Chapter05 - Chapter 5 Circular Motion 5.1 Circular Motion...

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Unformatted text preview: Chapter 5 Circular Motion 5.1 Circular Motion 5.2 Rotation with Constant Angular Acceleration 5.3 The Centripetal Acceleration 5.4 The Banked Curve 5.5 The Rotor 5.1 Circular Motion When a rigid body rotates around a fixed axis (called the axis of rotation), each part of the body undergoes circular motion. For example, consider a point particle moving in a circle of radius r as shown in the figure. If the particle moves through an arc of distance s , it will trace out an angle of: θ = s r Note that the angle θ defined in this way is given in units of radians (rad) where π rad = 180 ° . In general, the circular motion of an object can be described by the following angular quantities: (a) Average angular velocity ϖ av t t t ∆ ∆ =-- = θ θ θ ϖ 1 2 1 2 av SI unit: rad/s where ∆ θ is the angular displacement of the particle during the time interval ∆ t . (b) Instantaneous angular velocity ϖ ϖ θ θ = = → lim ∆ ∆ ∆ t t d dt SI unit: rad/s When a rigid body rotates, the angular velocity is the same for each point of the body. In advanced study, ϖ is quite often defined as a vector with its direction determined by the right-hand rule. This is common in the area of science and engineering. (c) Average angular acceleration α av t t t ∆ ∆ =-- = ϖ ϖ ϖ α 1 2 1 2 av SI unit: rad/s 2 1 θ O mg s = r θ where ∆ ϖ is the change of the angular velocity of the particle during the time interval ∆ t . (d) Instantaneous angular acceleration α 2 2 lim dt d dt d t t θ ϖ ϖ α = = ∆ ∆ = → ∆ SI unit: rad/s 2 It is again a vector in advanced study! 5.2 Rotation with Constant Angular Acceleration We can derive the kinematic equations for the angular quantities using the methods we used to derive those for the linear quantities. Here we derive these angular equations for the case that the angular acceleration α is constant. (1) constant = = dt d ϖ α t α ϖ ϖ + = ∴ where ϖ ϖ = = ( )| t t ( ↔ v v at = + ) (2) dt d θ ϖ = ⇒ ∫ ∫ = t dt d ϖ θ θ θ 2 2 1 t t α ϖ θ θ + + = ∴ ( ↔ x x v t at = + + 2 1 2 ) t t t t ) ( 2 1 2 1 2 1 2 1 2 ϖ ϖ θ α ϖ ϖ θ + + = + + + = ) ( 1 ϖ ϖ α- = t α ϖ ϖ ϖ ϖ θ θ 1 ) )( ( 2 1- + =- ∴ ⇒ ϖ ϖ α θ θ 2 2 2 = +- ( ) ( ↔ ) ( 2 2 2 x x a v v- + = ) 2 2 v Q 2 T v ∆ Q 1 1 v O' θ ∆ For a circular motion, the arc length is given by s r = θ , where r is constant. Differentiate both sides, we obtain Since T v dt ds = is the instantaneous speed and d dt θ ϖ = is the instantaneous angular velocity, we obtain 5.3 The Centripetal Acceleration We can obtain the expression for the centripetal acceleration from the geometry of the circular motion. Consider the case of uniform circular motion . For such case, the speed is constant, e.g....
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This note was uploaded on 02/22/2012 for the course CHEM yscn0027 taught by Professor Drtong during the Fall '10 term at HKU.

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Chapter05 - Chapter 5 Circular Motion 5.1 Circular Motion...

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