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Linear vs Rotational Motion Linear Motion Rotational Motion Mass m Moment of Inertia I Linear velocity v 2 Translational KE 12 mv Linear momentum p = mv r r ∆pr Fnet = ∑ F = ∆t Angular velocity ω 2 1 I ω Rotational KE 2 Angular momentum L = Iω τ net ∆L = ∑τ = ∆t ∆L Iω − Iωo I∆ω = = = Iα (F=ma) Note: if I is constant, Στ = ∆t ∆t ∆t For constant α: ω = ω o + αt ∆θ = ωot + 12 αt 2 ω2 =ω2o +2α∆θ Your Requests: Torque Centripetal Acc. Similar to HS Angular Momentum Angular Motion eq.s Angular Kinetic Energy A Windmill In a light wind, a windmill experiences a constant torque of 255 N m. If the windmill is initially at rest, what is its angular momentum after 2.00 s? ∆L τ= ∆t 2 ∆L = τ ∆t = (255 N)(2.00 s) = 510 kg m /s Notice that you do not need to know the moment of inertia of the windmill to do this calculation. Mary is about to do a push-up. Her center of gravity lies directly above a point on the floor which is d1=1.0 m from her feet and d2=0.7 m from her hands. A) If her mass is 50 kg, what is the force exerted by the floor on her hands, assuming that she holds this position? B) What is the force exerted by the floor on her feet? How should we approach this problem? Find the centripetal acceleration and final rotational kinetic energy of a amusement park 1000 kg centrifuge (r=5 m) that starts from rest and after 5 second is going 3 2 rad/s. Treat like a hoop (I=MR ). A diver can reduce her moment of inertia by a factor of about 3.5 when changing form the straight position to the tuck position. If she makes two rotations in 1.5 s when in the tuck position, what is her angular speed (rev/s) when in the straight position? How should we approach this problem? Example: A Rotating Disk Disk 1 is rotating freely and has angular velocity ωi and moment of inertia I1 about its symmetry axis, as shown. It drops onto disk 2 of moment of inertia I2, initially at rest. Because of kinetic friction, the two disks eventually attain a common angular velocity ωf. (a) What is ωf? (b) What is the ratio of final to initial kinetic 2 L = L I ω ( ) energy? f i K = 1 Iω 2 = 2 ( I1 + I 2 )ω f = I1ωi I 1 ω f = 1 ωi = ωi I1 + I 2 1 + ( I 2 / I1 ) L2 = K i 2 I f Kf 2I L2 / 2Ii L2 = 2I Ii I1 = = I f I1 + I 2 A car initially traveling eastward turns north by traveling in a circular path at a uniform speed. The length of the arc ABC is 235 m, and the car completes the turn in 36 seconds. a) Determine the car’s speed. b) What is the magnitude and direction of the acceleration when the car is at point B? Signs in Equilibrium 50° Signs R Us 1m Wall What tension will be needed in the rope to support the sign’s 15 kg weight and keep it from falling off the wall? A 800 N merry-go-round of radius 1.5 m is started from rest by a constant horizontal force of 50.0 N applied tangentially to the merry-go-round. Find the kinetic energy of the merry-go-round after 3.00 s. Idisk= ½ mR2 50.0 N How should we approach this problem?