Rotational Inertia Rigid Body Real objects have mass
Rotational Inertia
![Rigid Body ] Real objects have mass at points other than the center of](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-2.jpg "Rigid Body ] Real objects have mass at points other than the center of")
Rigid Body ] Real objects have mass at points other than the center of mass. ri ] Each point in an object can be measured from an origin at the center of mass. ] If the positions are fixed compared to the center of mass it is a rigid body.
![Translation and Rotation ] ] The motion of a rigid body includes the motion](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-3.jpg "Translation and Rotation ] ] The motion of a rigid body includes the motion")
Translation and Rotation ] ] The motion of a rigid body includes the motion of its center of mass. This is translational motion v. CM ] ] A rigid body can also move while its center of mass is fixed. This is rotational motion.
![Circular Motion ] Objects in circular motion have kinetic energy. • K = ½](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-4.jpg "Circular Motion ] Objects in circular motion have kinetic energy. • K = ½")
Circular Motion ] Objects in circular motion have kinetic energy. • K = ½ m v 2 ] r m The velocity can be converted to angular quantities. • K = ½ m (r )2 • K = ½ ( m r 2 ) 2
![Integrated Mass ] The kinetic energy is due to the kinetic energy of the](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-5.jpg "Integrated Mass ] The kinetic energy is due to the kinetic energy of the")
Integrated Mass ] The kinetic energy is due to the kinetic energy of the individual pieces. ] The form is similar to linear kinetic energy. • KCM = ½ m v 2 • Krot = ½ I 2 ] The term I is the moment of inertia of a particle.
![Moment of Inertia Defined ] The moment of inertia measures the resistance to a](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-6.jpg "Moment of Inertia Defined ] The moment of inertia measures the resistance to a")
Moment of Inertia Defined ] The moment of inertia measures the resistance to a change in rotation. • Mass measures resistance to change in velocity • Moment of inertia I = mr 2 for a single mass ] The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.
![Two Spheres ] A spun baton has a moment of inertia due to each](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-7.jpg "Two Spheres ] A spun baton has a moment of inertia due to each")
Two Spheres ] A spun baton has a moment of inertia due to each separate mass. • I = mr 2 + mr 2 = 2 mr 2 m r m ] If it spins around one end, only the far mass counts. • I = m(2 r)2 = 4 mr 2
![Mass at a Radius ] ] Extended objects can be treated as a sum](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-8.jpg "Mass at a Radius ] ] Extended objects can be treated as a sum")
Mass at a Radius ] ] Extended objects can be treated as a sum of small masses. A straight rod (M) is a set of identical masses Dm. ] The total moment of inertia is ] Each mass element contributes ] The sum becomes an integral distance r to r+Dr length L axis
![Rigid Body Rotation ] The moments of inertia for many shapes can found by](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-9.jpg "Rigid Body Rotation ] The moments of inertia for many shapes can found by")
Rigid Body Rotation ] The moments of inertia for many shapes can found by integration. • Ring or hollow cylinder: I = MR 2 • Solid cylinder: I = (1/2) MR 2 • Hollow sphere: I = (2/3) MR 2 • Solid sphere: I = (2/5) MR 2
![Point and Ring ] The point mass, ring and hollow cylinder all have the](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-10.jpg "Point and Ring ] The point mass, ring and hollow cylinder all have the")
Point and Ring ] The point mass, ring and hollow cylinder all have the same moment of inertia. ] • I = MR 2 ] The rod and rectangular plate also have the same moment of inertia. • I = (1/3) MR 2 All the mass is equally far away from the axis. ] The distribution of mass from the axis is the same. M R M M length R axis length R
![Parallel Axis Theorem ] ] Some objects don’t rotate about the axis at the](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-11.jpg "Parallel Axis Theorem ] ] Some objects don’t rotate about the axis at the")
Parallel Axis Theorem ] ] Some objects don’t rotate about the axis at the center of mass. The moment of inertia depends on the distance between axes. ] The moment of inertia for a rod about its center of mass: h = R/2 M axis
![Spinning Energy ] How much energy is stored in the spinning earth? ] The](http://slidetodoc.com/presentation_image/31b8835d8ff90e0171831832159bef75/image-12.jpg "Spinning Energy ] How much energy is stored in the spinning earth? ] The")
Spinning Energy ] How much energy is stored in the spinning earth? ] The earth spins about its axis. • The moment of inertia for a sphere: I = 2/5 M R 2 • The kinetic energy for the earth: Krot = 1/5 M R 2 2 • With values: K = 2. 56 x 1029 J The energy is equivalent to about 10, 000 times the solar energy received in one year.
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