轉動力學 (Rotational Motion) Chapter 10 Rotation

10.2 The Rotational Variables A rigid body is a body that can rotate with all its parts locked together and without any change in its shape. A fixed axis means that the rotation occurs about an axis that does not move. Figure skater Sasha Cohen in motion of pure rotation about a vertical axis. (Elsa/Getty Images, Inc.)

Rigid Object ( 剛體 ) A rigid object is one that is nondeformable The relative locations of all particles making up the object remain constant. All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible Q

Angular Position The arc length s and r are related: s =   r

Conversions Comparing degrees and radians Converting from degrees to radians

Angular Displacement and Angular Velocity Angular Displacement Average Angular Speed Instantaneous Angular Speed ( rad/s or s -1 )

Angular Acceleration Average angular acceleration Instantaneous angular acceleration ( rad/s 2 or s -2 )

Directions of  and  The directions are given by the right-hand rule

10.4: Rotation with Constant Angular Acceleration Just as in the basic equations for constant linear acceleration, the basic equations for constant angular acceleration can be derived in a similar manner. The constant angular acceleration equations are similar to the constant linear acceleration equations.

Comparison Between Rotational and Linear Equations

Relating Linear and Angular Variables Displacements Tangential Speeds The period of revolution T for the motion of each point and for the rigid body itself is given by ;

Tangential Acceleration Centripetal Acceleration Relating Linear and Angular Variables

10.6: Kinetic Energy of Rotation An object rotating about some axis with an angular speed, , has rotational kinetic energy even though it may not have any translational kinetic energy Each particle has a kinetic energy of K i = 1/2 m i v i 2 Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute v i =  i r

10.6: Kinetic Energy of Rotation For an extended rotating rigid body, treat the body as a collection of particles with different speeds, and add up the kinetic energies of all the particles to find the total kinetic energy of the body: (m i is the mass of the i th particle and v i is its speed). (  is the same for all particles). The quantity in parentheses on the right side is called the rotational inertia (or moment of inertia I ) of the body with respect to the axis of rotation. It is a constant for a particular rigid body and a particular rotation axis. (That axis must always be specified.) Therefore,

Calculating the Rotational Inertia (moment of inertia I)

此系統的轉動慣量 I = 2mr 2 此系統的轉動慣量 ( 環 ) 質點的轉動慣量 mm rr R M

Moment of Inertia of a Uniform Solid Cylinder Radius R, mass M and length L 質量均勻分佈 Divide the cylinder into concentric shells with radius r, thickness dr and length L M

Moment of Inertia of a Uniform Solid Cylinder 整個系統的轉動慣量 M

10.7: Calculating the Rotational Inertia If a rigid body consists of a great many adjacent particles (it is continuous, like a Frisbee), we consider an integral and define the rotational inertia of the body as

Moments of Inertia of Various Rigid Objects

Parallel-Axis Theorem ( 平行軸定理 ) For an arbitrary axis, the parallel-axis theorem often simplifies calculations I p = I CM + MD 2 I p is about any axis parallel to the axis through the center of mass of the object I CM is about the axis through the center of mass D is the distance from the center of mass axis to the arbitrary axis C.M. P D

P x y x y P x y x y △m△m rPrP r R rPrP r R Proof of the Parallel-Axis Theorem

C.M. P x y x y △m△m rPrP r R rPrP r R

P x y x y △m1△m1 r1r1 r1 r1 R △m2△m2 r2r2 r1r1 r1 r1 R r2r2 r2 r2 r2 r2

P x y x y 0 I P = I CM + MR 2 (Parallel-Axis Theorem) R

Perpendicular-Axis Theorem ( 垂直軸定理 ) I z = I x + I y riri Prove it Hint :

R M 圓盤 利用垂直軸定理，系統的軸之轉動慣量 = ?

A rigid body is made of three identical thin rods, each with length L, fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal. What is the angular speed of the body when the plane of the H is vertical ? O L M M L/2 CM = ? I O = ? i f C.M O CM M M M M Ex 1

A rigid body is made of three identical thin rods, each with length L, fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal. What is the angular speed of the body when the plane of the H is vertical ? O L M M L/2 CM = ? I O = ? i f C.M O CM M M mass of this leg is zero Ex 2

The Center of Mass There is a special point in a system or object, called the center of mass, that moves as if all of the mass of the system is concentrated at that point The system will move as if an external force were applied to a single particle of mass M located at the center of mass M is the total mass of the system

Center of Mass, position The center of mass can be located by its position vector, is the position of the i th particle, defined by

Center of Mass, Coordinates The coordinates of the center of mass are where M is the total mass of the system

Center of Mass, Example Both masses are on the x-axis The center of mass is on the x-axis The center of mass is closer to the particle with the larger mass

An object can be considered a distribution of small mass elements,  m i The center of mass is located at position Center of Mass, Object with a continuous mass distribution

The coordinates of the center of mass of the object are Center of Mass, Object with a continuous mass distribution

Forces In a System of Particles The acceleration can be related to a force If we sum over all the internal forces, they cancel in pairs and the net force on the system is caused only by the external forces

The system will move as if an external force were applied to a single particle of mass M located at the center of mass Forces In a System of Particles

A rigid body is made of three identical thin rods, each with length L, fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal. What is the angular speed of the body when the plane of the H is vertical ? O L M M L/2 CM = ? I O = ? i f C.M O CM M M M M Ex 1

A rigid body is made of three identical thin rods, each with length L, fastened together in the form of a letter H (Figure). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal. What is the angular speed of the body when the plane of the H is vertical ? O L M M L/2 CM = ? I O = ? i f C.M O CM M M mass of this leg is zero Ex 2

Torque ( 力矩 )

Torque, , is the tendency of a force to rotate an object about some axis. Torque is a vector  = r F sin  = F d

Vector Product, General Given any two vectors, and The vector product is defined as a third vector, whose magnitude is The direction of C is given by the right-hand rule

10.8: Torque The ability of a force F to rotate the body depends on both the magnitude of its tangential component F t, and also on just how far from O, the pivot point, the force is applied. To include both these factors, a quantity called torque  is defined  as: OR, where is called the moment arm of F.

10.9: Newton’s Law of Rotation For more than one force, we can generalize:

Torque and Angular Acceleration on a Particle ΔmΔm F sin  F cos 

Torque and Angular Acceleration on a Rigid Object ΔmiΔmi F i sin  F i cos 

Torque and Angular Acceleration on a Rigid Object

Work in Rotational Motion Find the work done by a force on the object as it rotates through an infinitesimal distance ds = r d  The radial component of the force does no work because it is perpendicular to the displacement FtFt τ = rF t

Work in Rotational Motion, cont Work is also related to rotational kinetic energy: This is the same mathematical form as the work-kinetic energy theorem for translation If an object is both rotating and translating, W =  K +  K R

Power in Rotational Motion The rate at which work is being done in a time interval dt is the power This is analogous to P = Fv in a linear system

Homework Chapter 10 ( page 267 ) 7, 29, 41, 51, 54, 63, 66, 78, 79, 98, 103