MOMENT OF INERTIA • Today’s Objectives: • Students will be able to: • Determine the mass moment of inertia of a rigid body or a system of rigid bodies. In-Class Activities: • Mass Moment of Inertia • Parallel-Axis Theorem • Composite Bodies
APPLICATIONS The flywheel on this tractor engine has a large mass moment of inertia about its axis of rotation. Once the flywheel is set into motion, it will be difficult to stop. This tendency will prevent the engine from stalling and will help it maintain a constant power output. Does the mass moment of inertia of this flywheel depend on the radius of the wheel? Its thickness?
APPLICATIONS (continued) The crank on the oil-pump rig undergoes rotation about a fixed axis that is not at its mass center. The crank develops a kinetic energy directly related to its mass moment of inertia. As the crank rotates, its kinetic energy is converted to potential energy and vice versa. Is the mass moment of inertia of the crank about its axis of rotation smaller or larger than its moment of inertia about its center of mass?
Understanding the Mass Moment of Inertia a) This ballerina is rotating about an axis passing through her center of mass. b) This ballerina is rotating about an axis which is not passing through her center of mass. In which case the rotation is easier and there is less resistance against angular acceleration? Case a
The mass moment of inertia is a measure of an object’s resistance to rotation. Thus, the object’s mass and how it is distributed both affect the mass moment of inertia. Mathematically, it is the integral I = r2 dm = r2r dV In this integral, r acts as the moment arm of the mass element and r is the density of the body. Thus, the value of I differs for each axis about which it is computed. m V MOMENT OF INERTIA (Section 17.1) In Section 17.1, the focus is on obtaining the mass moment of inertia via integration.
MOMENT OF INERTIA (continued) The figures below show the mass moment of inertia formulations for two flat plate shapes commonly used when working with three dimensional bodies. The shapes are often used as the differential element being integrated over the entire body.
PROCEDURE FOR ANALYSIS When using direct integration, only symmetric bodies having surfaces generated by revolving a curve about an axis will be considered. Shell element • If a shell element having a height z, radius r = y, and thickness dy is chosen for integration, then the volume element is dV = (2py)(z)dy. • This element may be used to find the moment of inertia Iz since the entire element, due to its thinness, lies at the same perpendicular distance y from the z-axis. Disk element • If a disk element having a radius y and a thickness dz is chosen for integration, then the volume dV = (py2)dz. • Using the moment of inertia of the disk element, we can integrate to determine the moment of inertia of the entire body.
READING QUIZ 1. Mass moment of inertia is a measure of the resistance of a body to A) translational motion. B) deformation. C) angular acceleration. D) impulsive motion. 2. Mass moment of inertia is always A) a negative quantity. B) a positive quantity. C) an integer value. D) zero about an axis perpendicular to the plane of motion.
Solution: The moment of inertia of a disk about an axis perpendicular to its plane is I = 0.5 m r2. Thus, for the disk element, we have dIy = 0.5 (dm) x2 where the differential mass dm = r dV = rpx2 dy. 1 1 rpx4 rp p(5) 8 ò ò = = = = Iy dy y dy 0 . 873 slug•ft2 2 2 18 0 0 EXAMPLE Given: The volume shown with r = 5 slug/ft3. Find: The mass moment of inertia of this body about the y-axis. Plan: Find the mass moment of inertia of a disk element about the y-axis, dIy, and integrate.
PARALLEL-AXIS THEOREM If the mass moment of inertia of a body about an axis passing through the body’s mass center is known, then the moment of inertia about any other parallelaxis may be determined by using the parallel axis theorem, I = IG + md2 where IG = mass moment of inertia about the body’s mass center m = mass of the body d = perpendicular distance between the parallel axes
Radius of Gyration The mass moment of inertia of a body about a specific axis can be defined using the radius of gyration (k). The radius of gyration has units of length and is a measure of the distribution of the body’s mass about the axis at which the moment of inertia is defined. I = m k2 or k = (I/m) PARALLEL-AXIS THEOREM (continued) Composite Bodies If a body is constructed of a number of simple shapes, such as disks, spheres, or rods, the mass moment of inertia of the body about any axis can be determined by algebraically adding together all the mass moments of inertia, found about the same axis, of the different shapes.
Solution: The center of mass is located relative to the pin at O at a distance y, where 10 10 + 1( ) 2( ) å miyi 32.2 32.2 = = = y 1.5 ft å 10 10 mi + 32.2 32.2 EXAMPLE II Given: Two rods assembled as shown, with each rod weighing 10 lb. Find: The location of the center of mass G and moment of inertia about an axis passing through G of the rod assembly. Plan: Find the centroidal moment of inertia for each rod and then use the parallel axis theorem to determine IG.
The moment of inertia IG may then be calculated by using the parallel axis theorem. IG = [I + m(y-1)2]OA + [I + m(2-y)2]BC IG = [0.104 + (10/32.2)(0.5)2] + [0.104 + (10/32.2)(0.5)2] IG = 0.362 slug·ft2 EXAMPLE II (continued) The mass moment of inertia of each rod about an axis passing through its center of mass is calculated by using the equation I = (1/12)ml2 = (1/12)(10/32.2)(2)2 = 0.104 slug·ft2
2. The mass moment of inertia of a thin ring of mass m and radius R about the Z axis is ______ . A) (1/2) m R2 B) m R2 C) (1/4) m R2 D) 2 m R2 CONCEPT QUIZ 1. The mass moment of inertia of a rod of mass m and length L about a transverse axis located at its end is _____ . A) (1/12) m L2 B) (1/6) m L2 C) (1/3) m L2 D) m L2
GROUP PROBLEM SOLVING Given: The density (r) of the object is 5 Mg/m3. Find: The radius of gyration, ky. Plan: Use a disk element to calculate Iy, and then find ky. Solution: Using a disk element (centered on the x-axis) of radius y and thickness dx yields a differential mass dm of dm = r p y2 dx = r p (50x) dx The differential moment of inertia dIy’ about the y-axis passing through the center of mass of the element is dIy’ = (1/4)y2 dm = 625 r p x2 dx
Integrate to determine Iy: Iy = 21.67x109rp 200 625 50 ò ò = = + = + Iy dIy rp(625x2 50x3)dx rp[( )(2003) ( )(2004)] 3 4 0 The mass of the solid is Therefore Iy = 21.67x103 m and ky = Iy /m = 147.2 mm 200 ò ò = = = = m dm rp(50x)dx rp(25)(200 )2 1x106 r p 0 GROUP PROBLEM SOLVING (continued) Using the parallel axis theorem, the differential moment of inertia about the y-axis is then dIy = dIy’ + dm(x2) = rp(625x2 + 50x3) dx
ATTENTION QUIZ 1. The mass moment of inertia of any body about its center of mass is always A) maximum. B) minimum. C) zero. D) None of the above. 2. If the mass of body A and B are equal but kA = 2kB, then A) IA = 2IB . B) IA = (1/2)IB . C) IA = 4IB . D) IA = (1/4)IB .