1. Theoretical Mechanics DYNAMICS * Navigation: Right (Down) arrow – next slide Left (Up) arrow – previous slide Esc – Exit Notes and Recommendations: ruschev@tu-plovdiv.bg Technical University of Sofia Branch Plovdiv

2. Lecture 8 Mass Moment of Inertia The mass moment of inertia of a system of particles or a body is a property that measures the resistance of the system or the body to angular acceleration in the same way that mass is a measure of the body's resistance to acceleration. Definition. The mass moment of inertia of a body about an axis is the integral of the products of all the elements of mass dm which compose the body by the squares of its shortest distances from this axis. The integration must be performed throughout the entire volume of the body. In the special case of  being a constant, this term may be factored out of the integral, and the integration is then purely a function of geometry

3. Lecture 8 Mass Moment of Inertia Parallel-axis theorem If the moment of inertia of the body about an axis passing through the body's mass center is known, then the moment of inertia about any other parallel axis can be determined by using the parallel-axis theorem.

4. Lecture 8 Mass Moment of Inertia Radius of gyration This is a geometrical property which has units of length. When it and the body's mass М are known, the body's moment of inertia is determined from the equation. Composite bodies The mass moment of inertia of a composite body is the sum of the moments of inertia of the individual parts about the same axis. It is often convenient to treat a composite body as defined by positive volumes and negative volumes. The moment of inertia of a negative element, such as the material removed to from a hole, must be considered a negative quantity.

5. Lecture 8 Mass Moment of Inertia Mass inertia moments of some simple homogeneous bodies 1. Slender rod x dx l x y Determine the moment of inertia J y for the slender rod. The rod's density  and cross-sectional area A are constant. Express the result in terms of the rod's total mass M

6. Lecture 8 Mass Moment of Inertia Mass inertia moments of some simple homogeneous bodies 2. Homogeneous cylinder Determine the moment of inertia and radius of gyration of a homogeneous right-circular cylinder of mass M and radius R about its central axis z

7. Lecture 8 Mass Moment of Inertia Mass inertia moments of some simple homogeneous bodies 3. Homogeneous solid sphere Determine the moment of inertia and radius of gyration of a homogeneous sphere of mass M and radius R about a diameter. Solution. A circular slice of radius y and thickness dx is chosen as the volume element. The moment of inertia about the x -axis of the elemental cylinder is

8. Lecture 8 Mass Moment of Inertia Mass inertia moments of some simple homogeneous bodies 4. Homogeneous rectangular parallelepiped Determine the moment of inertia of a homogeneous rectangular parallelepiped of mass M about the centroidal x 0 -axis and z -axis and about the x -axis through one end. See J.L. Meriam, L.G. Kraige, Engineering Mechanics. Dynamics. 2002. (p. 669)

10. Lecture 8 Mass Moment of Inertia Determine the mass moments of inertia and the radius of gyration of the steel machine element shown with respect to the y axes. (The density of steel is 7850 kg/m3.) Sample Problem

11. Lecture 8 Principle of Work and Energy for a Rigid Body Kinetic Energy of a Rigid Body in Translation We will consider the rigid body as a large number n of particles of mass  m i. When a rigid body is in translation, all the points of the body have the same velocity v. The particle has a velocity v and the particle's kinetic energy is The kinetic energy of the entire body is

12. Lecture 8 Principle of Work and Energy for a Rigid Body Kinetic Energy of a Rigid Body in Rotation When a rigid body is in rotation, the particle has a velocity v i and the particle's kinetic energy is The kinetic energy of the entire body is If the body has an angular velocity  the particle has a velocity v i

13. Lecture 8 Principle of Work and Energy for a Rigid Body Kinetic Energy of a Rigid Body in Plane Motion We will consider the plane motion of a rigid body shown in figure as sum of translation and rotation about center of mass G. If at the instant shown the particle has a velocity v i. The particle's kinetic energy is

14. The first integral represents the total mass M of the body. The second integral is the kinetic energy of the body in rotation about the mass center G. Lecture 8 Principle of Work and Energy for a Rigid Body Kinetic Energy of a Rigid Body in Plane Motion The kinetic energy of the entire body is

15. We therefore write Lecture 8 Principle of Work and Energy for a Rigid Body Kinetic Energy of a Rigid Body in Plane Motion The third integral is equal to and thus to zero, since, which represents the velocity of G relative to the frame, is clearly zero.