07%20-%20Rotational%20Motion

07%20-%20Rotational%20Motion - 7 - ROTATIONAL MOTION Page 1...

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7 - ROTATIONAL MOTION Page 1 Introduction In this chapter, rotational motion of a rigid body about a fixed axis of rotation is discussed. A rigid body is a system of particles in which interparticle distances do not change and the body cannot be deformed no matter how large a force is applied to it. Although a solid body is not a rigid body, it can be so considered for most of the practical applications. 7.1 Rotational Kinematics and Dynamics In rotational motion of a body, all its particles move on circular paths having centres on a definite straight line, called the axis of rotation. Kinematics deals with motion without considering its cause, whereas dynamics deals with motion alongwith its cause and properties of the body. 7.2 Relations between variables of rotational and linear motion ( a ) Angular displacement: The figure shows a rigid body rotating about a fixed axis OZ normal to the plane of the figure. P and P’ are the positions of a particle of the body at time t and t + t. Angle θ made by the line joining the particle to the centre of its rotation with a reference line OX shows its angular position at time t. Similarly, angle θ + ∆θ is its angular position at time t + t. The change in angular position, of a particle is called its angular displacement. The angular displacement of the particle P is ∆θ in time t. As the interparticle distances do not change in a rigid body, all its particles will have the same angular displacement in a given time. Hence, the angular displacement, ∆θ , of the particle P can be considered as the angular displacement of the rigid body. ( b ) Angular speed and angular velocity: The average angular speed of a particle or of the rigid body is defined as < ω > = interval time nt displaceme angular = t θ The instantaneous angular speed of a particle or of the rigid body is given by ω = t θ lim 0 t The unit of ω is radian / s or rotation / s. The direction of angular velocity is given by the right handed screw rule. When a right handed screw is kept parallel to the axis of rotation and rotated in the direction of rotation of the body, the direction of advancement of screw gives the direction of angular velocity. ( c ) Scalar relation between angular velocity and linear velocity: As shown in the figure, the particle P covers a linear distance equal to the arc length PP’
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7 - ROTATIONAL MOTION Page 2 in time t. Hence, average linear speed, < v > = t ' PP length arc = t θ r = r < ω >, where r is the radius of the circular path. The instantaneous linear speed is given by v = t θ r lim 0 t = t d θ d r = r ω ( Note that the angular velocity of all particles of the rigid body rotating about a fixed axis of rotation is the same for all the particles, whereas the linear speed of a particle depends upon its distance ( r ) from the axis of rotation. ) Linear velocity is a vector quantity and its direction at any point on the path of motion
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This note was uploaded on 11/28/2011 for the course PHYSICS 300 taught by Professor Smith during the Spring '06 term at ITT Tech Flint.

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07%20-%20Rotational%20Motion - 7 - ROTATIONAL MOTION Page 1...

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