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Unformatted text preview: Chapter 10 Rotation In this chapter we will study the rotational motion of rigid bodies about a fixed axis. To describe this type of motion we will introduce the following new concepts: Angular displacement Average and instantaneous angular velocity (symbol: ω ) Average and instantaneous angular acceleration (symbol: α ) Rotational inertia also known as moment of inertia (symbol I ) Torque (symbol τ ) We will also calculate the kinetic energy associated with rotation, write Newton’s second law for rotational motion, and introduce the workkinetic energy for rotational motion (101) The Rotational Variables In this chapter we will study the rotational motion of rigid bodies about fixed axes. A rigid body is defined as one that can rotate with all its parts locked together and without any change of its shape. A fixed axis means that the object rotates about an axis that does not move. We can describe the motion of a rigid body rotating about a fixed axis by specifying just one parameter. Consider the rigid body of the figure. We take the the zaxis to be the fixed axis of rotation. We define a reference line which is fixed in the rigid body and is perpendicular to the rotational axis. A top view is shown in the lower picture. The angular position of the reference line at any time t is defined by the angle θ(t) that the reference lines makes with the position at t = 0. The angle θ(t) also defines the position of all the points on the rigid body because all the points are locked as they rotate. The angle θ is related to the arc length s traveled by a point at a distance r from the axis via the equation: Note: The angle θ is measured in radians s r θ = (102) 1 2 1 2 2 1 In the picture we show the reference line at a time and at a later time . Between and the body undergoes an angular displacement An . gular Displace All the points of th ment e rigid t t t t θ θ θ ∆ = body have the same angular displacement because they rotate locked together. ( 29 1 2 1 2 1 2 We define as average angular velocity for the time interval , the ratio: We define as the instantaneous ang The SI unit for angular velocity is An rad gular Velocity ians/second avg t t t t t θ θ θ ϖ ∆ = = ∆ ular velocity the limit of as lim This is the definition of the first deriva Algerbraic sign of angular f tive with If a rigid body rot re at quen es cy: counterclockwise (CC W t t t t t θ θ ϖ ∆ → ∆ ∆ → ∆ ∆ = ∆ ) has a positive sign. If on the other hand the rotation is clockwise (CW) has a negative sign ϖ ϖ (103) t 1 t 2 d dt θ ϖ = If the angular velocity of a rotating rigid object changes with time we can describe the time rate of change of by defining the Angula angula r Accelerati r acelera on tion ϖ 1 2 1 1 2 2 In the figure we show the reference line at a time and at a later time ....
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This note was uploaded on 11/21/2011 for the course PHYS 2425 taught by Professor . during the Spring '11 term at San Jacinto.
 Spring '11
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 Physics

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