R-1Rotational MotionWe are going to consider the motion of a rigid body about a fixed axis of rotation. The angle of rotation is measured in radians: s(rads) (dimensionless)rq �Notice that for a given angle , the ratio s/r is independent of the size of the circle.Example: How many radians in 180o? Circumference C = 2 r rs = radsr rpq = =p rads = 180o, 1 rad = 57.3o Angle of a rigid object is measured relative to some reference orientation, just like 1D position x is measured relative to some reference position (the origin).Angle is the "rotational position". Like position x in 1D, rotationalposition has a sign convention.Positive angles are CCW (counter-clockwise). Definition of angular velocity: d = (rad/s)dt tq Dqw � wD, (like dx xv , v dt tD� =D)units [ ]rad = swIn 1D, velocity v has a sign (+ or –) depending on direction. Likewise, for fixed-axis rotation, has a sign convention, depending on the sense of rotation.1/18/2019 ©University of Colorado at Boulderv :(+)(–)(+) (–)srsrs = rrx0x +x R-2More generally, when axis not fixed, we define vector angular velocity wvwith direction = the direction of the axis + "right hand rule". Curl fingersof right hand around rotation, thumb points in direction of vector.For rotational motion, there is a relation between tangential velocity v(velocity along the rim) and angular velocity . s s = r rDDq = � D Dq,rsv = = rt tDqD= wD Dv = rDefinition of angular acceleration : 2d (rad/s )dt tw Dwa � a =D,( like dv va a dt tD� =D,) Units: [ ]2rad = sa = rate at which is changing. = constant = 0 speed v along rim = constant = r Equations for constant :Recall from Chapter 2: We defined dx dvv = , a = dt dt ,and then showed that, if a = constant, 02120 02 20 0v = v a tx x v t a tv v 2 a x x�+��= + +��= + -��( )Now, in Chapter 9, we define d d = , = dt dtq ww a. So, if = constant, 02120 02 20 0 = tt t2�w w +a��q =q +w + a��w = w + a q- q��( )Same equations, just different symbols.Example: Fast spinning wheel with 0 = 50 rad/s ( 0 = 2f f 8 rev/s ). Apply brake and wheel slows at = 10 rad/s. How many revolutions before the wheel stops? 1/18/2019 ©University of Colorado at Boulders in time trR-3Use 2 202w =w + a Dq, final = 0 22200500 2 125 rad2 2 10w=w + a Dq � Dq =- =- =a -( )1 rev125 rad 19 9 rev2 rad� =p.Definition of tangential acceleration atan = rate at which speed v along rim is changingd rdv da = rdt dt dtww� =tan( ) atan = r atan is different than the radial or centripetal acceleration 2rvar=ar is due to change in direction of velocity vatan is due to change in magnitude of velocity, speed vatan and ar are the tangential and radial components of the acceleration vector a.2 2tan r| a | a a a= = +rAngular velocity also sometimes called angular frequency.Difference between angular velocity and frequency f:# radianssecw = , # revolutionsfsec=T = period = time for one complete revolution (or cycle or rev) 2 rad 2T Tp pw = =, 1 rev 1fT T= =2 fw = pUnits of frequency f = rev/s = hertz (Hz) . Units of angular velocity = rad /s = s-1Example: An old vinyl record disk with radius r = 6 in = 15.2 cm is spinning at 33.3 rpm (revolutions per minute). What is the period T? 33 3 rev 33 3 rev 60 s 60 33 3 s1 80 s/rev1min 60 s 33 3 rev 1rev. . ( / . )..= � = ; period T = 1.80 s1/18/2019 ©University of Colorado at BoulderatanaarR-4 What is the frequency f ? f = 1 / T = 1 rev / (1.80 s) = 0.555 Hz What is the angular velocity ? 12 f 2 0 555 s 3 49 rad s( . ) . /-w = p = p ; What is the speed v of a bug hanging on to the rim of the disk? v = r = (15.2 cm)(3.49 s-1) = 53.0 cm/s What is the angular acceleration of the bug? = 0 , since = constant What is the magnitude of the acceleration of the bug? The acceleration has only a radial component ar , since the tangential acceleration atan = r = 0. a = 222r0 530 m/sva 1 84 m/sr 0 152 m( . )..= = = (about 0.2 g's)For every quantity in linear (1D translational) motion, there is corresponding quantity in rotational motion:Translation Rotation x dxvdt=d = dtqwdvadt=d = dtwa F (?) M (?)F = Ma (?) = (?) KE = (1/2) m v2 KE = (1/2) (?) 2The rotational analogue of force is torque.Force F causes acceleration a Torque causes angular acceleration The torque (pronounced "tork") is a kind of"rotational force". magnitude of torque: r F r Fsin^t � � = q [ ] [ ] [ ]r F m Nt = =1/18/2019 ©University of Colorado at BoulderaxisrFFF = F sinF||R-5r = "lever arm" = distance from axis to point of application of forceF = component of force perpendicular to lever armExample: Wheel on a fixed axis:Notice that only the perpendicular component of the force F will rotate the wheel. The component of the force parallel to the lever arm (F||) has no effect on the rotation of the wheel.If you want to easily rotate an object about an axis, you want a large lever arm r and a large perpendicular force F:Example: Pull on a door handle a distance r = 0.8 m from the hinge with a force of magnitude F = 20 N at an angle = 30o from the plane of the door, like so: = r F = r F sin = (0.8 m)(20 N)(sin 30o) = 8.0 mNFor fixed axis, torque has a sign (+ or –) :Positive torque causes counter-clockwise CCW rotation.Negative torque causes clockwise (CW) rotation.If several torques are applied, the net torque causes angular acceleration: nett = t � a�Aside: Torque, like force, is a vector quantity. Torque has a direction. Definition of vector torque : r Ft = �vvv = cross product of r and F: "r cross F" Vector Math interlude: The cross-product of two vectors is a third vector A B C� =v vv defined likethis: The magnitude of A B�vvis A B sin . The direction of A B�vvis the direction perpendicular to the plane defined by the vectors A and B plus right-hand-rule. (Curl fingers from first vector A to second vector B, thumb points in direction of A B�vv1/18/2019 ©University of Colorado at Boulderaxisno good!(r = 0)badbetterbestno good!(F =
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