PHYS 1441 Section 002 Lecture #19 Monday, Nov. 22, 2010 Dr. Jaehoon Yu Equations of Rotational Kinematics Relationship Between Angular and Linear Quantities Rolling Motion of a Rigid Body Todays homework is homework #11, due 10pm, Tuesday, Nov. 30!! Monday, Nov. 22, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 1 Announcements Quiz 4 results Class average: 30/70 Equivalent to: 43/100 Previous quizzes: 53/100, 56/100 and 64/100 Top score: 69/70 This Wednesday Review with problem solving exercise Monday, Nov. 22, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon

Yu 2 Reminder: Extra-Credit Special Project Derive the formula for the final velocity of two objects which underwent an elastic collision as a function of known quantities m1, m2, v01 and v02 in page 12 of this lecture note in a far greater detail than the note. 20 points extra credit Show mathematically what happens to the final velocities if m1=m2 and describe in words the resulting motion. 5 point extra credit Due: Start of the class Monday, Nov. 29 Monday, Nov. 22, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 3 Angular Displacement, Velocity, and Acceleration Angular displacement is defined as if f i ii

f i How about the average angular velocity, the rate of change of angular displacement? t f ti t Unit? rad/s Dimension? [T-1] By the same token, the average angular acceleration, rate of change of the angular velocity, is defined as Unit? rad/s2 Dimension? [T-2] f i t t f ti When rotating about a fixed axis, every particle on a rigid object rotates through the same angle and has the same angular speed and angular acceleration. Monday, Nov. 22, 2010

PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 4 Ex. Gymnast on a High Bar A gymnast on a high bar swings through two revolutions in a time of 1.90 s. Find the average angular velocity of the gymnast. What is the angular displacement? 2 rad 2.00 rev 1 rev 12.6 rad Why negative? Because he is rotating clockwise!! 12.6 rad 6.63rad s 1.90 s Monday, Nov. 22, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 5 Ex. A Jet Revving Its Engines As seen from the front of the engine, the fan blades are rotating with an angular speed of -110 rad/s. As the plane takes off, the angular velocity of the blades reaches -330 rad/s in a

time of 14 s. Find the angular acceleration, assuming it to be constant. f i t t f ti 330 rad s 110 rad s 14 s Monday, Nov. 22, 2010 16 rad s 2 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 6 Rotational Kinematics The first type of motion we have learned in linear kinematics was under the constant acceleration. We will learn about the rotational motion under constant angular acceleration, because these are the simplest motions in both cases. Just like the case in linear motion, one can obtain Angular velocity under constant = + f 0 angular acceleration: Linear

v vo + t kinematics Angular displacement under = + 0t + f 0 constant angular acceleration: 2 x f x0 + vot + 12 t Linear kinematics One can also obtain Linear kinematics 2 f 2 o v v + 2 x f xi Monday, Nov. 22, 2010

2 f t 1 2 t 2 = + 2 f 0 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 2 0 7 Problem Solving Strategy Visualize the problem by drawing a picture. Write down the values that are given for any of the five kinematic variables and convert them to SI units. Remember that the unit of the angle must be radians!! Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. When the motion is divided into segments, remember that the final angular velocity of one segment is the initial velocity for the next. Keep in mind that there may be two possible answers to a kinematics problem. Monday, Nov. 22, 2010

PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 8 Ex. Blending with a Blender The blade is whirling with an angular velocity of +375 rad/s when the puree button is pushed in. When the blend button is pushed, the blade accelerates and reaches a greater angular velocity after the blade has rotated through an angular displacement of +44.0 rad. The angular acceleration has a constant value of +1740 rad/s2. Find the final angular velocity of the blade. o +44.0rad +1740rad/s2 ? +375rad/s Which kinematic2 eq? t 2 o2 + 2

o + 2 2 375 rad s + 2 1740 rad s 2 44.0rad 542 rad s +542 rad s Why Because the blade is accelerating in Which Monday, Nov. 22, 2010 PHYS Fall 2010 Dr. Jaehoon 9 sign? ? 1441-002,counter-clockwise! Yu Relationship Between Angular and Linear Quantities What do we know about a rigid object that rotates about a fixed axis of rotation? Every particle (or masslet) in the object moves in a circle centered at the same axis of rotation with the same angular velocity. When a point rotates, it has both the linear and angular components in its motion. What is the linear component of the motion you see? Linear velocity along the tangential direction. How do we related this linear component of the motion with angular component? The arc-length is l r So the tangential speed v is v The direction of

follows the right-hand rule. l r r t t t r What does this relationship tell you about Although every particle in the object has the same the tangential speed of the points in the angular speed, its tangential speed differs and is object and their angular speed? proportional to its distance from the axis of rotation. Monday, Nov. 22, 2010 The away the particle is from the center of PHYS 1441-002, Fallfarther 2010 Dr. Jaehoon Yu the higher the tangential speed. rotation, 10 Is the lion faster than the horse? A rotating carousel has one child sitting on a horse near the outer edge and another child on a lion halfway out from the center. (a) Which child has the greater linear speed? (b) Which child has the greater angular speed?

(a) Linear speed is the distance traveled divided by the time interval. So the child sitting at the outer edge travels more distance within the given time than the child sitting closer to the center. Thus, the horse is faster than the lion. (b) Angular speed is the angle traveled divided by the time interval. The angle both the children travel in the given time interval is the same. Thus, both the horse and the lion have the same angular speed. Monday, Nov. 22, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 11 How about the acceleration? How many different linear acceleration components do you see in a circular motion and what are they? Two Tangential, at, and the radial acceleration, aSince the tangential speed v is v r r. t What does this relationship tell you? The magnitude of tangential a vtf vt0 r f r 0 f 0 r t r t t acceleration at is

t Although every particle in the object has the same angular acceleration, its tangential acceleration differs proportional to its distance from the axis of rotation. v 2 ( r )2 ar r r 2 The radial or centripetal acceleration ar r is What does The father away the particle is from the rotation axis, the more radial this tell you? acceleration it receives. In other words, it receives more centripetal force. Total linear acceleration is Monday, Nov. 22, 2010 2 2 a a + a ( r ) + ( r ) r 2 + 4 2 t 2 r 2 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 12

Ex. A Helicopter Blade A helicopter blade has an angular speed of 6.50 rev/s and an angular acceleration of 1.30 rev/s2. For point 1 on the blade, find the magnitude of (a) the tangential speed and (b) the tangential acceleration. rev 2 rad 6.50 40.8 rad s s 1 rev vT rv 3.00 m 40.8 rad s 122 m s rev 2 rad 2 8.17 rad s 1.30 2 s 1 rev aT r 3.00 m 8.17 rad s 2 24.5 m s 2 Monday, Nov. 22, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 13