Section10 - Physics 204A Class Notes Section 10 Rotation of...

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Physics 204A Class Notes 10-1 Section 10 – Rotation of Rigid Bodies What do objects do and why do they do it? We have a very thorough explanation of this in terms of kinematics, forces, energy and momentum. This includes three laws of motion and two conservation laws. However, we have only considered the motion of objects as a whole (translational motion). In general, objects rotate as well as translate. We need to revisit the questions “what do objects do?” and “why do they do it?” The good news is that we will be able to understand rotation by using our explanation of translational motion as a model. In fact, the model is so good that we will only need to introduce one new law (in the next chapter). Section Outline 1. The Definitions of the Rotational Variables 2. Rotational Kinematics 3. The Laws of Rotational Motion 4. Some Applications of the Laws of Rotational Motion 5. Rotational Kinetic Energy 1. The Definitions of the Rotational Variables The rotational variable will be defined by analogy to the translational variables. Translational Variables Rotational Variables Position : The location of an object with respect to a coordinate system. Angle : The rotational location of an object with respect to a coordinate system. Displacement : A change in position. Angular Displacement : A change in angle. Velocity : The rate of displacement. Angular Velocity : The rate of angular displacement. Acceleration : The rate of change of velocity. Angular Acceleration : The rate of change of angular velocity. These definitions can be rewritten mathematically, Translational Variables Rotational Variables Relationship Position : x Angle : θ s = r ! Displacement : dx Angular Displacement : d θ ds = rd ! Velocity : v ! dx dt Angular Velocity : ! " d # dt v ! ds dt = r d " dt # v t = r $ Acceleration : a ! dv dt Angular Acceleration : ! " d # dt a ! dv dt " a t = dv t dt = r d # dt " a t = r $ a c = v t 2 r = (r ! ) 2 r " a c = ! 2 r y x x r ! y x s
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Physics 204A Class Notes 10-2 Example 10.1: A CD 12.0cm in diameter is placed in a drive. It reaches 200rpm in 1.20s. Find (a)the average angular acceleration and (b)the translational acceleration when the rotation rate is 100rpm. Given: r = 0.0600m, ω i = 0, ω = 100rpm, ω f = 200rpm, and t = 1.20s Find: α = ? and r a = r a c + r a t The rotation rates in rpm must be converted to angular speeds in rad/s, ! f = 200 rev min ( ) 2 " rad rev ( ) min 60s # $ % & = 20.9rad / s and ! = 100 rev min ( ) 2 " rad rev ( ) min 60s # $ % & = 10.5rad / s . (a)Using the definition of angular acceleration, ! " d # dt $ ! = %# % t = # f & # i % t = 20.9 & 0 1.20 $ ! = 17.4rad / s 2 . (b)The angular acceleration is related to the tangential acceleration, a t = r ! = (0.0600)(17.4) " a t = 1.05m / s 2 . The centripetal acceleration is related to the angular velocity, a c = ! 2 r = (10.5) 2 (0.0600) " a c = 6.62m / s 2 . These are the perpendicular components of the total acceleration. Using the Pythagorean Theorem, a = c 2 + a t 2 = (6.62) 2 + (1.05) 2 ! a = 6.70m / s 2 . 2. Rotational Kinematics Recall that when and object is translating with a constant acceleration, we developed a set of equations to describe the motion called the kinematic equations.
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Section10 - Physics 204A Class Notes Section 10 Rotation of...

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