R-1 Rotational Motion We 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)rθ≡ Notice that for a given angle θ, the ratio s/r is independent of the size of the circle. s s r θ θ r Example: How many radians in 180o? Circumference C = 2 π r s = π r rs =radsrrπθ= =πθ π rads = 180o, 1 rad = 57.3o Angle θ 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 θ is the "rotational position". Like position x in 1D, rotational position θ has a sign convention. Positive angles are CCW (counter-clockwise). Definition of angular velocity: d = (rad/s)dt tθ∆θω≡ (like ω∆,dx xv , v dt t ∆≡=∆) r units []rad = sω In 1D, velocity v has a sign (+ or –) depending on direction. Likewise, for fixed-axis rotation, ω has a sign convention, depending on the sense of rotation. v : (+) (–) ω : (+) (–) x +θ 0 −θ θ = 0 x − x + 3/19/2009 ©University of Colorado at BoulderR-2 More generally, when axis not fixed, we define vector angular velocity ω with direction = the direction of the axis + "right hand rule". Curl fingers oright hand around rotation, thumb points in direction of vector. f For rotational motion, there is a relation between tangential velocity v (velocity along the rim) and angular velocity θ. s s=rr∆∆θ = ⇒ ∆ ∆θ , rsv = = rtt∆θ∆=ω∆∆ v = r ω Definition of angular acceleration : 2d (rad/s )dt tω∆ωα≡ α=∆, ( like dv va ) Units: []a dt t∆≡=∆,2rad = sα α = rate at which ω is changing. ω = constant ⇔ α = 0 ⇒ speed v along rim = constant = r ω Equations for constant α: Recall from Chapter 2: We defined dx dvv = , a = dt dt , and then showed that, if a = constant, 0212002200v = v a txx vt atvv2axx⎧+⎪⎪=+ +⎨⎪=+ −⎪⎩() Now, in Chapter 9, we define dd = , = dt dtθωωα. So, if α = constant, 0212002200 = ttt2⎧ωω+α⎪⎪θ=θ +ω + α⎨⎪ω=ω+ αθ−θ⎪⎩() Same equations, just different symbols. Example: Fast spinning wheel with ω0 = 50 rad/s ( ω0 = 2πf ⇒ f ≈ 8 rev/s ). Apply brake and wheel slows at α = −10 rad/s. How many revolutions before the wheel stops? ∆s in time ∆t r ∆θ ω ω 3/19/2009 ©University of Colorado at BoulderR-3 Use , ω2202ω=ω+ α∆θfinal = 0 ⇒ 222005002 1252210ω=ω + α∆θ ⇒ ∆θ=− =− =α−() rad 1 rev125 rad 19 9 rev2 rad×=π. Definition of tangential acceleration atan = rate at which speed v along rim is changing drdv da = rdt dt dtωω≡tan()= ⇒ atan = r α atan is different than the radial or centripetal acceleration 2rvar= ar is due to change in direction of velocity v atan is due to change in magnitude of velocity, speed v atan and ar are the tangential and radial components of the acceleration vector a. a atan ar22tan r|a| a a a== + Angular velocity ω also sometimes called angular frequency. Difference between angular velocity ω and frequency f: # radianssecω= , # revolutionsfsec= T = period = time for one complete revolution (or cycle or rev) ⇒ 2 rad 2TTππω= = , 1 rev 1f ⇒ TT==2fω=π Units of frequency f = rev/s = hertz (Hz) . Units of angular velocity = rad /s = s-1 Example: 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 60s 60 33 3 s180s/rev1min 60s 33 3rev 1rev.. (/.)..=⇒ =  ⇒ period T = 1.80 s 3/19/2009 ©University of Colorado at BoulderR-4 • What is the frequency f ? f = 1 / T = 1 rev / (1.80 s) = 0.555 Hz • What is the angular velocity ω ? 12 f 2 0 555 s 3 49 rad s(. ) . /−ω= π = π • What is the speed v of a bug hanging on to the rim of the disk? v = r ω = (15.2 cm)(3.49 s-1) = 53.0 cm/s• What is the angular acceleration α of the bug? α = 0 , since ω = 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 α = 0. a = 222r0 530 m/sva1r 0 152 m(. )..== =84m/s (about 0.2 g's) For every quantity in linear (1D translational) motion, there is corresponding quantity in rotational motion: Translation ↔ Rotation x ↔ θ dxvdt= ↔ d = dtθω dvadt= ↔ d = dtωα F ↔ (?) M ↔ (?) F = Ma ↔ (?) = (?) α KE = (1/2) m v2 ↔ KE = (1/2) (?) ω2 The rotational analogue of force is torque. Force F causes acceleration a ↔ Torque τ causes angular acceleration α . The torque (pronounced "tork") is a kind of "rotational force". magnitude of torque: rF rFsin⊥τ≡ ⋅ = θ [] [][]rF mNτ= =axis r F F⊥= F sinθ θ F⊥ F||3/19/2009 ©University of Colorado at BoulderR-5 r = "lever arm" = distance from axis to point of application of force F⊥ = component of force perpendicular to lever arm Example: 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 θ = 30o from the plane of the door, like so: τ = r F⊥ = r F sin θ = (0.8 m)(20 N)(sin 30o) = 8.0 m⋅N For 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: netτ=τ∝∑α Aside: Torque, like force, is a vector quantity. Torque has a direction. Definition of vector torque : r Fτ= × = cross product of r and F: "r cross F" Vector Math interlude: The cross-product of two vectors is a third vector defined like this: The magnitude of is A B sinθ . The direction of A B C×=A B× A B×is the direction perpendicular to the plane defined by the vectors A and B plus right-hand-rule. (Curl fingers from