Monday, Nov. 8, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu11. Torque2. Torque and Vector Product3. Moment of Inertia4. Torque and Angular Acceleration5. Parallel Axis Theorem6. Rotational Kinetic EnergyPHYS 1443 – Section 003Lecture #18Monday, Nov. 8, 2004Dr. Jaehoon YuToday’s homework is HW #10 on the class web page!!Monday, Nov. 8, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu2Announcements• Homework site will be down for at least two weeks– Homework problems will be posted on the class web page one HW at a time per week till HW site turns back on. – These problems will not be identical to your final HW problems but all the procedures will be identical. So I suggest you to work out the formula for your problem so that you can plug in numbers for your actual problems.– I have your HW grade up to HW9.• Class Average went up to 47.3 after the fix• Will spend about an hour today to discuss your mid-semester progress discussionsMonday, Nov. 8, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu3TorqueTorque is the tendency of a force to rotate an object about an axis. Torque, τ, is a vector quantity.≡τMagnitude of torque is defined as the product of the force exerted on the object to rotate it and the moment arm.FφdLine of ActionConsider an object pivoting about the point Pby the force F being exerted at a distance r. PrMoment armThe line that extends out of the tail of the force vector is called the line of action.The perpendicular distance from the pivoting point P to the line of action is called Moment arm.When there are more than one force being exerted on certain points of the object, one can sum up the torque generated by each force vectorially. The convention for sign of the torque is positive if rotation is in counter-clockwise and negative if clockwise. d2F221τττ+=∑2211dFdF−==φsinrF FdMonday, Nov. 8, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu4xyzOTorque and Vector ProductThe magnitude of torque given to the disk by the force F isLet’s consider a disk fixed onto the origin O and the force F exerts on the point p. What happens?φτsinFr=BAC ×≡The disk will start rotating counter clockwise about the Z axisThe above operation is called Vector product or Cross productFθτ=rxFrpBut torque is a vector quantity, what is the direction? How is torque expressed mathematically? Fr ×≡τWhat is the direction? The direction of the torque follows the right-hand rule!!What is the result of a vector product?Another vectorWhat is another vector operation we’ve learned?Scalar productθcosBABAC =⋅≡Result? A scalarθsinBABAC =×=Monday, Nov. 8, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu5Properties of Vector Product()dtBAd ×Vector Product is Non-commutative What does this mean?If the order of operation changes the result changesABBA×≠×ABBA×−=×Following the right-hand rule, the direction changesVector Product of two parallel vectors is 0.BAC ×=0=× AAThus,If two vectors are perpendicular to each otherBA×Vector product follows distribution law()CBA +×The derivative of a Vector product with respect to a scalar variable is θsinBA= 00sin == BAθsinBA=D90sinBA=ABBA ==CABA ×+×=dtBdABdtAd×+×=Monday, Nov. 8, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu6More Properties of Vector ProductThe relationship between unit vectors, kji and ,kkjjii ×=×=×=× BAVector product of two vectors can be expressed in the following determinant form 0=ji × ij ×−=k=kj ×jk ×−=i=ik ×ki ×−=j=zyxzyxBBBAAAkjizyzyBBAAi=zxzxBBAAj−yxyxBBAAk+()iBABAyzzy
View Full Document
Unlocking...