Rotation of Rigid Bodies Rotational Motion: in close analogy with linear motion (distance/displacement, velocity, acceleration) Angular measure in “natural units” Angles and Rotation in radians r Angle = arc length / radius s q from one complete circuit, 360o = 2p rad 45o = p/4 rad 90o = p/2 rad 180o = p rad 1 rad = 57.30o

instantaneous angular velocity is given by: an object which rotates about a fixed axis has an average angular velocity wav : usually rad/s but sometime rpm, rps instantaneous angular velocity is given by: s r q

Angular Acceleration: the rate of change of angular speed ac=w2r aT=ar

Example: the angular position of a flywheel is given by q = (2 Example: the angular position of a flywheel is given by q = (2.00 rad/s3) t3. The diameter of this flywheel is .360 m. Find the angular displacement at 2.00s and at 5.00s. Find the average angular velocity between 2.00s and 5.00s. Find the instantaneous angular velocity at 2.00s, 3.50s and 5.00s. Find the average angular acceleration between 2.00s and 5.00s. Find the instantaneous angular acceleration at 3.50s. Calculate the speed of a point on the edge of the flywheel at 3.50s. Calculate the tangential and radial acceleration of a point on the edge of the flywheel at 3.50s.

Rotation with constant angular acceleration (just like linear 1-d) watch units!!!

Example: A wheel with an initial angular velocity of 4 Example: A wheel with an initial angular velocity of 4.00 rad/s undergoes a constant acceleration of -1.20 rad/s2. What is the angular displacement and angular velocity at t= 3.00s? How many rotations does the wheel make before coming to rest? Example: A discus thrower turns with an angular acceleration of 50 rad/s2, moving the discus around a constant radius of .800 m. Find the tangential and centripetal acceleration when the discus has an angular velocity of 10 rad/s.

Example: An airplane propeller is to rotate at maximum of 2400 rpm while the aircraft’s forward velocity is 75.0 m/s. How big can the propeller be if the the speed of the tips relative to the air is not to exceed 270 m/s? At this speed, what is the acceleration of the propeller tip? Example: discus chain-linked gears, belt drives etc: linear velocity vs angular velocity.

Rotational Kinetic Energy for a single point particle v r m v1 m1 v3 for a solid rotating object, piece by piece r1 r2 m2 r3 m3 v2

Example: Three masses are connected by light bracing as shown Example: Three masses are connected by light bracing as shown. What is the moment of inertia about each of the axis shown? What would the kinetic energy for be for rotation at 4.00 rad/s about each of the axis shown? .10kg .50m .30m .20kg .30kg .40m axis perpendicular to plane

Moments of inertia for some common geometric solids b a R b R

the work done by the force, the kinetic energy of the drum, A cord is wrapped around a solid 50 kg cylinder which has a diameter of 0.120 m, and which rotates (frictionlessly) about an axis through its center. A 9.00 N force is applied to the end of the cable, causing the cable to unwind and the drum (initially at rest) to rotate. After the cable has unwound a distance 2.00m, determine the work done by the force, the kinetic energy of the drum, the rotational velocity of the drum, and the speed of the unwinding cable. 9.00 N

Combining Translation and Rotation KE = KEtranslation + KErotation = ½mv2 + ½Iw2 A connection for rolling without slipping: s = q r v = w r a = a r, a : angular acceleration Gravitational Potential Energy for an extended object use center of mass: U = mgycm The Great Race: 2 objects rolling (from rest) down the same incline lost PE = gained KE same radius, object with the smallest I has most v => wins race

A mass m is suspended by a string wrapped around a pulley of radius R and moment of inertia I. The mass and pulley are initially at rest. After the mass has dropped a height h, determine the relation between the final speed of the mass and the given parameters (m, I, h). Examine the special case where the pulley is a uniform disk of mass M.

The Parallel Axis Theorem The moment of Inertia about an axis is related in a simple way to the moment of inertia about a parallel axis which runs through the center of mass: Ip = Icm + MR2 p a b cm

Example: An 3. 6 kg object is found to have a moment of inertia of Example: An 3.6 kg object is found to have a moment of inertia of .132 kg m2 about an axis which is found to be .15m from is center of mass. What is the moment of inertia of this object about a parallel axis which does go through the objects center of mass? Example: Find the moment of inertia of a thin uniform disk about an axis perpendicular to its flat surface, running along the edge of the disk.

Beyond discrete masses: adventures in vector calculus!