU = = = + = 2 2 1 1 G G G G G G 1 2 2 1 2 1 θθ = = M d M U if M is constant.
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Work of Forces Acting on a Rigid Body Forces acting on rigid bodies which do no work: • Forces applied to fixed points: - reactions at a frictionless pin when the supported body rotates about the pin. • Forces acting in a direction perpendicular to the displacement of their point of application: - reaction at a frictionless surface to a body moving along the surface - weight of a body when its center of gravity moves horizontally • Friction force at the point of contact of a body rolling without sliding on a fixed surface. ( ) 0 = = = dt v F ds F dU c C
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Kinetic Energy of a Rigid Body in Plane Motion • Consider a rigid body of mass m in plane motion. () 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2 2 1 Δ Δ ω I v m m r v m v m v m T i i i i + = + = + = • Kinetic energy of a rigid body can be separated into: - the kinetic energy associated with the motion of the mass center G and - the kinetic energy associated with the rotation of the body about G .
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chapt11 - Plane Motion of Rigid Bodies: Energy and Momentum...

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