Phys07 - Unit 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Rotational...

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1 Unit 7 Rotational Motions 7.1 Rotational kinematics 7.2 A comparison of linear kinematics and rotational kinematics 7.3 Rotational kinetic energy and moment of inertia 7.4 Torque and angular acceleration 7.5 Static equilibrium 7.6 Rotational work 7.7 Angular momentum 7.1 Rotational kinematics Definitions of some useful quantities (a) Angular position θ = Angle measured from the reference line SI Unit: radian, which is dimensionless. > 0 counterclockwise rotation from reference line < 0 clockwise rotation from reference line 1 revolution = 360 o = 2 π rad 1 rad ~57.3 o (b) Instantaneous velocity ω = Rate of change of angular displacement t t = ω 0 lim SI Unit: rad / s > 0 counterclockwise rotation < 0 clockwise rotation (c) Period T = The time to complete one revolution s=r r O
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2 ω π 2 = T SI Unit: second, s (d) Frequency f = Number of oscillation per second. T f 1 = SI Unit: s -1 (e) Angular acceleration α = Rate of change of angular velocity t t = 0 lim SI Unit: s -2 7.2 A comparison of linear kinematics and rotational kinematics Linear Quantity Angular Quantity x v a θ Linear Equation Angular Equation at v v + = 0 2 0 0 2 1 at t v x x + + = ) ( 2 0 2 0 2 x x a v v + = t + = 0 2 0 0 2 1 t t + + = ) ( 2 0 2 0 2 + = Example Find the speeds of points A, B and C at the wheel, if the wheel is rotating without slipping with a uniform speed on the horizontal plane. v B A C r
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3 Answer: The speed of the wheel is v = ω r. The speed of point A : v A = r + r =2 r. The speed of point B : r r r v B 2 ) ( ) ( 2 2 = + = . The speed of point C : v C = r r = 0 . 7.3 Rotational kinetic energy and moment of inertia The kinetic energy of the rotating particle is given by 2 2 2 2 2 2 1 ) ( 2 1 ) ( 2 1 2 1 I mr r m mv K = = = = where I = mr 2 is defined as the moment of inertia . For a rigid body, there are many particles rotating at the same time with the same angular velocity , the kinetic energy of it is given by 2 2 2 2 2 2 1 ) ( 2 1 ) ( 2 1 2 1 I r m r m v m K K i i i i i i i i i i i = = = = = Axis of rotation Massless rod v=r m r m i r i Axis of rotation v i
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4 where = i i i r m I 2 is the moment of inertia of a rigid body. Rigid Objects of Various Shapes Axis of Rotation Moment of Inertia Ring or cylindrical hollow Along the axis of cylinder I = MR 2 Disk or solid cylinder Along the axis of cylinder 2 2 1 MR I = Hollow sphere Along the axis of sphere 2 3 2 MR I = Solid sphere Along the axis of sphere 2 5 2 MR I = Long thin rod Axis through the center of rod 2 12 1 ML I = Long thin rod Axis through the rim of rod 2 3 1 ML I = Solid plate ( L : length of plate) Axis through center, in plane of plate 2 12 1 ML I = Solid plate ( L
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Phys07 - Unit 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Rotational...

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