Transcript Class26
Class 26 - Rotation Chapter 10 - Monday October 25th •Review Kinetic energy, moment of inertia, parallel axis theorem •Newton's second law for rotation •Work, power and rotational kinetic energy •Sample problems Reading: pages 241 thru 263 (chapter 10) in HRW Read and understand the sample problems Assigned problems from chapter 10 (due Sunday October 31st at 11pm): 2, 10, 28, 30, 36, 44, 48, 54, 58, 64, 78, 124 Kinetic energy of rotation K 12 I 2 I mi ri 2 Therefore, for a continuous rigid object: I r 2 dm 2 r dV Parallel axis theorem Icom h rotation c.o.m. axis axis •If moment of inertia is known about an axis though the center of mass (c.o.m.), then the moment of inertia about any parallel axis is: I I com Mh 2 •It is essential that these axes are parallel; as you can see from table 10-2, the moments of inertia can be different for different axes. Some rotational inertia Torque •The ability to rotate an object about an axis depends not only on the force you apply, but also where and in what direction you apply the force. •In particular, the further away from the axis that you push, the easier it is to rotate the object. •We define a quantity called torque, : r F sin •The SI unit for torque is N.m, which is the same as that for work (joule). •However, torque and work are quite different and should not be confused. Torque •There are two ways to compute torque: r F sin rFt r sin F r F •The direction of the force vector is called the line of action, and r is called the moment arm. •The first equation shows that the torque is equivalently given by the component of force tangential to the line joining the axis and the point where the force acts. •In this case, r is the moment arm of Ft. Torque •Torque is actually a vector quantity given by the following vector product: rF •Thus, torques add like vectors, i.e. if several torques act on an object, the net torque net is given by the vector sum. •In this chapter, you will not need to worry about the vector character of , since we shall only consider rotational motion about a fixed axis. Newton's second law for rotation •We can relate Ft to the tangential acceleration using Newton's second law: Ft mat •The torque is then given by: Ft r mat r m r r mr Substituting r for at. 2 I For a rigid body which is free to rotate about a fixed axis, Fr cannot affect the motion. Work and rotational kinetic energy K K f K i 12 I 2f 12 Ii2 W (Work-kinetic energy theorem) •By now, you should be noticing a pattern in the connections between linear and angular equations: x ; v ; a ; m I; F , etc.. •Then, kinetic energy is calculated by substituting the linear variables by their corresponding angular variables. •We can do the same for work and power. f W d i W f i dW P dt (Work, rotation about a fixed axis) (Work at constant torque) (Power, rotation about a fixed axis) Summarizing relations for translational and rotational motion •Note: work obtained by multiplying torque by an angle - a dimensionless quantity. Thus, torque and work have the same dimensions, but you see that they are quite different.