MODULE 5. PLANE MOTION OF RIGID BODIES: ENERGY AND MOMENTUM METHODS

In this module, the method of work and energy in analyzing plane motion of rigid bodies will be considered. The work of a force and of a couple will be defined, and the expression for the kinetic energy of a rigid body in plane motion is obtained. The principle of work and energy to solve problems involving displacements and velocities will also be discussed and the application of the principle of conservation of energy to solve a variety of engineering problems. In the second section, applying the principle of impulse and momentum to solve problems involving velocities and time will be introduced, wherein the concept of conservation of angular momentum for rigid bodies in plane motion will also be discussed.

energy methods for a rigid body

principle of work and energy

The principle of work and energy to the motion of a rigid body can be emphasized by assuming first that the rigid body is made up of a large number n of particles of mass Δmi.
The method of work and energy, when applied to all of the particles forming a rigid body, yields the equation as shown in the left figure, where T1 and T2 are, respectively, the initial and final values of the total kinetic energy of the particles forming the body and U1→2 is the work done by the external forces exerted on the rigid body.
Expressing the work done by nonconservative forces as U^NC 1→2 and define the potential energy terms for conservative forces, the expression can be expressed into another formula where Vg1 and Vg2 are the initial and final gravitational potential energy of the center of mass of the rigid body and Ve1 and Ve2 are the initial and final values of the elastic energy associated with springs in the system.
Also, note that Ve = 1/2 kx^2 , where x is the deflection of the spring from its unstretched length. For a single rigid body, Vg = mgy, where y is the elevation of the center of mass from a reference plane or datum.

work of forces acting on a rigid body

Adding the expression for the work of a couple from the work of a force equation, the equation can be derived into another form as shown in the right figure, where F is the magnitude of the force, α is the angle it forms with the direction of motion of its point of application A, and s is the variable of integration that measures the distance traveled by A along its path.
When the moment of a couple is constant, the work of the couple as shown in the left where θ1 and θ2 are expressed in radians.

Kinetic Energy of a Rigid Body in Plane Motion

The kinetic energy of a rigid body in plane motion was found by considering the motion of the body as the sum of a translation with its mass center and a rotation about the mass center.
The equation for kinetic energy of rigid body is shown in the right figure, where v is the velocity of the mass center and ω is the angular velocity of the body

systems of rigid bodies

When a problem involves several rigid bodies, it is important to analyze all of the bodies together as a system instead of analyzing each individual rigid body separately. The equation (as shown in the right figure) will be used, where T is the sum of the kinetic energies of the bodies forming the system and U is the work done by all the forces acting on the bodies—internal as well as external.
The forces exerted on each other by pin-connected members or by meshed gears are equal and opposite, and since they have the same point of application, they undergo equal small displacements. Therefore, their total work is zero.

The forces exerted by an inextensible cord on the two bodies it connects have the same magnitude and their points of application move through equal distances, but the work of one force is positive and the work of the other is negative. Therefore, their total work is zero.
The forces exerted by a spring on the two bodies it connects also have the same magnitude, but their points of application generally move through different distances. Therefore, their total work is usually not zero

conservation of energy

The principle of conservation of energy can be expressed as shown in the left figure, where V represents the potential energy of the system.
The equation can also be written in terms of gravitational potential energy, Vg, and elastic potential energy, Ve.
The formulas indicate that when a rigid body, or a system of rigid bodies, moves under the action of conservative forces, the sum of the kinetic energy and of the potential energy of the system remains constant. Note that, in the case of the plane motion of a rigid body, the kinetic energy of the body should include both the translational term 1/2mv^2 and the rotational term 1/2Iω^2.

power

Power is defined as the time rate at which work is done. For a body acted upon by a force F and moving with a velocity v, we expressed the power as shown in the left figure.
In the case of a rigid body rotating with an angular velocity ω and acted upon by a couple of moment M parallel to the axis of rotation, the equation can also be expressed in its alternative form.

momentum methods for a rigid body

Principle of Impulse and Momentum

The principle of impulse and momentum for the plane motion of a rigid body is expressed by the vector equation as shown in the left figure, where Syst Momenta represents the system of the momenta of the particles forming the rigid body and Syst Ext Imp represents the system of all the external impulses exerted during the motion.

The system of the momenta of a rigid body is equivalent to a linear momentum vector mv attached at the mass center of the body and an angular momentum couple about the center of mass Iω.
By using the impulse–momentum diagram, the components in any direction and sum moments about any point can be added, where d⊥ is the perpendicular distance from point P to the line of action of the linear velocity of G .

When choosing to sum moments about the center of gravity of the body, the equation will then reduce to another formula as shown in the left figure.
When choosing to sum moments about a fixed point O,this also reduces to another equation, where IO is the mass moment of inertia about point O.

systems of rigid bodies

The motion of several rigid bodies can be analyzed by applying the principle of impulse and momentum to each body separately. These can be done by simply following the steps below:

Draw impulse–momentum diagrams for the entire system of bodies. For each moving part of the system, the diagrams of momenta should include a linear momentum vector and a momentum couple.
Sum and equate successively the x components, y components, and moments of all vectors involved, you obtain three relations expressing that the momenta at time t1 and the impulses of the external forces form a system equipollent to the system of the momenta at time t2.
Check each equation to make sure that consistent units are used. Take care not to add linear and angular momenta indiscriminately.

conservation of angular momentum

Conservation of angular momentum about a given axis occurs when, for a system of rigid bodies, the sum of the moments of the external impulses about that axis is zero.
In many engineering applications, the linear momentum is not conserved, yet the angular momentum HP of the system about a given point P is conserved.
Problems involving the conservation of angular momentum about a point P using the general method of impulse and momentum, i.e., by drawing impulse-momentum diagrams. The equation shown in the left figure can be obtained by summing and equating moments about P.

reflection:

The last module contains the method of work and energy in the method of impulse and momentum. The topics helped me to garner knowledge on how to use work and energy methods and its application in the analysis of motion of the particles. I learned that the total work of internal forces acting on the particles of a given rigid body is always zero. The work of a couple acting on rigid body is equivalent to inegral of moment of couple with respect to the body's angular displacement. Also, I found out that the friction force acting on a point of contact between the rigid body and surface does no work if a rigid body rolls without sliding on a fixed surface. Thus, I learned that the equation used in solving kinetic energy in rotation about fixed axis can be used in analyzing motion of systems of rigid bodies, as long as the forces acting on bodies are included in the computation of work of external forces.

The latter part discusses the principle of impulse and momentum in solving problems involving velocities and time, and the concept of conservation of angular momentum of rigid bodies in plane motion. Here, I realized that the system of momenta is equal to the linear momentum vector that is attached to the mass center and to the angular momentum couple. In addition, the principle of impulse and momentum can be used in solving problems with three or more unknowns. This could be made possible by including a linear momentum vector and momentum couple for each moving part of the system.

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