Rigid Body Dynamics chapter 10 continues around and around we go Rigid Body Rotation 1 KE i mi vi2 2 vi ri 1 1 KE i mi 2 ri 2 mi ri 2 2 2 2 1 1 KE total mi ri 2 2 I 2 2 i 2 Moment of Inertia Rotation Axis I mi ri 2 i 2 CORRESPONDENCE 1 2 1 2 mv I 2 2 v m I Rotational Kinetic Energy We there have an analogy between the kinetic energies associated with linear motion K mv 2 and the kinetic energy associated with rotational motion KR I 2 Rotational kinetic energy is not a new type of energy the form is different because it is applied to a rotating object The units of rotational kinetic energy are also Joules J Important Concept Moment of Inertia The The definition of moment of inertia is I ri 2mi i dimensions of moment of inertia are ML2 and its SI units are kg m2 We can calculate the moment of inertia of an object more easily by assuming it is divided into many small volume elements each of mass mi Moment of Inertia cont We can rewrite the expression for I in terms of m lim 2 2 I Dmi r D m r 0 i i dm i With the small volume segment assumption If is constant the integral can be evaluated with I rotherwise r 2dV known geometry its variation with position must be known Question WHAT IS THE MOMENT OF INERTIA OF THIS OBJECT Let s Look at the possibilities c d Two balls with masses M and m are connected by a rigid rod of length L and negligible mass as in Figure P10 22 For an axis perpendicular to the rod show that the system has the minimum moment of inertia when the axis passes through the center of mass Show that this moment of inertia is I L2 where mM m M Remember the Various Densities Volumetric Mass Density mass per unit volume m V Face Mass Density mass per unit thickness of a sheet of uniform thickness t t Linear Mass Density mass per unit length of a rod of uniform cross sectional area m L Moment of Inertia of a Uniform Thin Hoop Since this is a thin hoop all mass elements are the same distance from the center I r dm R 2 I MR 2 2 dm Moment of Inertia of a Uniform Rigid Rod The shaded area has a mass dm dx Then the moment of inertia is M I r dm x dx L 2 L 1 I ML2 12 2 L 2 2 Moment of Inertia of a Uniform Solid Cylinder Divide the cylinder into concentric shells with radius r thickness dr and length L Then for I dV L 2 rdr I r 2dm r 2 2pr Lr dr 1 I z MR 2 2 Moments of Inertia of Various Rigid Objects Parallel Axis Theorem In the previous examples the axis of rotation coincided with the axis of symmetry of the object For an arbitrary axis the parallel axis theorem often simplifies calculations The theorem states I ICM MD 2 I is about any axis parallel to the axis through the center of mass of the object ICM is about the axis through the center of mass D is the distance from the center of mass axis to the arbitrary axis Howcome The ri mi L Not in same plane new axis is parallel to the old axis of rotation Assume that the object rotates about an axis parallel to the z axis The new axis is parallel to the original axis mi From the top L NEW ri OLD Cen I new mi L ri 2 i note a a a 2 I new mi L L 2L ri ri 2 i I new ML m r 2L mi ri 2 2 i 1 i 2 I new ML I old i 2L mi ri i t er of M ass Remember the Center of Mass rCM 1 M mr i i i Since for our problem the sum is ABOUT the center of mass rCM must be zero rCM 0 So 2 I new ML I old ZERO Inew ICM ML2 2L mi ri i Parallel Axis Theorem Example The axis of rotation goes through O The axis through the center of mass is shown The moment of inertia about the axis through O would be IO ICM MD 2 Moment of Inertia for a Rod Rotating Around One End The moment of inertia of the rod about its center is 1 I CM ML2 12 D is L Therefore I I CM MD 2 2 1 L 1 I ML2 M ML2 12 2 3 Many machines employ cams for various purposes such as opening and closing valves In Figure P10 29 the cam is a circular disk rotating on a shaft that does not pass through the center of the disk In the manufacture of the cam a uniform solid cylinder of radius R is first machined Then an off center hole of radius R 2 is drilled parallel to the axis of the cylinder and centered at a point a distance R 2 from the center of the cylinder The cam of mass M is then slipped onto the circular shaft and welded into place What is the kinetic energy of the cam when it is rotating with angular speed about the axis of the Torque Another Vector F Torque is the tendency of a force to rotate an object about some axis Torque Torque is a vector r F sin Fd rXF F is the force is the angle the force makes with the horizontal d is the moment arm or lever arm More Torqueing The moment arm d is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d r sin Torque horizontal component of F F cos has no tendency to produce a rotation Torque will have direction The If the turning tendency of the force is counterclockwise the torque will be positive If the turning tendency is clockwise the torque will be negative Right Hand Screw Rule Net Torque The force F1 will tend to cause a counterclockwise rotation about O The force F2 will tend to cause a clockwise rotation about O F1d1 F 2d2 Torque vs Force Forces motion Described by Newton s Second Law Forces motion can cause a change in linear can cause a change in rotational The effectiveness of this change depends on the force and the moment arm The change in rotational motion depends on the torque Torque Units The SI units of torque are N m Although torque is a force multiplied by a distance it is very different from work and energy The units for torque are reported in N m and not changed to Joules Torque and Angular Acceleration Consider a particle of mass m rotating in a circle of radius r under the influence of tangential force Ft The tangential force provides a tangential acceleration Ft mat …